Stabilization of a Rigid Body in Two Fluids With Surface Tension

Do, K. D.

doi:10.1115/1.4067515

Abstract

This paper poses and solves a stabilization problem of a rigid body governed by nonlinear differential equations in two viscous incompressible fluids governed by Navier–Stokes equations (NSEs), where surface tension of the interface between the two fluids is considered, in a bounded domain in three dimensional space. Since only weak solutions of the NSEs exist globally while global existence of their strong/smooth solutions is a millennium problem, point-wise fluid forces and moments acting on the rigid body are not able to bound. This difficulty is overcome by designing an appropriate control law and performing stability analysis of the closed-loop system including the NSEs and surface tension, where “work and power of the two fluids” instead of forces and moments on the rigid body are used. A simulation is included to illustrate the results.

1 Introduction

The most common rigid bodies, which are operating between two fluids, are floating ones, where one fluid is water and the other fluid is air. Such rigid bodies include surface ships [1,2], buoys of wave energy converters [3,4], floating wind turbines [5], floating breakwater and jacket platform [6]. Stabilization of rigid bodies operating between two fluids is an important and challenging area due to their applications and complex loads of fluids on them. An appropriate determination of the fluid loads on the rigid bodies is essential for control design. There are two main approaches to determine loads of fluids on the rigid bodies: (1) approximation approach, where the forces and moments of fluids on the rigid bodies are approximated by using linear (Stokes) and/or potential flow theory (e.g., Refs. [1,2,7,8]); and (2) fundamental approach based on Navier–Stokes equations (NSEs) by using the original NSEs (e.g., Refs. [9–23]).

In the approximation approach, the loads of fluids are referred to as fluid forces and fluid moments induced by air (wind) and water (waves, currents), see Refs. [1,2,7,8] and references therein. For example, the fluid forces are approximated by the sum of Froude-Krylov force, diffraction force, radiation force, hydrostatic force, and viscous drag while wind force is approximately calculated from wind spectra. The fluid forces and fluid moments are assumed to be point-wise bounded. This requires a strong solution of the NSEs for a viscous incompressible fluid. Currently, existence of a strong/smooth solution of NSEs is local in either time or small initial fluid velocity [24,25] under sufficient regularity of initial fluid velocity. Global existence of a strong/smooth solution of NSEs is a millennium problem [26].

In the fundamental NSEs approach, fully coupled dynamics of both rigid bodies and fluids, where motion of rigid bodies are governed by nonlinear differential equations (ODEs) and motion of the fluids are governed by the NSEs in three dimensional space. This approach is much more complex than the approximation approach but gives insides of fluid–structure interactions. The complexities include: (i) existence of an appropriate solution of the fluid and rigid body system; (ii) time-varying domain of the fluids due to motion of the rigid bodies; (iii) interaction between fluids with tension on their interface; (iv) interaction between the fluids and rigid bodies; (v) bound of the forces and moments induced by the fluids on the rigid body. There are several works related to the fundamental NSEs approach. Existence of a weak solution for a system of a fluid and multiple rigid bodies was proved in Refs. [13–21], where dynamics of the fluid and rigid bodies is written as a global fluid and the initial value of the fluid velocity is assumed in $H_{0}^{1} (Ω)$ ⁠, see “Notation of functional spaces” below for a definition of space $H_{0}^{1} (Ω)$ ⁠, and a coordinate transformation is used to handle difficulty caused by the fluid time-varying domain. Several works on stabilization of a rigid body in a fluid include: [27] on a linear control for stabilization of a rigid ball and [28–30] on stabilization of a rigid ball in one-dimensional, two-dimensional, and three-dimensional with the initial value of the fluid velocity in space $H_{0}^{1} (Ω)$ ⁠.

The most relevant references to the present paper are [22] and [23]. In Ref. [22], a control law is designed to stabilize a rigid body completely submerged in a single incompressible fluid based on Lyapunov's direct method, where the initial fluid velocity is assumed in space H. In Ref. [23], a control law is designed to stabilize a rigid body on the interface between two incompressible fluids. The initial fluid velocity of the fluids is also assumed in the space H. The surface tension between two fluid is set to zero. The control design is based on the forwarding method [31] with some modification because the dynamics of a rigid body is not of a strict feed-forward form.

Existence of a weak solution of the closed-loop system in both [22] and [23] is carried out based on a penalized method [32] while stability is analyzed by using fluid work and fluid power to overcome difficulties caused by less regularity of the initial fluid velocity. However, proof of existence of a weak solution and stability analysis of the closed-loop system in Ref. [23] is much more involved than in Ref. [22] due to composition of the two fluids.

The present paper proposes new contributions in design of a control law to almost globally practically exponentially stabilize a rigid body on the interface between two fluids with surface tension in a domain Ω, where “almost global” means that there is no collision between the rigid body and the boundary $\partial Ω$ of the domain Ω, existence of a weak solution and analysis of stability and convergence of the closed-loop system. The initial velocity of the fluids is only assumed in space H, which is less regularity than space $H_{0}^{1}$ ⁠, see Eq. (12) for a definition of space $H (Ω)$ ⁠. Since the initial velocity of the fluids is only assumed in space H, there is no information on (point-wise) bound of the forces and moments induced by the fluids on the rigid body. Thus, we consider the effect of the work and power of the fluids (instead of the fluid forces and moments) on the rigid body, see the paragraph just below Eq. (74) for the concept of the work and power of the fluids, which was introduced in Ref. [22]. The control design is based on the backstepping method in Ref. [33] and the forwarding method [31].

In comparison with the works in Refs. [22] and [23], the present paper is significant from both practical and theoretical points of view. Practically, the work in Ref. [22] is only applicable to a rigid body in a single incompressible fluid (such as in water or in air) while the work in Ref. [23] allows a rigid body on the interface between two incompressible fluids such as surface ships and floating off-shore structures. The work in Ref. [23] covers the work in Ref. [22] but not vice versa but the surface tension is set to zero in Ref. [23]. In practice, surface tension of a fluid is never zero. Hence, setting surface tension of a fluid to zero significantly simplifies the real scenario. This deteriorates modeling accuracy and performance of a control system. The work in the present paper considers a rigid body on the interface between two incompressible fluid, where nonzero surface tension is allowed. Hence, the present paper covers both [22] and [23] but not vice versa. In addition, since surface tension is allowed to be nonzero, it reflects reality in modeling and hence improves performance of a control system.

Theoretically, nonzero surface tension (see the second equation in Eq. (4)) makes the normal component of the stress tensor discontinuous. This is the main source that makes formulation and existence proof of a weak solution and analysis stability of the closed-loop system difficult, see also Remark 2.1. In fact, formulation of a weak solution of the closed-loop system in Sec. 4.1 is much harder than those in Refs. [22] and [23], see the Eq. (42) with the extra term $κ 〈 H, φ_{f} 〉$ defined in Eq. (43) due to nonzero surface tension. Nonzero surface tension also makes stability analysis of the closed-loop system in Sec. 5 very involved, see the term $B_{2} = κ \int_{Γ^{*} (t)} H n \cdot X_{s} d H^{2} (x)$ in Eq. (90).

The rest of the paper is organized as follows: In Sec. 2, a problem of stabilizing a rigid body in two fluids in a domain Ω is posed. In Sec. 3, a control law is designed to achieve almost global practical exponential stabilization of the rigid body, and is of an inverse pre-optimal form, which can be amended to be inverse optimal [34] without having to solve a Hamilton-Jacobi-Bellman or a Hamilton-Jacobi-Isaacs equation. In Sec. 4, we formulate the closed-loop system as a global fluid, of which existence of weak solution is shown via a penalizing method. In Sec. 5, almost global practical exponential stability of the closed-loop system is proved. A simulation is given in Sec. 6 to illustrate the results.

1.1 Notation of Functional Spaces.

Let Ω be an open bounded set in $ℝ^{3}$ ⁠, and a time constant T > 0. $L^{p} (Ω)$ ⁠, where $1 \leq p < \infty$ ⁠, denotes the standard Lebesgue space of measurable p-integrable functions; $L^{\infty} (Ω)$ denotes the space of essentially bounded functions; $H^{1} (Ω)$ is the usual Sobolev space of order 1, see Ref. [35]; $H_{0}^{1} (Ω)$ denotes $H^{1} (Ω)$ with compact support; ${(a, b)}_{Ω} = \int_{Ω} a \cdot b d x$ with $a \cdot b = \sum_{i} (a_{i} b_{i})$ and $| | a | |_{Ω}^{2} = {(a, a)}_{Ω}$ ⁠; $L^{p} (0, T; X)$ ⁠, where $1 \leq p < \infty$ and X is a Banach space with the norm denoted by $| | \cdot | |_{X}$ ⁠, denotes a Bochner space with the norm $| | u | |_{L^{p} (0, T; X)} = {(\int_{0}^{T} | | u | |_{X}^{p} d t)}^{1 / p}$ ⁠. Let $Y = X'$ be the dual space of a Banach space X. The space $L_{ω}^{\infty} (Ω; Y)$ is defined as all weak- $*$ measurable functions $ν : Ω \to Y$ ⁠, i.e., $x \mapsto (ν_{x}, F (x, \cdot)) = {(ν_{x}, F (x, \cdot))}_{X', X}$ ⁠, where $ν_{x}, x \in Ω$ denotes a family of probability measures, is measurable for each $F \in L^{1} (Ω; X)$ ⁠, such that $| | ν | |_{L_{Ω}^{\infty}} (Ω; Y) : = ess \sup_{x \in Ω} | | ν_{x} | |_{Y} < \infty$ ⁠. The space of k-times differentiable functions on a set Ω is denoted by $C^{k} (Ω)$ ⁠. We also use $| | \cdot | |_{E}$ to denote the Euclidean (point-wise) norm, i.e., $| | a | |_{E} = \sqrt{a \cdot a}$ ⁠. For a scalar, we use $| \cdot |_{E}$ to denote the absolute value.

2 Problem Statement

Let $Ω \subset ℝ^{3}$ be a $C^{1}$ domain occupied by fluid 1 (which is usually the air), fluid 2 (which is usually the water), and a rigid body. The fluid 1 and fluid 2 occupy bounded open domains $Ω_{1} (t)$ and $Ω_{2} (t)$ at time $t \geq 0$ ⁠, see Fig. 1. These fluid are separated by the interface $Γ (t)$ and surround a rigid body represented by $Ω_{s} (t)$ ⁠, which is a bounded open connected subdomain of Ω. We have $Ω = Ω_{1} \cup Ω_{2} \cup Ω_{s}$ ⁠. The domain $Ω_{s}^{*} (t)$ will be defined and used in Sec. 5. The time-dependence of the domains $Ω_{i}, i = 1, 2$ and Ω_s is due to motion of the rigid body. We assume that $Ω_{s} (0) ⋐ Ω$ ⁠. We will drop the argument t of $Ω_{i}, i = 1, 2$ ⁠, Ω_s, and Γ when there is no confusion.

Fig. 1

View large Download slide

Domain definition

For the fluids, we denote by

ρ_{i} > 0, μ_{i} > 0, u_{i}, p_{i}, f_{i}

with i = 1, 2 the density, dynamic viscosity, velocity, pressure, body force of fluid i, respectively. Motion of each fluid is governed by the NSEs

\begin{matrix} ρ_{i} (\partial_{t} u_{i} + (u_{i} \cdot \nabla) u_{i}) + \nabla p_{i} - μ_{i} Δ u_{i} = ρ_{i} f_{i}, in D' (Q_{i}) \\ div (u_{i}) = 0 in Q_{i} \end{matrix}

(1)

where

Q_{i} = (0, T) \times Ω_{i}

with the time constant

T < \infty

⁠, and

D' (Q_{i})

denotes the space of distribution on Q_i. We assume that

\begin{matrix} f_{i} \in L^{2} (0, T; L^{2} (Ω_{i})) \end{matrix}

(2)

Denoting by

\partial Ω_{s}

and

\partial Ω_{i}

the boundary of Ω_s and Ω_i, respectively, we impose homogeneous Dirichlet boundary conditions on the boundaries

\begin{matrix} u_{i} = 0 on (0, T) \times \cup_{i = 1}^{2} \partial Ω_{i} ∖ \partial Ω_{s} \end{matrix}

(3)

At the interface Γ between

Ω_{1}

and

Ω_{2}

⁠, the transmission condition is expressed as continuity of the velocity and discontinuity of the normal component of the stress tensor due to surface tension, and immiscibility condition is expressed as equality of velocity of fluids normal to Γ and normal velocity of Γ

\begin{matrix} {\begin{matrix} u_{1} = u_{2} \\ σ_{2} n - σ_{1} n = κ H n, \\ V = u_{i} \cdot n \end{matrix} on (0, T) \times Γ \end{matrix}

(4)

where

V

and

H

denote the normal velocity and mean curvature, respectively, of Γ taken with respect to the exterior normal n of

\partial Ω_{2}

⁠, κ is the surface tension constant,

σ_{i}

is the fluid stress tensor of the fluid i given by

\begin{matrix} σ_{i} = 2 μ_{i} D (u_{i}) - p_{i} I_{3} \end{matrix}

(5)

with

D (u_{i}) = \frac{1}{2} (\nabla u_{i} + {(\nabla u_{i})}^{T})

being the rate tensor of the fluid,

•^{T}

denotes the transpose of

•

(this should not be confused with the time constant T), and

I_{3}

being the 3 × 3 identity matrix. On the interface Γ, the tangential divergence of a function

ϕ \in C^{1} (Γ)

is defined as

\begin{matrix} {div}^{Γ} ϕ : = div (ϕ) - n \otimes n : \nabla ϕ = P_{τ} : \nabla ϕ \\ P_{τ} = I_{3} - n \otimes n \end{matrix}

(6)

where $P_{τ}$ is the orthogonal projection onto the tangential space, which is defined point-wisely on Γ, the symbol ⊗ denotes the outer product operator, i.e., $a \otimes b = a b^{T}$ with $b^{T}$ being the transpose of b for $a, b \in ℝ^{3}$ ⁠, and $A : B = Tr (A B)$ for 3 × 3 symmetric matrices A, B with $Tr$ being the trace of matrices. The mean curvature is defined as $H = - {div}^{Γ} (ϕ)$ ⁠.

Remark 2.1. Due to the surface tension term $κ H n$ in Eq. (4), the normal component of the stress tensor is discontinuous. This introduces difficulties in proof of existence of a weak solution and stability analysis of the closed-loop system in Secs. 4 and 5 as discussed in Sec. 1. The surface tension term $κ H n$ also makes the present paper more general from both practical and theoretical points of view than the works in Refs. [22] and [23].

For the rigid body, we define the mass m_s, the density

ρ_{s} > 0

⁠, the vector of the center of gravity

x_{c s} (t)

(measured with respect to the origin, which coincides with the center of

Γ (0)

⁠) and its velocity vector

u_{c s} (t)

⁠, the modified Rodrigues parameter vector

η_{s} (t)

representing the rigid body orientation, see Ref. [36], (this vector is related to the principal axis e and the principal angle γ through

η_{s} (t) = e tan (\frac{γ}{4})

⁠, which is well-defined for all eigenaxis rotations in the range

[0, 2 π)

⁠, and should not be confusing with almost global stability in this paper), angular velocity vector

ω_{s} (t)

⁠, the inertia matrix

J_{s}

⁠, the transformation matrix

R_{s}

⁠, and the velocity vector field

u_{s} (t, x)

by

\begin{matrix} ρ_{s} = \frac{m_{s}}{| Ω_{s} (0) |_{E}}, x_{c s} = \frac{1}{| Ω_{s} |_{E}} \int_{Ω_{s}} x d x \\ u_{s} (t, x) = u_{c s} + ω_{s} \times r_{s} for x \in Ω_{s} \\ J_{s} = ρ_{s} \int_{Ω_{s} (0)} (| | r_{s} | |_{E}^{2} I_{3} - r_{s} \otimes r_{s}) d x, J_{s} = J_{s}^{T} > 0 \\ a^{T} J_{s} b = ρ_{s} \int_{Ω_{s} (0)} (a \times r_{s}) \cdot (b \times r_{s}) d x, \forall a, b \in ℝ^{3} \\ R_{s} = \frac{1}{2} (I_{3} - G (η_{s}) + η_{s} η_{s}^{T} - \frac{1 + | | η_{s} | |_{E}^{2}}{2} I_{3}) \end{matrix}

(7)

where

r_{s} = x - x_{cs}

⁠, the symbol × denotes the cross vector product operator,

I_{3}

denotes the 3 × 3 identity matrix,

| Ω_{s} |_{E}

denotes the volume of Ω_s,

G (η_{s})

denotes the 3 × 3 skew-symmetric matrix of

η_{s}

⁠, it holds that

\begin{matrix} η_{s}^{T} R_{s} ω_{s} = \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) η_{s}^{T} ω_{s} \\ R_{s}^{T} R_{s} = \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) I_{3} \end{matrix}

(8)

for all

η_{s}, ω_{s} \in ℝ^{3}

⁠, and we drop the argument t of state variables of the rigid body for clarity. We assume that the rigid body is neutrally buoyant. Then, equations of motion of the rigid body are given by

\begin{matrix} \frac{d x_{cs}}{d t} = u_{cs} \frac{d η_{s}}{d t} = R_{s} ω_{s} \\ m_{s} \frac{d u_{cs}}{d t} = \sum_{i = 1}^{2} \int_{\partial Ω_{is}} σ_{i} n_{i} d τ + F_{s} + \sum_{i = 1}^{2} \int_{Ω_{is}} ρ_{s} f_{i} d τ \\ J_{s} \frac{d ω_{s}}{d t} = - ω_{s} \times (J_{s} ω_{s}) + \sum_{i = 1}^{2} \int_{\partial Ω_{is}} r_{s} \times (σ_{i} n_{i}) d τ \\ + M_{s} + \sum_{i = 1}^{2} \int_{Ω_{is}} ρ_{s} r_{s} \times f_{i} d τ \end{matrix}

(9)

where Ω_is is the part of Ω_s submerged in Ω_i, and hence

Ω_{s} = \cup_{i = 1}^{2} Ω_{is}

⁠, and

\partial Ω_{is}

is the interface between Ω_i and Ω_is;

n_{i}

is the normal unit vector pointing outside of Ω_is;

F_{k}

is the control force. The control force vector

F_{s}

and the control moment vector

M_{s}

are expressed in terms of actuator forces as follows:

\begin{matrix} F_{s} = \sum_{k = 1}^{N} F_{k} \\ M_{s} = \sum_{k = 1}^{N} r_{F k} \times F_{k} \end{matrix}

(10)

where $r_{F k} = x_{F_{k}} - x_{c s}$ with $x_{F_{k}}$ being the position vector of actuator forces $F_{k}$ such that $F_{s} \in ℝ^{3}$ and $M_{s} \in ℝ^{3}$ (i.e., the rigid body is fully actuated).

At the interface between Ω_is and Ω_i, the transmission condition is expressed as continuity of the velocity

\begin{matrix} u_{is} = u_{i} on \partial Ω_{is} \end{matrix}

(11)

In derivation of Eq. (9), we have used a transmission condition that the stress is continuous in normal direction on the interface, i.e., $σ_{i} n_{i} = σ_{s} n_{i}$ on $\partial Ω_{is}$ with $σ_{s}$ being the Cauchy stress tensor, i.e., $- σ_{s} n_{i}$ is the force applied by the rigid body on the fluid i.

For use in the rest of the paper, we denote

Q = (0, T) \times Ω

⁠, and introduce the following function spaces:

\begin{matrix} V (Ω) = {v \in H_{0}^{1} (Ω), div (v) = 0} \\ H (Ω) = {v \in L^{2} (Ω), div (v) = 0, v |_{\partial Ω} = 0} \\ H (\bar{Ω}) = {v |_{Ω}, v \in H (ℝ^{3})} \\ K (t) = {v \in V, \exists (v_{v}, ω_{v}) \in ℝ^{3} \times ℝ^{3} \\ v |_{Ω_{s}} = v_{v} + ω_{v} \times r_{s}} \\ I_{T} = {v \in C (0, T; H), \exists v_{f} \in C_{0} (0, T; H (\bar{Ω})) and \\ v_{s} \in C_{0} (0, T; K) such that v_{f} = v_{s} on Ω_{f} \\ and v = v_{s} on Ω_{s}} \end{matrix}

(12)

where $Ω_{f} = \cup_{i = 1}^{2} Ω_{i}$ ⁠. Note that the elements of K(t) are given by the rigid body velocity in Ω_s. One can prove the following lemma on the space K(t), see Refs. [32,37].

Lemma 2.1. The space K(t) is equivalent to

\begin{matrix} K (t) = {v \in V, D (v) = 0 in Ω_{s}} \end{matrix}

(13)

In this paper, we address the following control objective.

Control Objective 2.1. Under the initial data

\begin{matrix} Ω_{s} (0) ⋐ Ω \\ u_{i} (0, x) \in H \\ (x_{c s} (0), η (0), u_{cs} (0), ω_{s} (0)) \in \underline{Ω} \times D_{η_{s}} \times ℝ^{3} \times ℝ^{3} \end{matrix}

(14)

where $\underline{Ω} ⋐ Ω$ and $D_{η_{s}} = {(- \infty, \infty)}^{3}$ , design the control force vector $F_{s}$ and control moment vector $M_{s}$ to almost globally practically exponentially stabilize the rigid body at the origin. Moreover, the designed control forces and control moments must ensure existence of a weak solution, which is defined in Definition 4.1, of the closed-loop system consisting of both the rigid body and the fluids.

Note that the initial values $x_{cs} (0) \in \underline{Ω}$ was imposed to be compatible with $Ω_{s} (0) ⋐ Ω$ ⁠, and $η_{s} (0) \in D_{η_{s}}$ was made due to the domain of the modified Rodrigues parameter vector $η_{s}$ ⁠.

In the rest of this section, we write Eqs. – as single NSEs for a global fluid, and rewrite Eq. accordingly. We define an indicator function and global variables

\begin{matrix} χ_{f} (t, x = 1_{Ω_{1} (t)} (x) \\ {\begin{matrix} ρ_{f} (t, x) = ρ_{i}, μ_{f} (t, x) = μ_{i} \\ u_{f} (t, x) = u_{i} \\ p_{f} (t, x) = p_{i} σ_{f} (t, x) = σ_{i} \\ f (t, x) = f_{i} \end{matrix} for x \in Ω_{i} \end{matrix}

(15)

We will drop the arguments

(t, x)

of all the above variables when there is no confusion. Clearly, we have

\begin{matrix} ρ_{f} = χ_{f} (ρ_{1} - ρ_{2}) + ρ_{2} \\ μ_{f} = χ_{f} (μ_{1} - μ_{2}) + μ_{2} \end{matrix}

(16)

We now can write the NSEs for the global fluid that describes the motion of incompressible fluids 1 and 2, and the transport equations for χ_f, ρ_f, and μ_f based on [38] as

\begin{matrix} {\begin{matrix} \partial_{t} χ_{f} + div (χ_{f} u_{f}) = 0 \\ \partial_{t} ρ_{f} + div (ρ_{f} u_{f}) = 0, in D' (Q_{f}) \\ \partial_{t} μ_{f} + div (μ_{f} u_{f}) = 0 \\ \partial_{t} (ρ_{f} u_{f}) + div (ρ_{f} u_{f} \otimes u_{f}) - div (σ_{f}) = κ H \nabla χ_{f} + ρ_{f} f \\ div (u_{f}) = 0 \end{matrix} \\ u_{f} = 0 on [0, T) \times \partial Ω \\ u_{f} |_{t = 0} : = u_{f 0} in Ω \end{matrix}

(17)

where $Q_{f} = [0, T] \times Ω_{f}$ ⁠. Here, $\nabla χ_{f}$ is a finite Radon measure and $| | \nabla χ_{f} | |_{M (Ω_{f})} = H^{2} (Γ (t))$ ⁠, see Ref. [39]. There are no boundary conditions for χ_f, ρ_f, and μ_f. As shown in Refs. [13,38,39], the nonmiscibility condition in Eq. (4) is equivalent to a transport equation for $μ (t, x)$ ⁠, where the surface tension is taken into account due to the term $H \nabla χ_{f}$ in Eq. (17), see also [40], and the transmission conditions in Eqs. (4) and (11) will hold as soon as $u (t, x)$ belong to $L^{2} (0, T; H^{1} (Ω))$ ⁠.

The rigid body dynamics Eq. are now rewritten as

\begin{matrix} \frac{d x_{cs}}{d t} = u_{cs} \\ \frac{d η_{s}}{d t} = R_{s} ω_{s} \\ m_{s} \frac{d u_{cs}}{d t} = \int_{\partial Ω_{s}} σ_{f} n_{s} d τ + F_{s} + \int_{Ω_{s}} ρ_{s} f d τ \\ J_{s} \frac{d ω_{s}}{d t} = - ω_{s} \times (J_{s} ω_{s}) + \int_{\partial Ω_{s}} r_{s} \times (σ_{f} n_{s}) d τ \\ + M_{s} + \int_{Ω_{s}} ρ_{s} r_{s} \times f d τ \end{matrix}

(18)

where

n_{s}

is the normal unit vector pointing outside of Ω_s, and the transmission condition is rewritten as

\begin{matrix} u_{s} = u_{f} on \partial Ω_{s} \end{matrix}

(19)

3 Control Design

We extend the backstepping method in Ref. [33] and the forwarding method [31] to design the controls $F_{s}$ and $M_{s}$ with necessary modifications so that these controls are of a form that can be amended to be inverse optimal. Existence of a weak solution of the closed-loop system will be shown in Sec. 4. Analysis of stability and convergence of the closed-loop system will be given in 5. Since the rigid body dynamics Eq. (18) is of a second-order system, the control design consists of two steps.

3.1 Step 1.

Define

\begin{matrix} u_{cse} = u_{cs} - α_{u_{cs}} \\ ω_{se} = ω_{s} - α_{ω_{s}} \end{matrix}

(20)

where

α_{u_{cs}}

and

α_{ω_{s}}

are virtual controls of

u_{cs}

and

ω_{s}

⁠, respectively. At this step, we consider the following Lyapunov function candidate to design

α_{u_{cs}}

and

α_{ω_{s}}

⁠:

\begin{matrix} U_{11} = \frac{1}{2} | | x_{cs} | |_{E}^{2} + \frac{1}{2} | | η_{s} | |_{E}^{2} \end{matrix}

(21)

Differentiating Eq. and using Eqs. and yields

\begin{matrix} \frac{d U_{11}}{d t} = x_{cs} \cdot (α_{u_{cs}} + u_{cse}) + η_{s} \cdot R_{s} (α_{ω_{s}} + ω_{se}) \end{matrix}

(22)

From Eq. , we design the virtual controls

α_{u_{cs}}

and

α_{ω_{s}}

as

\begin{matrix} α_{u_{cs}} = - K_{11} x_{cs} \\ α_{ω_{s}} = - K_{12} η_{s} \end{matrix}

(23)

where

K_{11}

and

K_{12}

are positive definite matrices. Substituting Eq. into Eq. and using Eq. gives

\begin{matrix} \frac{d U_{11}}{d t} = - x_{cs} \cdot K_{11} x_{cs} - η_{s} \cdot R_{s} K_{12} η_{s} \\ + x_{cs} \cdot u_{cse} + η_{s} \cdot R_{s} ω_{se} \\ = - x_{cs} \cdot K_{11} x_{c s} - \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) η_{s} \cdot K_{12} η_{s} \\ + x_{cs} \cdot u_{cse} + \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) η_{s} \cdot ω_{se} \end{matrix}

(24)

Multiplying the first equation of Eq. by m and the second equation of Eq. by

J_{s}

⁠, then differentiating the results and using Eqs. and gives

\begin{matrix} m_{s} \frac{d u_{cse}}{d t} = F_{s} + m_{s} K_{11} (u_{cse} - K_{11} x_{cs}) + \int_{\partial Ω_{s}} σ_{f} n_{s} d τ \\ + \int_{Ω_{s}} ρ_{s} f d τ \\ J_{s} \frac{d ω_{se}}{d t} = - ω_{s} \times (J_{s} ω_{s}) + M_{s} + J_{s} K_{12} R_{s} (ω_{se} - K_{12} η_{s}) \\ + \int_{\partial Ω_{s}} r_{s} \times (σ_{f} n_{s}) d τ + \int_{Ω_{s}} ρ_{s} r_{s} \times f d τ \end{matrix}

(25)

3.2 Step 2.

To design the actual controls

F_{s}

and

M_{s}

⁠, we consider the following Lyapunov function candidate:

\begin{matrix} U_{1} = U_{11} + \frac{1}{2} m_{s} | | u_{cse} | |_{E}^{2} + \frac{1}{2} ω_{se} \cdot J_{s} ω_{se} \end{matrix}

(26)

Differentiating Eq. and using Eqs. and gives

\begin{matrix} \frac{d U_{1}}{d t} = - x_{cs} \cdot K_{11} x_{cs} - \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) η_{s} \cdot K_{12} η_{s} \\ + u_{cse} \cdot F_{s} + ω_{se} \cdot M_{s} + ϖ_{0} + ϖ + Ξ \end{matrix}

(27)

where

\begin{matrix} ϖ_{0} = x_{cs} \cdot u_{cse} + \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) η_{s} \cdot ω_{se} \\ + u_{cse} \cdot [m_{s} K_{11} (u_{cse} - K_{11} x_{cs})] \\ + ω_{se} \cdot [ω_{s} \times (J_{s} ω_{s}) + J_{s} K_{12} R_{s} (ω_{se} - K_{12} η_{s})] \end{matrix}

(28)

\begin{matrix} ϖ = u_{cse} \cdot \int_{\partial Ω_{s}} σ_{f} n_{s} d τ + ω_{se} \cdot \int_{\partial Ω_{s}} r_{s} \times (σ_{f} n_{s}) d τ \end{matrix}

(29)

and

\begin{matrix} Ξ = u_{cse} \cdot \int_{Ω_{s}} ρ_{s} f d τ + ω_{se} \cdot \int_{Ω_{s}} ρ_{s} r_{s} \times f d τ \\ \leq ε_{01} | | u_{cse} | |_{E}^{2} + ε_{02} | | ω_{se} | |_{E}^{2} + Ξ_{10} \end{matrix}

(30)

with

\begin{matrix} Ξ_{10} = \frac{1}{4 ε_{01}} | | ρ_{s} f | |_{Ω_{s}}^{2} + \frac{1}{4 ε_{02}} | | ρ_{s} r_{s} \times f | |_{Ω_{s}}^{2} \end{matrix}

(31)

ε₀₁ and ε₀₂ being positive constants to be chosen.

Using Eqs. and , and Young's inequality, we can bound terms in

ϖ_{0}

as follows:

\begin{matrix} x_{cs} \cdot u_{cse} \leq ε_{03} | | x_{cs} | |_{E}^{2} + \frac{| | u_{cse} | |_{E}^{2}}{4 ε_{03}} \\ \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) η_{s} \cdot ω_{se} \leq \frac{1 + | | η_{s} | |_{E}^{2}}{4} (ε_{04} | | η_{s} | |_{E}^{2} + \frac{| | ω_{se} | |_{E}^{2}}{4 ε_{04}}) \\ u_{cse} \cdot m_{s} K_{11} u_{cse} \leq m_{s} λ_{M} (K_{11}) | | u_{cse} | |_{E}^{2} \\ - u_{cse} \cdot m_{s} K_{11}^{2} x_{cs} \leq m_{s} λ_{M} (K_{11}^{2}) (ε_{05} | | x_{cs} | |_{E}^{2} + \frac{| | u_{cse} | |_{E}^{2}}{4 ε_{05}}) \\ ω_{se} \cdot (ω_{s} \times (J_{s} ω_{s})) = ω_{se} \cdot ((ω_{se} + K_{12} η_{s}) \times (J_{s} ω_{s})) \\ = ω_{se} \cdot ((K_{12} η_{s}) \times (J_{s} ω_{s})) \\ = ω_{se} \cdot ((K_{12} η_{s}) \times (J_{s} (ω_{se} + K_{12} η_{s}))) \\ \leq \frac{1}{2} λ_{M} (K_{12}) λ_{M} (J_{s}) (1 + | | η_{s} | |_{E}^{2}) | | ω_{se} | |_{E}^{2} \\ + λ_{M}^{2} (K_{12}) | | η_{s} | |_{E}^{2} (ε_{06} + \frac{| | ω_{se} | |_{E}^{2}}{4 ε_{06}}) \end{matrix}

\begin{matrix} ω_{se} \cdot (J_{s} K_{12} R_{s} ω_{se}) \leq λ_{M} (J_{s} K_{12}) \frac{1 + | | R_{s} | |_{E}^{2}}{2} | | ω_{se} | |_{E}^{2} \\ \leq λ_{M} (J_{s} K_{12}) (\frac{1}{2} + \frac{1 + | | η_{s} | |_{E}^{2}}{8}) | | ω_{se} | |_{E}^{2} \\ - ω_{se} \cdot (J_{s} K_{12} R_{s} K_{12} η_{s}) \\ \leq λ_{M} (J_{s}) λ_{M}^{2} (K_{12}) \frac{1 + | | R_{s} | |_{E}^{2}}{2} | ω_{se} \cdot η_{s} |_{E} \\ \leq λ_{M} (J_{s}) λ_{M}^{2} (K_{12}) (\frac{1}{2} + \frac{1 + | | η_{s} | |_{E}^{2}}{8}) (ε_{07} | | η_{s} | |_{E}^{2} + \frac{| | ω_{se} | |_{E}^{2}}{4 ε_{07}}) \end{matrix}

(32)

where

ε_{0 i}, i = 5, \dots, 7

are positive constants to be chosen, and

λ_{M} (•)

denotes the maximum eigenvalue of

•

⁠. Using Eqs. and , we can bound the sum

ϖ_{0} + Ξ

as follows:

\begin{matrix} ϖ_{0} + Ξ \leq ξ_{01} | | x_{cs} | |_{E}^{2} + ξ_{02} | | u_{cse} | |_{E}^{2} + ξ_{031} | | η_{s} | |_{E}^{2} \\ + ξ_{032} (1 + | | η_{s} | |_{E}^{2}) | | η_{s} | |_{E}^{2} + ξ_{041} | | ω_{se} | |_{E}^{2} \\ + ξ_{042} (1 + | | η_{s} | |_{E}^{2}) | | ω_{se} | |_{E}^{2} + Ξ_{10} \end{matrix}

(33)

where

\begin{matrix} ξ_{01} = ε_{03} + m_{s} λ_{M} (K_{11}^{2}) ε_{05} \\ ξ_{02} = ε_{01} + \frac{1}{4 ε_{03}} + m_{s} λ_{M} (K_{11}) + m_{s} λ_{M} (K_{11}^{2}) \frac{1}{4 ε_{05}} \\ ξ_{031} = λ_{M}^{2} (K_{12}) ε_{06} + λ_{M} (J_{s}) λ_{M}^{2} (K_{12}) \frac{1}{2} ε_{07} \\ ξ_{032} = \frac{1}{4} ε_{04} + λ_{M} (J_{s}) λ_{M}^{2} (K_{12}) \frac{1}{8} ε_{07} \\ ξ_{041} = ε_{02} + λ_{M} (J_{s} K_{12}) \frac{1}{2} + λ_{M} (J_{s}) λ_{M}^{2} (K_{12}) \frac{1}{8 ε_{07}} \\ ξ_{042} = \frac{1}{16} ε_{04} + \frac{1}{2} λ_{M} (K_{12}) λ_{M} (J_{s}) + λ_{M}^{2} (K_{12}) \frac{1}{4 ε_{06}} \\ + λ_{M} (J_{s} K_{12}) \frac{1}{8} + λ_{M} (J_{s}) λ_{M}^{2} (K_{12}) \frac{1}{32 ε_{07}} \end{matrix}

(34)

Substituting Eq. into Eq. gives

\begin{matrix} \frac{d U_{1}}{d t} \leq - x_{cs} \cdot K_{11} x_{cs} - \frac{1}{4} (1 + | | η_{s} | |_{E}^{2}) η_{s} \cdot K_{12} η_{s} \\ + ξ_{01} | | x_{cs} | |_{E}^{2} + ξ_{02} | | u_{cse} | |_{E}^{2} + ξ_{031} | | η_{s} | |_{E}^{2} \\ + ξ_{032} (1 + | | η_{s} | |_{E}^{2}) | | η_{s} | |_{E}^{2} + ξ_{041} | | ω_{se} | |_{E}^{2} \\ + ξ_{042} (1 + | | η_{s} | |_{E}^{2}) | | ω_{se} | |_{E}^{2} + u_{cse} \cdot F_{s} \\ + ω_{se} \cdot M_{s} + ϖ + Ξ_{10} \end{matrix}

(35)

From Eq. , we design

F_{s}

and

M_{s}

as follows:

\begin{matrix} F_{s} = - K_{21} u_{cse} \\ M_{s} = - (1 + | | η_{s} | |_{E}^{2}) K_{22} ω_{se} \end{matrix}

(36)

where $K_{21}$ and $K_{22}$ are positive definite matrices to be chosen. Note that the nonlinear term $(1 + | | η_{s} | |_{E}^{2})$ is due to the nonlinear terms $R_{s} ω_{s}$ and $- ω_{s} \times (J_{s} ω_{s})$ in the rigid body dynamics Eq. (18). From Eqs. (36) and (10), we can solve for the actuator forces $F_{k}$ ⁠. The controls Eq. (36) do not cancel any system functions. This is important for robustness.

Substituting Eq. into Eq. gives

\begin{matrix} \frac{d U_{1}}{d t} \leq - k_{11} | | x_{cs} | |_{E}^{2} - k_{12} | | u_{cse} | |_{E}^{2} - k_{21} | | η_{s} | |_{E}^{2} \\ - k_{22} | | ω_{se} | |_{E}^{2} + ϖ + Ξ_{10} \end{matrix}

(37)

where we have chosen the control gain matrices

(K_{11}, K_{21}, K_{22})

⁠, and positive constants

ε_{0 i}, i = 1, \dots, 7

such that

\begin{matrix} k_{11} = λ_{m} (K_{11}) - ξ_{01} > 0 \\ k_{12} = λ_{m} (K_{21}) - ξ_{02} > 0 \\ k_{21} = \frac{1}{4} λ_{m} (K_{12}) - ξ_{031} - ξ_{032} > 0 \\ k_{22} = λ_{m} (K_{22}) - ξ_{041} - ξ_{042} > 0 \end{matrix}

(38)

It can be easily shown that there always exist sufficiently large control gain matrices $K_{11}, K_{12}, K_{21}$ ⁠, and $K_{22}$ such that all the inequalities in Eq. (38) hold for given positive constants $ε_{0 i}, i = 1, \dots, 7$ ⁠.

From Eqs. and , we can write Eq. as follows:

\begin{matrix} \frac{d U_{1}}{d t} \leq - k_{1} U_{1} + ϖ + Ξ_{10} \end{matrix}

(39)

where

\begin{matrix} k_{1} = \frac{2 \min (k_{11}, k_{12}, k_{21}, k_{22})}{\max (1, m_{s}, λ_{M} (J_{s}))} \end{matrix}

(40)

Remark 3.1. Let us define the state vector

Y_{s e} = col (x_{cs}, u_{cse}, η_{s}, ω_{se})

⁠. Then, we can write Eq. as

\begin{matrix} [\begin{matrix} F_{s} \\ M_{s} \end{matrix}] = - W {(\frac{\partial U_{1}}{\partial Y_{se}})}^{T} \end{matrix}

(41)

where W is a positive definite matrix. This form means that the controls $F_{s}$ and $M_{s}$ are inverse pre-optimal and can be easily extended to be inverse optimal by using the Legendre-Fenchel transform, see Ref. [34], which has many desired properties such as an infinite gain margin and minimization of a cost function without having to solve a Hamilton-Jacobi-Bellman equation or a Hamilton-Jacobi-Isaacs equation. However, we do not detail this issue here as we focus on the interaction of the controlled rigid body and two phase flows of two viscous incompressible fluids in this paper.

Remark 3.2. We cannot obtain a bound for the fluid force $\int_{\partial Ω_{s}} σ_{f} n_{s} d τ$ and the fluid moment $\int_{\partial Ω_{s}} r_{s} \times (σ_{f} n_{s}) d τ$ ⁠, and hence the term ϖ defined in Eq. (29), i.e., we cannot conclude any stability of the controlled rigid body dynamics Eq. (18) with actuator forces $F_{k}$ obtained from Eqs. (10) and (36) based on Eqs. (26) and (39) at this point. This is a major difficulty compared with the approximation approach, see Refs. [1,2,7,8] and references therein. Therefore, we will analyze stability of the rigid body dynamics and convergence of its states in Sec. 5 after we show existence of the weak solution of the dynamics of the fluids and the controlled rigid body in Sec. 4.

4 Existence of a Weak Solution of the Closed-Loop System

In this section, we show the closed-loop system consisting of the dynamics of the fluids Eq. (17) and the controlled rigid body dynamics Eq. (18) with actuator forces $F_{k}$ obtained from Eqs. (10) and (36) has at least one weak solution.

4.1 Formulation of a Weak Solution.

Multiplying the fourth equation in Eq. by a test function

φ \in I_{T}

and integrating over Ω_f, and then integrating by parts gives

\begin{matrix} \frac{d}{d t} \int_{Ω_{f}} ρ_{f} u_{f} \cdot φ_{f} d x - \int_{Ω_{f}} ρ_{f} u_{f} \cdot \partial_{t} φ_{f} d x - κ 〈 H, φ_{f} 〉 \\ - \int_{Ω_{f}} ρ_{f} u_{f} \otimes u_{f} : \nabla φ_{f} d x + \int_{Ω_{f}} 2 μ_{f} D (u_{f}) : D (φ_{f}) d x \\ = - \int_{\partial Ω_{s}} (σ_{f} n_{s}) \cdot φ_{f} d τ + \int_{Ω_{f}} ρ_{f} f \cdot φ_{f} d x \end{matrix}

(42)

where

\begin{matrix} 〈 H, φ_{f} 〉 : = \int_{Ω_{f}} H \nabla χ_{f} \cdot φ_{f} d x = \int_{Γ (t)} H n \cdot φ_{f} d H^{2} (x) \end{matrix}

(43)

with $H^{2}$ being the two-dimensional Hausdorff measure.

Next, we derive from Eq. with controls

F_{k}

obtained from Eqs. and for

φ \in I_{T}

that

\begin{matrix} \frac{d}{d t} \int_{Ω_{s}} ρ_{s} u_{s} \cdot φ_{s} d x - \int_{Ω_{s}} ρ_{s} u_{s} \cdot \partial_{t} φ_{s} d x = \int_{\partial Ω_{s}} (σ_{f} n_{s}) \cdot φ_{s} d τ \\ + \int_{Ω_{s}} \sum_{k = 1}^{N} (δ_{Fk} F_{k}) \cdot φ_{s} d x + \int_{Ω_{s}} ρ_{s} f \cdot φ_{s} d x \end{matrix}

(44)

where

δ_{Fk}

denotes the diagonal matrix of Dirac delta function of

x - x_{Fk}

with

x_{Fk}

being the position of actuator forces

F_{k}

⁠, i.e.,

x_{Fk} = x_{c s} + r_{Fk}

⁠. In derivation of Eq. , we have used

φ_{s} = v_{φ_{s}} + ω_{φ_{s}} \times r_{s}

and

\begin{matrix} v_{φ_{s}} \cdot F_{s} + ω_{φ_{s}} \cdot M_{s} = \sum_{i = 1}^{N} (v_{φ_{s}} \cdot F_{k} + ω_{φ_{s}} \cdot (r_{Fk} \times F_{k})) \\ = \sum_{k = 1}^{N} \int_{Ω_{s}} (v_{φ_{s}} \cdot δ_{Fk} F_{k} + ω_{φ_{s}} \cdot (r_{s} \times δ_{Fk} F_{k})) d x \\ = \sum_{k = 1}^{N} \int_{Ω} (v_{φ_{s}} \cdot δ_{Fk} F_{k} + ω_{φ_{s}} \cdot (r_{s} \times δ_{Fk} F_{k})) d x \\ = \sum_{k = 1}^{N} \int_{Ω} φ_{s} \cdot δ_{Fk} F_{k} d x \end{matrix}

(45)

due to $ω_{φ_{s}} \cdot (r_{s} \times δ_{Fk} F_{k}) = (ω_{φ_{s}} \times r_{s}) \cdot δ_{Fk} F_{k}$ ⁠.

Adding Eqs. and yields

\begin{matrix} \frac{d}{d t} \int_{Ω_{f}} ρ_{f} u_{f} \cdot φ_{f} d x - \int_{Ω_{f}} ρ_{f} u_{f} \cdot \partial_{t} φ_{f} d x - κ 〈 H, φ_{f} 〉 \\ - \int_{Ω_{f}} ρ_{f} u_{f} \otimes u_{f} : \nabla φ_{f} d x + \int_{Ω_{f}} 2 μ_{f} D (u_{f}) : D (φ_{f}) d x \\ + \frac{d}{d t} \int_{Ω_{s}} ρ_{s} u_{s} \cdot φ_{s} d x - \int_{Ω_{s}} ρ_{s} u_{s} \cdot \partial_{t} φ_{s} d x \\ = \int_{Ω_{s}} \sum_{k = 1}^{N} (δ_{F k} F_{k}) \cdot φ_{s} d x + \int_{Ω_{s}} ρ_{s} f \cdot φ_{s} d x \\ + \int_{Ω_{f}} ρ_{f} f \cdot φ_{f} d x \end{matrix}

(46)

Equation is a global weak formulation of the momentum equations in Eqs. and with actuator forces

F_{k}

obtained from Eqs. and , where the surface tension is taken into account. This goes together with the conservation of mass, which represents the transport of Ω_s by the rigid vector field

u_{s}

⁠, i.e., the indicator function

\begin{matrix} χ_{s} (t, x) = {\begin{matrix} 1 if x \in Ω_{s} \\ 0 otherwise \end{matrix} \end{matrix}

(47)

satisfies

\begin{matrix} \partial_{t} χ_{s} + div (u_{s} χ_{s}) = 0 in D' (Q) \end{matrix}

(48)

Now, it is seen that Eqs. and involve with the fields

u_{f}, u_{s}, φ_{f}

⁠, and

φ_{s}

defined over Ω such that

\begin{matrix} u = (1 - χ_{s}) u_{f} + χ_{s} u_{s} \\ φ = (1 - χ_{s}) φ_{f} + χ_{s} φ_{s} \end{matrix}

(49)

Setting

φ_{f} = u_{f}

and

φ_{s} = u_{s}

in Eq. yields

\begin{matrix} \frac{1}{2} \frac{d}{d t} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x + \frac{1}{2} \frac{d}{d t} \int_{Ω_{s}} ρ_{s} | | u_{s} | |_{E}^{2} d x + κ \frac{d}{d t} H^{2} (Γ) \\ + \int_{Ω_{f}} 2 μ_{f} | | D (u_{f}) | |_{E}^{2} d x = \int_{Ω_{s}} \sum_{k = 1}^{N} (δ_{Fk} F_{k}) \cdot u_{s} d x \\ + \int_{Ω_{s}} ρ_{s} f \cdot u_{s} d x + \int_{Ω_{f}} ρ_{f} f \cdot u_{f} d x \end{matrix}

(50)

where we have used [39]

\begin{matrix} \int_{Ω_{f}} H \nabla χ_{f} \cdot u_{f} d x = \int_{Γ} H n \cdot u_{f} d H^{2} = - \frac{d}{d t} H^{2} (Γ) \end{matrix}

(51)

Hence, the distributional gradient

\nabla χ_{f}

is a finite Radon measure and

| | \nabla χ_{f} | |_{M (Ω_{f})} = H^{2} (Γ)

⁠, where

M (Ω_{f})

denotes the space of finite Randon measures.

We consider the “energy”

E

defined by

\begin{matrix} E = \frac{1}{2} x_{cs}^{T} K_{21} K_{11} x_{c s} + 2 η_{s}^{T} K_{22} K_{12} η_{s} + κ H^{2} (Γ) \\ + \frac{1}{2} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x + \frac{1}{2} \int_{Ω_{s}} ρ_{s} | | u_{s} | |_{E}^{2} d x \end{matrix}

(52)

Differentiating Eq. along the solutions of Eq. and using Eqs. and and the Hölder inequality, we obtain formally

\begin{matrix} \frac{d E}{d t} = - \int_{Ω_{f}} 2 μ_{f} | | D (u_{f}) | |_{E}^{2} d x - u_{cs}^{T} K_{21} u_{cs} - ω_{s}^{T} K_{22} ω_{s} \\ + \int_{Ω_{s}} ρ_{s} f \cdot u_{s} d x + \int_{Ω_{f}} ρ_{f} f \cdot u_{f} d x \end{matrix}

(53)

Using

\int_{Ω_{s}} ρ_{s} | | u_{s} | |_{E}^{2} d x = m_{s} | | u_{cs} | |_{E}^{2} + ω_{s}^{T} J_{s} ω_{s}

and the Poincaré inequality, we have

\begin{matrix} \frac{d E}{d t} \leq - \int_{Ω_{f}} 2 μ_{f} | | D (u_{f}) | |_{E}^{2} d x - u_{cs}^{T} K_{21} u_{c s} - ω_{s}^{T} K_{22} ω_{s} \\ + ε_{10} \int_{Ω_{s}} ρ_{s} | | u_{s} | |_{E}^{2} d x + ε_{11} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x + Ξ_{20} \\ \leq - \int_{Ω_{f}} 2 (μ_{f} - ε_{12}) | | D (u_{f}) | |_{E}^{2} d x \\ - (λ_{m} (K_{11}) - m_{s} ε_{10}) | | u_{c s} | |_{E}^{2} \\ - (λ_{m} (K_{12}) - ε_{11} λ_{M} (J_{s})) | | ω_{s} | |_{E}^{2} \\ - \int_{Ω_{f}} (2 ε_{12} - ε_{11} \max (ρ_{1}, ρ_{2}) c_{p}) | | D (u_{f}) | |_{E}^{2} d x + Ξ_{20} \end{matrix}

(54)

where

\begin{matrix} Ξ_{20} = (\frac{1}{4 ε_{10}} + \frac{1}{4 ε_{11}}) \int_{Ω_{s}} ρ_{s} | | f | |_{E}^{2} d x \end{matrix}

(55)

c_p is the Poincaré constant, and

ε_{1 i}, i = 0, 1, 2

are positive constants chosen such that

\begin{matrix} μ_{f} - ε_{12} > 0 \\ λ_{m} (K_{11}) - m_{s} ε_{10} > 0 \\ λ_{m} (K_{12}) - ε_{11} λ_{M} (J_{s}) > 0 \\ 2 ε_{12} - ε_{11} \max (ρ_{1}, ρ_{2}) c_{p} \geq 0 \end{matrix}

(56)

Then, we have from Eq. that

\begin{matrix} \frac{d E}{d t} \leq - \int_{Ω_{f}} 2 (μ_{f} - ε_{12}) | | D (u_{f}) | |_{E}^{2} d x \\ - (λ_{m} (K_{11}) - m_{s} ε_{10}) | | u_{c s} | |_{E}^{2} \\ - (λ_{m} (K_{12}) - ε_{11} λ_{M} (J_{s})) | | ω_{s} | |_{E}^{2} + Ξ_{20} \end{matrix}

(57)

Integrating Eq. from 0 to t gives

\begin{matrix} E (t) \leq E (0) - \int_{0}^{t} \int_{Ω_{f}} 2 (μ_{f} - ε_{12}) | | D (u_{f}) | |_{E}^{2} d x d s \\ - (λ_{m} (K_{11}) - m_{s} ε_{10}) \int_{0}^{t} | | u_{cs} | |_{E}^{2} d s \\ - (λ_{m} (K_{12}) - ε_{11} λ_{M} (J_{s})) \int_{0}^{t} | | ω_{s} | |_{E}^{2} d s + \int_{0}^{t} Ξ_{20} d s \end{matrix}

(58)

Due to Eq. , which ensures that

\int_{0}^{t} Ξ_{20} d s

is bounded, we expect from Eqs. , , , , and , and the first three equations in Eq. that

\begin{matrix} x_{c s} \in L^{\infty} (0, T; ℝ^{3}), η_{s} \in L^{\infty} (0, T; ℝ^{3}) \\ u_{f} \in L^{\infty} (0, T; H) \cap L^{2} (0, T; V) \\ u_{s} \in L^{\infty} (0, T; K) \\ χ_{f} \in L^{\infty} (0, T; B V (Ω_{f}; {0, 1})) \\ ρ_{f} \in L^{\infty} (0, T; B V (Ω_{f}; {ρ_{1}, ρ_{2}})) \\ μ_{f} \in L^{\infty} (0, T; B V (Ω_{f}; {μ_{1}, μ_{2}})) \\ χ_{s} \in L^{\infty} (0, T; (Ω; {0, 1}) \\ V \in L^{\infty} (0, T; M (Ω \times S^{2})) \\ p \in L^{2} (Q) \end{matrix}

(59)

where $χ_{f} \in L^{\infty} (0, T; B V (Ω_{f}; {0, 1}))$ means that $χ_{f} \in L^{\infty} (0, T; B V (Ω_{f}))$ and $χ_{f} \in {0, 1}$ with $B V (Ω_{f}) = {v \in L^{1} (Ω_{f}) : \nabla v \in M (Ω_{f})$ being the space of functions with bounded variation [41]; $ρ_{f} \in L^{\infty} (0, T; B V (Ω_{f}; {ρ_{1}, ρ_{2}}))$ means that $ρ_{f} \in L^{\infty} (0, T; B V (Ω_{f}))$ and $ρ_{f} \in {ρ_{1}, ρ_{2}}$ ⁠; $μ_{f} \in L^{\infty} (0, T; B V (Ω_{f}; {μ_{1}, μ_{2}}))$ means that $μ_{f} \in L^{\infty} (0, T; B V (Ω_{f}))$ and $μ_{f} \in {μ_{1}, μ_{2}}$ ⁠; and $χ_{s} \in L^{\infty} (0, T; (Ω; {0, 1})$ means $χ_{s} \in L^{\infty} (0, T; Ω)$ and $χ_{s} \in {0, 1}$ ⁠.

The above derivations motivate the following weak solution definition.

Definition 4.1. Under the initial data Eq. , the tuple

(x_{c s}, η_{s}

, χ_f, ρ_f, μ_f,

u_{f}, u_{s}, χ_{s}, V, p)

is a weak solution of the closed-loop system consisting of Eqs. , , and if they satisfy Eqs. and , the first three equations in Eq. in

D' (Q_{f})

and Eq. in

D' (Q)

⁠, Eq. , and the compatibility condition

\begin{matrix} - \int_{Ω_{f} \times S^{2}} s \cdot ψ (x) d V (x, s) = \int_{Ω_{f}} ψ d \nabla χ_{f} (t) \end{matrix}

(60)

for all $ψ \in C_{0}^{3} (Ω_{f})$ ⁠.

The above definition of a weak solution is similar to a definition of a weak solution to a two phase flow of two viscous incompressible fluids, [39] except for the fact that we take the constraint of rigidity into account.

4.2 Penalized System.

In order to find the tuple $(x_{c s}, η_{s}$ ⁠, χ_f, ρ_f, μ_f, $u_{f}, u_{s}, χ_{s}, V)$ such that they satisfy Definition 4.1, we use a penalization approach. Due to Lemma 2.1, there are three methods to penalize the rigidity constraint: (i) penalizing the difference between the fluid velocity and rigid body velocity [32]; (ii) penalizing the spatial derivative of the rigid body velocity [37]; (iii) combination of methods i and ii [42]. We use the third method.

Let

\frac{1}{n} > 0

be a penalized parameter. Given the initial data

\begin{matrix} x_{csn} (0) = x_{cs} (0), η_{sn} (0) = η_{s} (0), μ_{fn} (0) = μ_{f} (0) \\ u_{fn} (0) = u_{f} (0), u_{sn} (0) = u_{s} (0), χ_{fn} (0) = χ_{f} (0) \\ ρ_{fn} (0) = ρ_{f} (0), χ_{sn} (0) = χ_{s 0} \end{matrix}

(61)

we wish to find the tuple

(x_{csn}, η_{sn}

⁠, χ_fn, ρ_fn, μ_fn,

u_{n}, χ_{sn}, V_{n}, p_{n})

such that they satisfy

\begin{matrix} x_{csn} \in L^{\infty} (0, T; ℝ^{3}), η_{s n} \in L^{\infty} (0, T; ℝ^{3}) \\ u_{n} \in L^{\infty} (0, T; H) \cap L^{2} (0, T; V) \\ χ_{fn} \in L^{\infty} (0, T; B V (Ω_{f}; {0, 1})) \\ ρ_{fn} \in L^{\infty} (0, T; B V (Ω_{f}; {ρ_{1}, ρ_{2}})) \\ μ_{fn} \in L^{\infty} (0, T; B V (Ω_{f}; {μ_{1}, μ_{2}})) \\ χ_{sn} \in L^{\infty} (0, T; (Ω; {0, 1}) \\ V_{n} \in L^{\infty} (0, T; M (Ω \times S^{2})) \\ p_{n} \in L^{2} (Q) \end{matrix}

(62)

and are a solution to the penalized system

\begin{matrix} {\begin{matrix} ρ_{n} (\partial_{t} u_{n} + (u_{n} \cdot \nabla) u_{n}) - div (σ_{n}) - κ H_{n} \nabla χ_{fn} \\ + n ρ_{n} h_{sn} (u_{n} - u_{n, s}) - \sum_{k = 1}^{N} (n_{k} F_{k}) = ρ_{n} f \\ div (u_{n}) = 0 \\ \partial_{t} χ_{fn} + u_{n} \cdot \nabla χ_{fn} = 0 \\ \partial_{t} ρ_{fn} + u_{n} \cdot \nabla ρ_{fn} = 0 \\ \partial_{t} μ_{fn} + u_{n} \cdot \nabla μ_{fn} = 0 \\ \partial_{t} χ_{sn} + u_{n, s} \cdot \nabla χ_{sn} = 0 \end{matrix} in Q \\ {\begin{matrix} \frac{d x_{csn}}{d t} = u_{csn} \\ \frac{d η_{sn}}{d t} = R_{sn} ω_{sn} \end{matrix} in ℝ^{3} \end{matrix}

(63)

where

\begin{matrix} ρ_{n} = (1 - χ_{sn}) ρ_{fn} + χ_{sn} ρ_{s} \\ μ_{n} = μ_{fn} + \frac{1}{n^{2}} χ_{sn}, H_{n} = - {div}^{Γ} (u_{n}) \\ u_{n} = (1 - χ_{sn}) u_{fn} + χ_{sn} u_{sn} \\ σ_{n} = 2 μ_{n} D (u_{n}) - p_{n} I_{3} \\ u_{n, s} = \frac{1}{m_{sn}} \int_{Ω} ρ_{n} u_{n} χ_{sn} d x + (J_{sn}^{- 1} \int_{Ω} ρ_{n} r_{n} \times u_{n} χ_{sn} d x) \times r_{sn} \\ R_{n} = \frac{1}{2} (I - G (η_{sn}) + η_{sn} η_{sn}^{T} - \frac{1 + | | η_{sn} | |_{E}^{2}}{2} I_{3}) \end{matrix}

(64)

with

\begin{matrix} r_{sn} = x - x_{csn}, x_{csn} = \frac{1}{m_{sn}} \int_{Ω} ρ_{sn} h_{sn} x d x \\ m_{sn} = \int_{Ω} ρ_{n} h_{sn} d x \\ J_{sn} = \int_{Ω} ρ_{n} (| | r_{sn} | |_{E}^{2} I_{3} - r_{sn} \otimes r_{sn}) h_{sn} d x \end{matrix}

(65)

In addition, we impose the homogeneous Dirichlet boundary condition

\begin{matrix} u_{n} = 0 on (0, T) \times \partial Ω \end{matrix}

(66)

and define $Ω_{sn} = {x \in Ω_{s}, χ_{sn} (t, x) = 1}$ ⁠. Several observations on the penalized system Eq. (63) are given in the following remark.

Remark 4.1. The penalized system Eq. (63) is based on [32], where action of the controls $F_{s}$ and $M_{s}$ given by Eq. (36) and nonhomogeneous viscosity and surface tension are taken into account.

It is clear that

| Ω_{sn} |_{E} = \int_{Ω} χ_{sn} d x = | Ω_{s} (0) |_{E}

because

u_{n, s}

is divergence free and

h_{sn}

vanishes on

\partial Ω

as we assume there is no collision between the rigid body and

\partial Ω

⁠. We also have

\begin{matrix} m_{sn} = \int_{Ω_{sn}} ρ_{n} d x \\ J_{sn} = \int_{Ω_{sn}} ρ_{n} (| | r_{sn} | |_{E}^{2} I_{3} - r_{sn} \otimes r_{sn}) d x \end{matrix}

(67)

Since $ρ_{n} \geq \min (ρ_{1}, ρ_{2}, ρ_{s}) > 0, m_{sn}$ is positive, and the matrix $J_{s n}$ is symmetric and positive definite, see Eq. (7), and hence is invertible.

In the first equation of Eq. , the term

u_{n, s}

defined in Eq. is the projection of

u_{n}

onto the velocity fields which are rigid on

Ω_{s}^{n}

because one can prove that, see Ref. [32]

\begin{matrix} \int_{Ω} ρ_{n} χ_{sn} (u_{n} - u_{n, s}) \cdot ζ d x = 0 \end{matrix}

(68)

where $ζ$ is a rigid velocity field, i.e., there exist $(v_{ζ}, ω_{ζ}) \in ℝ^{3} \times ℝ^{3}$ such that $ζ = v_{ζ} + ω_{ζ} \times r_{s} (t, x)$ ⁠, and $u_{n, s}$ is given by Eq. (64). Hence, the penalized term $n ρ_{n} χ_{sn} (u_{n} - u_{n, s})$ in the first equation of Eq. (63) is the difference between $u_{n}$ and its projection onto velocity fields rigid in the rigid body domain, i.e., $u_{n, s}$ ⁠. Moreover, the viscosity term μ_n vanishes asymptotically in the solid part. Hence, there will be no uniform bound in V for $u_{n}$ ⁠. This does not conflict with the penalized term $n ρ_{n} χ_{sn} (u_{n} - u_{n, s})$ because of the space K.

The density, viscosity, and fluid character function χ_fn are transported with the velocity field $u_{n}$ ⁠. This eases calculations in estimating bounds for the penalized system.

Existence of a weak solution to the penalized system Eq. (63) is stated in the following lemma.

Lemma 4.1. There is at least one weak solution $(x_{csn}, η_{sn}$ ⁠, χ_fn, ρ_fn, μ_fn, $u_{n}, χ_{sn}, V_{n}, p_{n})$ to the penalized system Eq. (63) that satisfies Eq. (62) for all $t \in [0, T]$ ⁠, where T is such that $Ω_{sn} (t) ⋐ Ω$ and $η_{sn} (t) \in D_{η_{s}}$ for all $t \in [0, T]$ ⁠.

Proof . Proof of this lemma principally follows the part of a priori estimates and convergence arguments in proof of Theorem 2.1 in Ref. [32] for a rigid body and fluid system, proof of Theorem 1.1 in Ref. [38] for a multifluid system (particularly, the renormalized solution of the transport equations is used), and proof of Theorem 1.6 in Ref. [39]. The only main difference is that a priori estimates should use the penalized energy $E_{n}$ as in Eq. (52) with $(x_{cs}, η_{s}, u)$ being substituted by $(x_{csn}, η_{sn}, u_{n})$ ⁠. This is due to inclusion of $(x_{cs}, η_{s})$ and controls $(F_{s}, M_{s})$ in this paper.▪

4.3 Existence of a Weak Solution.

Having obtained a weak solution of the penalized system Eq. (63) in Lemma 4.1, existence of a weak solution stated in Definition 4.1 is given in the following theorem.

Theorem 4.1. Let

(x_{csn}, η_{sn}

, χ_fn, ρ_fn, μ_fn,

u_{n}, χ_{sn}, V_{n}, p_{n})

be a weak solution to the penalized system Eq. (63). Then, under the initial data Eq. , there exists a subsequence of

(x_{csn}, η_{sn}

, χ_fn, ρ_fn, μ_fn,

u_{n}, χ_{sn}, V_{n})

such that

x_{csn} \to x_{cs}

strongly in

L^{\infty} (Q)

;

η_{sn} \to η_{s}

strongly in

L^{\infty} ((0, T) \times D_{η_{s}})

;

χ_{fn} \overset{*}{⇀} χ_{f}

in

L^{\infty} (Q)

;

ρ_{fn} \overset{*}{⇀} ρ_{f}

in

L^{\infty} (Q)

;

μ_{fn} \overset{*}{⇀} μ_{f}

in

L^{\infty} (Q)

;

χ_{sn} \to χ_{s}

strongly in

C (0, T; L^{q} (Ω)), q \geq 1

;

u_{n} \to u

strongly in

L^{2} (Q)

and weakly in

L^{\infty} (0, T; H) \cap L^{2} (0, T; V)

; and

V_{n} \overset{*}{⇀} V

in

L^{\infty} (0, T; H^{- ℓ} (Ω_{f} \times S^{2}))

, where

ℓ > \frac{5}{2}

, when

n \to \infty

. The tuple

(x_{c s}, η_{s}

, χ_f, ρ_f, μ_f,

u_{f}, u_{s}, χ_{s}, V)

is a weak solution of the closed-loop system consisting of Eqs. (1), (18), and (36)as defined in Definition 4.1. The constant T is such that

Ω_{s} (t) ⋐ Ω

and

η_{s} (t) \in D_{η_{s}}

for all

t \in [0, T]

. Moreover, there is

p \in W^{- 1, \infty} (0, T; L^{2} (Ω))

such that

\begin{matrix} \partial_{t} (ρ_{f} u_{f}) + div (ρ_{f} u_{f} \otimes u_{f}) - div (σ_{f}) = κ H \nabla χ_{f} + ρ_{f} f \end{matrix}

(69)

in $D' ((0, T) \times Ω_{f})$ , where $σ_{f} = 2 μ_{f} D (u_{f}) - p I_{3}$ ⁠.

Proof . Proof of this theorem can be readily obtained from that of Theorem 2.1 in Ref. [32], Theorem 1.1 in Ref. [38], and Theorem 1.6 in Ref. [39] with a note as in the proof of Lemma 4.1. For example, we show that

V_{n} \overset{*}{⇀} V

in

L^{\infty} (0, T; H^{- ℓ} (Ω_{f} \times S^{2}))

⁠, where

ℓ > \frac{5}{2}

⁠. We consider the sequence

V_{n} (t), t \in [0, \infty)

associated with

Γ_{n} (t)

⁠, i.e.,

\begin{matrix} (V_{n} (t), φ) : = \int_{Ω_{f}} φ (x, n_{n} (x)) d | \nabla χ_{fn} (t) |, φ \in C_{0} (Ω_{f} \times S^{2}) \end{matrix}

(70)

where C₀ denotes the closure of the usual space of continuous functions compactly supported on Ω_f, and

n_{n} (x)) = - \frac{\nabla χ_{fn}}{| \nabla χ_{fn} |}

⁠. Now, we set

\begin{matrix} (V_{n}, φ) = \int_{0}^{T} (V_{n} (t), φ (t)) d t, \forall φ \in L^{1} (0, T; C_{0} (Ω_{f} \times S^{2})) \end{matrix}

(71)

Then, since

V_{n} \in L_{ω}^{\infty} (0, T; M (Ω_{f} \times S^{2}))

⁠, see Sec. 1 for notation of

L_{ω}^{\infty} (0, T; M (Ω_{f} \times S^{2}))

⁠, is uniformly bounded and

M (Ω_{f} \times S^{2}) ↪ H^{- ℓ} (Ω_{f} \times S^{2})

⁠, where

ℓ > \frac{2 \times 3 - 1}{2} = \frac{5}{2}

⁠, we have

\begin{matrix} V_{n} \overset{*}{⇀} V in L^{\infty} (0, T; H^{- ℓ} (Ω_{f} \times S^{2})) as n \to \infty \end{matrix}

(72)

▪

5 Stability and Convergence of the Closed-Loop System

This section provides stability and convergence analysis of the closed-loop system, which can be based on Eqs. (26), (39), (52), and (53) once can handle the term ϖ in Eq. (29). All the calculations in this section use only properties of the weak solution shown in Theorem 4.1.

5.1 The Term ϖ.

Substituting Eqs. and into Eq. yields

\begin{matrix} ϖ = (u_{cs} + K_{11} x_{cs}) \cdot \int_{\partial Ω_{s}} σ_{f} n_{s} d τ \\ + (ω_{s} + K_{12} η_{s}) \cdot \int_{\partial Ω_{s}} r_{s} \times (σ_{f} n_{s}) d τ \\ = \sum_{i = 1}^{3} A_{i} \end{matrix}

(73)

where

\begin{matrix} A_{1} = \int_{\partial Ω_{s}} (σ_{f} n_{s}) \cdot u_{s} d τ \\ A_{2} = \int_{\partial Ω_{s}} (σ_{f} n_{s}) \cdot K_{11} x_{cs} d τ \\ A_{3} = \int_{\partial Ω_{s}} (σ_{f} n_{s}) \cdot ((K_{12} η_{s}) \times r_{s}) d τ \end{matrix}

(74)

The term A₁ can be considered as the power of the fluids acting on the floating rigid body while the terms A₂ and A₃ can be referred to as the work of the fluids acting on the floating rigid body. It is noted that from Sec. 4, we have

\begin{matrix} 0 < a_{10} \leq ρ_{f} (t, x) \leq a_{11} < \infty \\ 0 < a_{20} \leq μ_{f} (t, x) \leq a_{21} < \infty \end{matrix}

(75)

Since we already showed existence of a weak solution of the closed-loop system (including both the fluids and rigid body) in Theorem 4.1, the idea to handle the term ϖ is as follows. For the term A₁, we will multiply the fourth equation in Eq. (17) by $u_{f}$ to detail A₁, where we will use the transmission condition (19). For the terms A₂ and A₃, we proceed in two steps. In the first step, we use regularity of the Stokes problem, see Ref. [43], to obtain vector fields over Ω_s for $K_{11} x_{cs}$ and $(K_{12} η_{s}) \times r_{s}$ ⁠. In the second step, we use the extension theorem, see Ref. [44, Theorem 1, Sec. 5.4] to extend the vector fields to a virtual rigid body $Ω_{s}^{*}$ ⁠, see Fig. 1, to detail the terms A₂ and A₃.

5.1.1 Detail of A₁.

Multiplying the fourth equation in Eq. by

u_{f}

and integrating over Ω_f yields

\begin{matrix} \frac{1}{2} \frac{d}{d t} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x + κ \frac{d}{d t} H^{2} (Γ) + \int_{Ω_{f}} 2 μ_{f} | | D (u_{f}) | |_{E}^{2} d x \\ = - \int_{\partial Ω_{s}} (σ_{f} n_{s}) \cdot u_{s} d τ + \int_{Ω_{f}} ρ_{f} f \cdot u_{f} d x \end{matrix}

(76)

where we have used the transmission condition . Substituting Eq. into the first equation of Eq. gives

\begin{matrix} A_{1} = - \frac{1}{2} \frac{d}{d t} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x - κ \frac{d}{d t} H^{2} (Γ) \\ - \int_{Ω_{f}} 2 μ_{f} | | D (u_{f}) | |_{E}^{2} d x + \int_{Ω_{f}} ρ_{f} f \cdot u_{f} d x \end{matrix}

(77)

Using Hölder's inequality results in

\begin{matrix} A_{1} \leq - \frac{1}{2} \frac{d}{d t} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x - κ \frac{d}{d t} H^{2} (Γ) \\ - \int_{Ω_{f}} 2 μ_{f} | | D (u_{f}) | |_{E}^{2} d x + a_{11} ε_{20} \int_{Ω_{f}} | | u_{f} | |_{E}^{2} d x \\ + \frac{a_{11}}{4 ε_{20}} \int_{Ω_{f}} | | f | |_{E}^{2} d x \end{matrix}

(78)

where ε₂₀ is a positive constant to be chosen.

5.1.2 Detail of A₂ and A₃.

We define the domain $Ω_{s}^{*}$ ⁠, which is an extension of the domain Ω_s in the normal direction, such that $Ω_{s} ⋐ Ω_{s}^{*} ⋐ Ω$ ⁠, see Fig. 1. We cite the following results for use in detailing A₂ and A₃.

Lemma 5.1. Consider the Stokes problem

\begin{matrix} - μ Δ {\tilde{w}}_{1} + \nabla q_{1} = 0, in Ω_{s} \\ {\tilde{w}}_{1} = w_{1} on \partial Ω_{s} and \int_{\partial Ω_{s}} w_{1} \cdot n d τ = 0 \end{matrix}

(79)

Then, for any

w_{1} \in W^{k + 2 - 1 / r, r} (\partial Ω_{s})

, where

k \geq 0

is an integer and

1 < r < \infty

, the Stokes problem Eq. has a unique pair

({\tilde{w}}_{1}, {\tilde{q}}_{1})

with

{\tilde{q}}_{1}

being q₁ different from a constant, that satisfies

\begin{matrix} | | {\tilde{w}}_{1} | |_{k + 2, r; Ω_{s}} + | | {\tilde{q}}_{1} | |_{W^{k + 1, r} (Ω) / ℝ} \\ \leq c (Ω_{s}, k, r) | | w_{1} | |_{k + 2 - 1 / r, r; \partial Ω_{s}} \end{matrix}

(80)

where $c (Ω_{s}, k, r)$ is a constant depending only on Ω_s, k, r.

Proof . See Ref. [43].▪

Lemma 5.2. Suppose

1 \leq r \leq \infty

. Assume that Ω_s is bounded in

ℝ^{d}

and

\partial Ω_{s}

is

C^{1}

. Select a bounded open set

Ω_{s}^{*}

such that

Ω_{s} \subset \subset Ω_{s}^{*}

. Then, there exists a bounded linear operator

\begin{matrix} E : W^{1, r} (Ω_{s}) \to W^{1, r} (ℝ^{d}) \end{matrix}

(81)

such that for each ${\tilde{w}}_{1} \in W^{1, p} (Ω_{s})$ : (i) $E {\tilde{w}}_{1} = {\tilde{w}}_{1}$ a.e. in Ω_s, (ii) $E {\tilde{w}}_{1}$ has support in $Ω_{s}^{*}$ , and iii) $| | {\tilde{w}}_{1} | |_{W^{1, r} (ℝ^{d})} \leq c (r, Ω_{s}, Ω_{s}^{*}) | | {\tilde{w}}_{1} | |_{W^{1, r} (Ω_{s})}$ , where $c (r, Ω_{s}, Ω_{s}^{*})$ is a constant depending on only r, Ω_s, and $Ω_{s}^{*})$ ⁠.

Proof . See Ref. [44, Theorem 1, Sec. 5.4].▪

In what follows, we denote:

\begin{matrix} X_{s} (t, x) = E {\tilde{w}}_{1} (t, x), in Ω_{s}^{*} \end{matrix}

(82)

We will derive a general formula for detailing

A_{k}, k = 2, 3

with

w_{1}

playing the role of either

K_{11} x_{cs}, \partial_{t} (K_{11} x_{cs}) = K_{11} u_{cs}, (K_{12} η_{s}) \times r_{s}

⁠, or

\partial_{t} ((K_{12} η_{s}) \times r_{s}) = (K_{12} R_{s} ω_{s}) \times r_{s} - (K_{12} η_{s}) \times u_{cs}

⁠. This makes sense because on the boundary

\partial Ω_{s}, w_{1}

can take the value of either

K_{11} x_{cs}, K_{11} u_{cs}, (K_{12} η_{s}) \times r_{s}, or (K_{12} R_{s} ω_{s}) \times r_{s} - (K_{12} η_{s}) \times u_{cs}

Denote

\begin{matrix} B = \int_{\partial Ω_{s}} X_{s} \cdot (σ_{f} n) d τ \end{matrix}

(83)

Now, multiplying the fourth equation in Eq. by

X_{s}

and integrating over

Ω_{s}^{*}

yields

\begin{matrix} \int_{Ω_{s}^{*}} X_{s} \cdot \partial_{t} (ρ_{f} u_{f}) d x + \int_{Ω_{s}^{*}} X_{s} \cdot (div (ρ_{f} u_{f} \otimes u_{f})) d x \\ + \int_{Ω_{s}^{*}} X_{s} \cdot div (σ_{f}) d x - κ 〈 H, X_{s} 〉 = \int_{Ω_{s}^{*}} X_{s} \cdot (ρ_{f} f) d x \end{matrix}

(84)

Using integration by parts, the boundary condition

X_{s} = 0

on

\partial Ω_{s}^{*}

⁠, and the transmission condition , and noting that Ω_s is time-dependent, we have

\begin{matrix} \int_{Ω_{s}^{*}} X_{s} \cdot \partial_{t} (ρ_{f} u_{f}) d x = \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s} d x \\ - \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot \partial_{t} X_{s} d x \\ - \int_{\partial Ω_{s}} (ρ_{f} u_{f} \cdot X_{s}) u_{s} \cdot n d τ \\ \int_{Ω_{s}^{*}} X_{s} \cdot div (ρ_{f} u_{f} \otimes u_{f}) d x = \int_{\partial Ω_{s}} (ρ_{f} u_{f} \cdot X_{s}) u_{s} \cdot n d τ \\ - \int_{Ω_{s}^{*}} (ρ_{f} u_{f} \otimes u_{f}) : \nabla X_{s} d x \\ \int_{Ω_{s}^{*}} X_{s} \cdot div (σ_{f}) d x = \int_{\partial Ω_{s}} X_{s} \cdot (σ_{f} n) d τ \\ - \int_{Ω_{s}^{*}} μ_{f} \nabla (u_{f}) \cdot \nabla X_{s} d x \end{matrix}

(85)

Substituting Eq. into Eq. , and using Eq. gives

\begin{matrix} B = B_{1} + B_{2} \end{matrix}

(86)

where

\begin{matrix} B_{1} = \int_{Ω_{s}^{*}} μ_{f} \nabla (u_{f}) \cdot \nabla X_{s} d x - \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s} d x \\ - \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot \partial_{t} X_{s} d x - \int_{Ω_{s}^{*}} (ρ_{f} u_{f} \otimes u_{f}) : \nabla X_{s} d x \\ + \int_{Ω_{s}^{*}} X_{s} \cdot (ρ_{f} f) d x \\ B_{2} = κ {〈 H, X_{s} 〉}^{*} \end{matrix}

(87)

where

{〈 H, X_{s} 〉}^{*}

is

〈 H, X_{s} 〉

⁠, which is restricted to

Γ^{*}

with

Γ^{*}

being the interface between

Ω_{1}

and

Ω_{2}

belonging to

Ω_{s}^{*}

⁠. Using the Hölder inequality, we obtain

\begin{matrix} B_{1} \leq \frac{a_{21}}{4 ε_{21}} | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} + a_{21} ε_{21} | | \nabla X_{s} | |_{Ω_{s}^{*}}^{2} \\ - \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s} d x + \frac{a_{11}}{4 ε_{22}} | | u_{f} | |_{Ω_{s}^{*}}^{2} \\ + a_{11} ε_{22} | | \partial_{t} X_{s} | |_{Ω_{s}^{*}}^{2} + a_{11} c | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} | | \nabla X_{s} | |_{Ω_{s}^{*}} \\ + a_{11} ε_{23} | | X_{s} | |_{Ω_{s}^{*}}^{2} + \frac{a_{11}}{4 ε_{23}} | | f | |_{Ω_{s}^{*}}^{2} \end{matrix}

(88)

where

ε_{2 i}, i = 1, 2, 3

are positive constants to be chosen, and we have used

\begin{matrix} | \int_{Ω_{s}^{*}} (u_{f} \otimes u_{f}) : \nabla X_{s} d x |_{E} \leq c | | u_{f} | |_{L^{4} (Ω_{s}^{*})}^{2} | | \nabla X_{s} | |_{Ω_{s}^{*}} \\ \leq c | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} | | \nabla X_{s} | |_{Ω_{s}^{*}} \end{matrix}

(89)

due to the embedding $V \subset {(L^{6} (Ω_{s}^{*}))}^{3} \subset {(L^{4} (Ω_{s}^{*}))}^{3}$ because $Ω_{s}^{*}$ is a bounded domain, where c is a positive embedding constant.

Next, we calculate the bound of B₂. By definition of

〈 H, X_{s} 〉

⁠, see Eq. , we have

\begin{matrix} B_{2} = κ \int_{Γ^{*} (t)} H n \cdot X_{s} d H^{2} (x) \\ \leq κ \int_{Γ^{*} (t)} | H |_{E} | | X_{s} | |_{E} d H^{2} (x) \\ = κ \sum_{i = 1}^{N} \int_{Γ_{i}^{*} (t)} H_{i} | | X_{i s} | |_{E} d H^{2} (x) \\ = - κ \sum_{i = 1}^{N} \int_{Γ_{i}^{*} (t)} n_{i} \cdot \nabla | | X_{i s} | |_{E} d H^{2} (x) \\ \leq κ H^{2} (Γ) \sup_{x \in Γ^{*} (t)} | | \nabla | | X_{s} | |_{E} | |_{E} \\ \leq κ H^{2} (Γ) (ε_{24} \sup_{x \in Γ^{*}} | | \nabla | | X_{s} | |_{E} | |_{E}^{2} + \frac{1}{ε_{24}}) \end{matrix}

(90)

where we have partitioned

Γ^{*} (t)

to N infinitesimal regions such that in each region

H

does not change its sign (⁠

H_{i}

and

X_{i s}

denote

H

and

X_{s}

in the i region, respectively,

n_{i}

denotes the normal unit vector to

Γ_{i}^{*}

⁠), and ε₂₄ is a positive constant to be chosen. Substituting Eqs. and into Eq. yields

\begin{matrix} B \leq \frac{a_{21}}{4 ε_{21}} | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} + a_{21} ε_{21} | | \nabla X_{s} | |_{Ω_{s}^{*}}^{2} \\ - \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s} d x + \frac{a_{11}}{4 ε_{22}} | | u_{f} | |_{Ω_{s}^{*}}^{2} \\ + a_{11} ε_{22} | | \partial_{t} X_{s} | |_{Ω_{s}^{*}}^{2} + a_{11} c | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} | | \nabla X_{s} | |_{Ω_{s}^{*}} \\ + a_{11} ε_{23} | | X_{s} | |_{Ω_{s}^{*}}^{2} + \frac{a_{11}}{4 ε_{23}} | | f | |_{Ω_{s}^{*}}^{2} \\ + κ H^{2} (Γ) (ε_{24} \sup_{x \in Γ^{*}} | | \nabla | | X_{s} | |_{E} | |_{E}^{2} + \frac{1}{4 ε_{24}}) \end{matrix}

(91)

Now, setting

w_{1} = K_{11} x_{cs}

and

X_{s}^{x_{c s}} = E {\tilde{w}}_{1}

with

\partial_{t} w_{1} = K_{11} u_{cs}

in Eq. yields

\begin{matrix} A_{2} \leq - \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s}^{x_{cs}} d x + \frac{a_{21}}{4 ε_{21}} | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} \\ + c a_{21} ε_{21} λ_{M} (K_{11}) | | x_{cs} | |_{E}^{2} + \frac{a_{11}}{4 ε_{22}} | | u_{f} | |_{Ω_{s}^{*}}^{2} \\ + c a_{11} ε_{22} λ_{M} (K_{11}) | | u_{cs} | |_{E}^{2} \\ + c a_{11} λ_{M} (K_{11}) | | \nabla u_{f} | |^{2} | | x_{cs} | |_{E} \\ + c a_{11} ε_{23} λ_{M} (K_{11}) | | x_{cs} | |_{E}^{2} + \frac{a_{11}}{4 ε_{23}} | | f | |_{Ω_{s}^{*}}^{2} \\ + c ε_{24} H^{2} (Γ) | | x_{c s} | |_{E}^{2} + \frac{κ}{4 ε_{24}} H^{2} (Γ) \end{matrix}

(92)

On the other hand, setting

w_{1} = (K_{12} η_{s}) \times r_{s}

and noting that

\partial_{t} w_{1} = (K_{12} R_{s} ω_{s}) \times r_{s} - (K_{12} η_{s}) \times u_{cs}

in Eq. yields

\begin{matrix} A_{3} \leq - \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s}^{η_{s}} d x + \frac{a_{21}}{4 ε_{21}} | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} \\ + {ca}_{21} ε_{21} λ_{M}^{2} (K_{12}) | Ω_{s}^{*} |_{E}^{2} | | η_{s} | |_{E}^{2} + \frac{a_{11}}{4 ε_{22}} | | u_{f} | |_{Ω_{s}^{*}}^{2} \\ + {ca}_{11} ε_{22} λ_{M}^{2} (K_{12}) | | R_{s} ω_{s} | |_{E}^{2} + {ca}_{11} ε_{22} | | η_{s} \times ω_{s} | |_{E}^{2} \\ + {ca}_{11} λ_{M}^{2} (K_{12}) | Ω_{s}^{*} |_{E}^{2} | | \nabla u_{f} | |^{2} | | η_{s} | |_{E} \\ + {ca}_{11} ε_{23} λ_{M}^{2} (K_{12}) | Ω_{s}^{*} |_{E}^{2} | | η_{s} | |_{E}^{2} + \frac{a_{11}}{4 ε_{23}} | | f | |_{Ω_{s}^{*}}^{2} \\ + c ε_{24} H^{2} (Γ) | | η_{s} | |_{E}^{2} + \frac{κ}{4 ε_{24}} H^{2} (Γ) \end{matrix}

(93)

where $X_{s}^{η_{s}} = E {\tilde{w}}_{1}$ and $| Ω_{s}^{*} |_{E}$ denotes the size of $Ω_{s}^{*}$ ⁠.

5.1.3 Detail of ϖ.

Substituting Eqs. , , and into Eq. gives

\begin{matrix} ϖ \leq - \frac{1}{2} \frac{d}{d t} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x - κ \frac{d}{d t} H^{2} (Γ) \\ - \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s}^{x_{c s}} d x - \frac{d}{d t} \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s}^{η_{s}} d x \\ + ϖ_{1}^{*} + Ξ_{30} \end{matrix}

(94)

where

\begin{matrix} ϖ_{1}^{*} = - \int_{Ω_{f}} 2 μ_{f} | | D (u_{f}) | |_{E}^{2} d x + a_{11} ε_{20} \int_{Ω_{f}} | | u_{f} | |_{E}^{2} d x \\ + \frac{a_{21}}{4 ε_{21}} | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} + {ca}_{21} ε_{21} λ_{M} (K_{11}) | | x_{cs} | |_{E}^{2} \\ + \frac{a_{11}}{4 ε_{22}} | | u_{f} | |_{Ω_{s}^{*}}^{2} + {ca}_{11} ε_{22} λ_{M} (K_{11}) | | u_{cs} | |_{E}^{2} \\ + {ca}_{11} λ_{M} (K_{11}) | | \nabla u_{f} | |^{2} | | x_{cs} | |_{E} + \frac{a_{21}}{4 ε_{21}} | | \nabla u_{f} | |_{Ω_{s}^{*}}^{2} \\ + {ca}_{11} ε_{23} λ_{M} (K_{11}) | | x_{c s} | |_{E}^{2} + c ε_{24} H^{2} (Γ) | | x_{c s} | |_{E}^{2} \\ + {ca}_{21} ε_{21} λ_{M}^{2} (K_{12}) | Ω_{s}^{*} |_{E}^{2} | | η_{s} | |_{E}^{2} + \frac{a_{11}}{4 ε_{22}} | | u_{f} | |_{Ω_{s}^{*}}^{2} \\ + {ca}_{11} ε_{22} λ_{M}^{2} (K_{12}) | | R_{s} ω_{s} | |_{E}^{2} + {ca}_{11} ε_{22} | | η_{s} \times ω_{s} | |_{E}^{2} \\ + {ca}_{11} λ_{M}^{2} (K_{12}) | Ω_{s}^{*} |_{E}^{2} | | \nabla u_{f} | |^{2} | | η_{s} | |_{E} \\ + {ca}_{11} ε_{23} λ_{M}^{2} (K_{12}) | Ω_{s}^{*} |_{E}^{2} | | η_{s} | |_{E}^{2} + c ε_{24} H^{2} (Γ) | | η_{s} | |_{E}^{2} \\ Ξ_{30} = \frac{a_{11}}{4 ε_{20}} \int_{Ω_{f}} | | f | |_{E}^{2} d x + \frac{a_{11}}{4 ε_{23}} | | f | |_{Ω_{s}^{*}}^{2} + \frac{a_{11}}{4 ε_{23}} | | f | |_{Ω_{s}^{*}}^{2} + \frac{κ}{2 ε_{24}} H^{2} (Γ) \end{matrix}

(95)

5.2 Convergence of the Closed-Loop System.

With ϖ detailed by Eq. , we consider the following Lyapunov function candidate for the closed-loop system:

\begin{matrix} U = U_{1} + ε_{0} E + U_{2} \end{matrix}

(96)

where U₁ is given by Eq. ,

E

is given by Eq. , and ε₀ is a positive constant to be chosen, and

\begin{matrix} U_{2} = \frac{1}{2} \int_{Ω_{f}} ρ_{f} | | u_{f} | |_{E}^{2} d x + κ H^{2} (Γ) + \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s}^{x_{c s}} d x \\ + \int_{Ω_{s}^{*}} (ρ_{f} u_{f}) \cdot X_{s}^{η_{s}} d x \end{matrix}

(97)

Using Hölder's inequality, we can find the bound of U₂ as

\begin{matrix} | U_{2} |_{E} \leq ε_{31} | | x_{c s} | |_{E}^{2} + ε_{32} | | η_{s} | |_{E}^{2} + ε_{33} | | u_{f} | |_{Ω_{f}}^{2} + κ H^{2} (Γ) \end{matrix}

(98)

where ε₃₁ and ε₃₂ are positive constants chosen such that

\begin{matrix} ε_{31} < \frac{1}{2} λ_{\min} (K_{21} K_{11}) \\ ε_{32} < 2 λ_{\min} (K_{22} K_{12}) \end{matrix}

(99)

and

\begin{matrix} ε_{33} = \frac{1}{2} a_{11} + \frac{1}{4 ε_{31}} | Ω_{f} |_{E}^{2} a_{11}^{2} λ_{M} (K_{11}) \\ + \frac{1}{4 ε_{32}} | Ω_{f} |_{E}^{4} λ_{M} (K_{12}) a_{11}^{2} \end{matrix}

(100)

Therefore, we can find the bound for U as

\begin{matrix} U_{1} + {\underline{ε}}_{0} E \leq U \leq U_{1} + {\bar{ε}}_{0} E \end{matrix}

(101)

where ${\underline{ε}}_{0}$ and ${\bar{ε}}_{0}$ are positive constants by choosing a sufficiently large ε₀.

Now, differentiating Eq. and using Eqs. , , and yield

\begin{matrix} \frac{d U}{d t} = \frac{d U_{1}}{d t} + ε_{0} \frac{d E}{d t} + \frac{d U_{2}}{d t} \\ \leq - k_{1} U_{1} - ε_{0} \int_{Ω_{f}} 2 (μ_{f} - ε_{12}) | | D (u_{f}) | |_{E}^{2} d x \\ - ε_{0} (λ_{m} (K_{11}) - m_{s} ε_{10}) | | u_{cs} | |_{E}^{2} \\ - ε_{0} (λ_{m} (K_{12}) - ε_{11} λ_{M} (J_{s})) | | ω_{s} | |_{E}^{2} \\ - ε_{24} H^{2} (Γ) + ϖ_{1}^{*} + Ξ_{0} \end{matrix}

(102)

where ε₂₄ is a positive constant, and

\begin{matrix} Ξ_{0} = Ξ_{10} + ε_{0} Ξ_{20} + Ξ_{30} + ε_{24} H^{2} (Γ) \end{matrix}

(103)

Since we already proved the estimates in Eq. , by choosing a sufficiently large ε₀, where we note that

ϖ_{1}^{*}

is given by Eq. , there exist a positive constant

\bar{k}

and a non-negative constant

{\bar{k}}_{0}

(depending on the initial data due to

H^{2} (Γ)

in Eq. , which is bounded because of Eqs. and , such that

\begin{matrix} \frac{d U}{d t} \leq - \bar{k} U + {\bar{k}}_{0} | | f | |_{Ω}^{2} \end{matrix}

(104)

Next, we need the following lemma.

Lemma 5.3. Let

a, g : [0, T] \to ℝ

be given continuous functions. Then the solution of the nonhomogeneous ordinary differential equation

\begin{matrix} \frac{d x}{d t} = a (t) x + g (t) \end{matrix}

(105)

with the initial value

x (0) = x_{0} \in ℝ

, is defined for

t \in [0, T]

and is given by

\begin{matrix} x (t) = e^{\bar{a} (t)} x_{0} + e^{\bar{a} (t)} \int_{0}^{t} e^{- \bar{a} (s)} g (s) d s \end{matrix}

(106)

where $\bar{a} (t) = \int_{0}^{t} a (s) d s$ ⁠.

Proof . A direct substitution of Eq. (106) into Eq. (105) yields the proof.

Applying Lemma 5.3 to Eq. yields

\begin{matrix} U (t) \leq U (0) e^{- \bar{k} t} + e^{- \bar{k} t} \int_{0}^{t} e^{\bar{k} s} {\bar{k}}_{0} | | f (s) | |_{Ω}^{2} d s \\ \leq U (0) e^{- c t} + {\bar{k}}_{0} \sum_{i = 1}^{2} \int_{0}^{t} | | f (s) | |_{Ω}^{2} d s, \end{matrix}

(107)

which means that U(t) (hence

| | u | |_{Ω}, | | x_{cs} | |_{E}, | | η_{s} | |_{E}

⁠) exponentially converges to

{\bar{k}}_{0} \int_{0}^{t} | | f (s) | |_{Ω}^{2} d s

⁠, which is bounded because

f_{i}

satisfy Eq. and f is defined in Eq. . We summarize the main results in the following theorem.▪

Theorem 5.1. Under the initial data Eq. , the controls

F_{k}

, which are obtained from Eqs. and , solves Control Objective 2.1 for all

t \in [0, T]

, where T is such that

Ω_{s} (t) ⋐ Ω

and

η_{s} (t) \in D_{η_{s}}

for all

t \in [0, T]

. In particular, the closed-loop system consisting of Eqs. , , and has at least one weak solution, which is defined in Definition 4.1 for all

t \in [0, T]

. Moreover, the closed-loop system is almost globally practically exponentially stable at the origin, i.e.,

\begin{matrix} ϒ (t) \leq ϒ (0) e^{- c_{1} t} + c_{0} \end{matrix}

(108)

where $ϒ (t) = | | x_{cs} (t) | |_{E}^{2} + | | η_{s} (t) | |_{E}^{2} + | | u_{cs} | |_{E}^{2} + | | ω_{s} | |_{E}^{2} + \int_{Ω_{f}} | | u_{f} (t, x) | |_{E}^{2} d x$ , c₁ is a positive constant, and c₀ is a non-negative constant (depending on the initial data).

6 Simulations

A simulation is carried out to illustrate the effectiveness of the control law given by Eq. (36). For the rigid body, we take the physical shape of a neutrally buoyant cube with dimensions: $\frac{π}{10} m \times \frac{π}{10} m \times \frac{π}{10} m$ and the mass: m_s = 1 kg, which give $J_{s} = diag (0.01645, 0.01645, 0.01645) {kgm}^{2}$ ⁠. For the air with $μ_{1} = 1.81 \times 10^{- 5} kg / ms$ and $ρ_{1} = 1.225 kg / m^{3}$ and water with $μ_{2} = 1.793 \times 10^{- 3} kg / ms$ and $ρ_{2} = 980 kg / m^{3}$ ⁠, the domains $Ω_{1}$ and $Ω_{2}$ are also taken as cubes subtracting the rigid body. The dimensions of $Ω_{1}$ and $Ω_{2}$ are respectively taken as $([- 2 π, 2 π] m \times [- 2 π, 2 π] m \times [0, 2 π] m) ∖ Ω_{1 s}$ ⁠, where $Ω_{1 s}$ is the part of the rigid body submerged in the air, and $([- 2 π, 2 π] m \times [- 2 π, 2 π] m \times [- 2 π, 0] m) ∖ Ω_{2 s}$ ⁠, where $Ω_{2 s}$ is the part of the rigid body submerged in the water. The surface tension constant is $κ = 7.7 \times 10^{- 3} kg / m$ ⁠. Thus, the domain Ω is a cube with dimensions $L_{1} \times L_{2} \times L_{3} = [- 2 π, 2 π] m \times [- 2 π, 2 π] m \times [- 2 π, 2 π] m$ ⁠. We approximate all the sharp corners of Ω and Ω_s by rounding them off to make $\partial Ω$ and $\partial Ω_{f}$ Lipschitz, where recalling that $Ω_{f} = \cup_{i = 1}^{2} Ω_{i}$ ⁠. We assume that the rigid body is neutrally buoyant otherwise we can pre-eliminate the difference between buoyancy and gravity forces before applying Eq. (36). The body forces are taken as $f_{1} = 10^{- 2} \frac{1}{1 + t} {[sin (2 t), cos (1.5 t), sin (t)]}^{T} kg / m^{2} s^{2}$ and $f_{2} = 10^{- 1} e^{- 0.1 t} {[sin (1.5 t), cos (2 t), sin (1.5 t)]}^{T} kg / m^{2} s^{2}$ ⁠, which satisfy the condition (2).

We will use the semi-Galerkin method to the penalized system Eq. to obtain a numerical weak solution, where we approximate

\begin{matrix} u_{n} (t, x) = \sum_{l = 1}^{n} c_{l}^{n} (t) a_{l} (x) \end{matrix}

(109)

where $c_{l}^{n} (t)$ are scalar functions of time, $a_{l} (x)$ are eigenfunctions of the Stokes operator. We substitute Eq. (109) into the first equation of Eq. (63) and multiply it by $ξ = Spann {a_{l} (x); l = 1, \dots, n}$ to obtain a system of ODEs for $c_{l}^{n} (t)$ ⁠, which is numerically solvable. The transport equations (the third, fourth, and fifth equations of Eq. (63)) are solved by using the characteristic method. We now need to derive eigenfunctions for our domain Ω. To do so, we need the following lemma [45, Theorem III.2.3].

Lemma 6.1. If Ω is a bounded open set in $ℝ^{3}$ with Lipschitz boundary, then H coincides with the space of divergence free functions in $L^{2} (Ω)$ such that $u \cdot n = 0$ on $\partial Ω$ , wherenis the normal unit vector to $\partial Ω$ ⁠.

With this lemma, eigenfunctions of the Stokes problem are equivalent to those of the Laplace operator with the condition

u \cdot n = 0

on

\partial Ω

as we consider a weak solution in H. Hence, we look for

a_{l}

such that

\begin{matrix} 1) Δ a_{l} = - λ_{l} a_{l}, div (a_{l}) = 0 in Ω; a_{l} \cdot n = 0 on \partial Ω \\ 2) a_{l} is an orthonormal basis of H (Ω) \\ 3) a_{l} is an orthogonal basis of V (Ω) \\ 4) 0 < λ_{1} \leq λ_{2} \leq \dots and λ_{l} \to \infty as l \to \infty \end{matrix}

(110)

A nontrivial calculation gives

a_{l}

(we neglect the round off corners) as follows:

\begin{matrix} a_{l} = {\bar{a}}_{l} [\begin{matrix} (l_{2} + l_{3}) sin (l_{1} x_{1}) cos (l_{2} x_{2}) cos (l_{3} x_{3}) \\ (l_{3} - l_{1}) cos (l_{1} x_{1}) sin (l_{2} x_{2}) cos (l_{3} x_{3}) \\ - (l_{1} + l_{2}) cos (l_{1} x_{1}) cos (l_{2} x_{2}) sin (l_{3} x_{3}) \end{matrix}] \end{matrix}

(111)

where $(l_{1}, l_{2}, l_{3}) \in Z^{3} ∖ {0}$ such that $l^{2} = l_{1}^{2} + l_{2}^{2} + l_{3}^{2}$ ⁠, which are taken into account to have summing combination in calculating Eq. (109), and ${\bar{a}}_{l} = \frac{1}{\sqrt{2 π^{3} A}}$ with $A = l_{1}^{2} + l_{2}^{2} + l_{3}^{2} + l_{1} l_{2} - l_{1} l_{3} + l_{2} l_{3}$ ⁠. It is easy to show that A > 0 for all $(l_{1}, l_{2}, l_{3}) \in Z^{3} ∖ {0}$ ⁠, and that $a_{l}$ satisfies all the properties listed in Eq. (110) with $λ_{l} = l_{1}^{2} + l_{2}^{2} + l_{3}^{2}$ ⁠.

For the initial values of the fluid velocity, we take $c_{l}^{n} (0)$ to be random number in $\frac{1}{n^{2}} [- 1, 1]$ if $x \in Ω_{1} (0) \cup Ω_{1 s} (0)$ and in $\frac{1}{n^{2}} [- 2, 2]$ if $x \in Ω_{2} (0) \cup Ω_{2 s} (0)$ ⁠. The initial value of the interface between the air and water is neutral, i.e., $Ω_{1} (0) = ([- 2 π, 2 π] m \times [- 2 π, 2 π] m \times [0, 2 π] m) ∖ Ω_{1 s} (0)$ and $Ω_{2} (0) = ([- 2 π, 2 π] m \times [- 2 π, 2 π] m \times [- 2 π, 0] m) ∖ Ω_{2 s} (0)$ ⁠. This results in $μ (0) = μ_{1}$ if $x \in Ω_{1} (0), μ (0) = μ_{2}$ if $x \in Ω_{2} (0), μ (0) = 0$ if $x \in Ω_{s} (0)$ ⁠; and $ρ (0) = ρ_{1}$ if $x \in Ω_{1} (0), ρ (0) = ρ_{2}$ if $x \in Ω_{2} (0), ρ (0) = ρ_{s}$ if $x \in Ω_{s} (0)$ ⁠. The initial values of the rigid body are taken as $x_{c s} (0) = {[0.5, - 0.5, - 0.2]}^{T}$ m, $η_{s} (0) = {[1.6, 0.4, 2.5]}^{T}$ ⁠, which yields a principal axis/angle pair $e = {[0.4782, 0.2050, 0.8540]}^{T}$ and $γ = 4.9665$ rad. The initial values of the velocities $u_{c s} (0)$ and $ω_{s} (0)$ of the rigid body are determined via Eqs. (7) and (61), where $u (0, x)$ is substituted by $u_{δ}^{n} (0, x)$ ⁠. We perform open-loop and closed-loop simulations for comparison.

We first simulate the open-loop system, i.e., we set the controls $F_{s} = 0$ and $M_{s} = 0$ ⁠. The position vector $x_{c s} = {[x_{1 c s}, x_{2 c s}, x_{3 c s}]}^{T}$ ⁠, orientation vector $η_{s} = {[η_{1 s}, η_{2 s}, η_{3 s}]}^{T}$ ⁠, linear velocity vector $u_{c s} = {[u_{1 c s}, u_{2 c s}, u_{3 c s}]}^{T}$ ⁠, and angular velocity vector $ω_{s} = {[ω_{1 s}, ω_{2 s}, ω_{3 s}]}^{T}$ are plotted in Fig. 2. The H-norm of the global velocity $| | u | |_{Ω}$ ⁠, the control force vector $F_{s} = {[F_{1 s}, F_{2 s}, F_{3 s}]}^{T}$ ⁠, and control moment vector $M_{s} = {[M_{1 s}, M_{2 s}, M_{3 s}]}^{T}$ are plotted in Fig. 3. It is seen from these figures that $x_{1 c s}, x_{2 c s}, η_{3 s}$ tend to drift away from the origin due to forces and moments from the fluids; $x_{3 c s}, η_{1 s}, η_{2 s}$ tend to the origin due to (restoring) forces and moments from the fluids; $u_{c s}, | | u | |_{Ω}$ ⁠, and $ω_{s}$ decay slowly to zero (decay is due to resistance of the fluids); and indeed the controls $F_{s}$ and $M_{s}$ are zero.

Fig. 2

View large Download slide

Open-loop states: $x_{cs}, η_{s}, u_{cs}, ω_{s}$

Fig. 3

Open-loop velocity norm:||u||Ω=(∫Ω||u||E2dx)12 and open-loop controls: Fs and Ms

View large Download slide

Open-loop velocity norm: $| | u | |_{Ω} = {(\int_{Ω} | | u | |_{E}^{2} d x)}^{\frac{1}{2}}$ and open-loop controls: $F_{s}$ and $M_{s}$

In the closed-loop simulations, we choose the control gains as follows: $K_{11} = 5 I_{3}, K_{12} = 5 I_{3}, K_{21} = 15 I_{3}, K_{22} = 15 I_{3}$ ⁠. It can be checked that the conditions in Eq. (38) hold. The position vector $x_{c s} = {[x_{1 c s}, x_{2 c s}, x_{3 c s}]}^{T}$ ⁠, orientation vector $η_{s} = {[η_{1 s}, η_{2 s}, η_{3 s}]}^{T}$ ⁠, linear velocity vector $u_{c s} = {[u_{1 c s}, u_{2 c s}, u_{3 c s}]}^{T}$ ⁠, and angular velocity vector $ω_{s} = {[ω_{1 s}, ω_{2 s}, ω_{3 s}]}^{T}$ are plotted in Fig. 4. The H-norm of the global velocity $| | u | |_{Ω}$ ⁠, the control force vector $F_{s} = {[F_{1 s}, F_{2 s}, F_{3 s}]}^{T}$ ⁠, and control moment vector $M_{s} = {[M_{1 s}, M_{2 s}, M_{3 s}]}^{T}$ are plotted in Fig. 5. It is seen from these figures that all the states $x_{c s}, η_{s}, u_{c s}, ω_{s}$ ⁠, and the velocity norm $| | u | |_{Ω} = {(\int_{Ω} | | u | |_{E}^{2} d x)}^{\frac{1}{2}}$ ⁠; and the controls $F_{s}$ and $M_{s}$ converge to the origin (note that the body forces $f_{1}$ and $f_{2}$ decay to zero). In comparison with the open-loop simulation results, see Figs. 2 and 3, the effectiveness of the controls is clearly illustrated. Note that the overshoot in $| | u | |_{Ω}$ is higher for the closed-loop due to the control action.

Fig. 4

View large Download slide

Closed-loop states: $x_{cs}, η_{s}, u_{cs}, ω_{s}$

Fig. 5

Closed-loop velocity norm:||u||Ω=(∫Ω||u||E2dx)12 and closed-loop controls: Fs and Ms

View large Download slide

Closed-loop velocity norm: $| | u | |_{Ω} = {(\int_{Ω} | | u | |_{E}^{2} d x)}^{\frac{1}{2}}$ and closed-loop controls: $F_{s}$ and $M_{s}$

7 Conclusions

Almost global practical exponential stabilization problem of a rigid body between two fluids with surface tension was posed and solved. The main ingredients include (i) formulation of two fluids as a single fluid; (ii) design of an appropriate feedback stabilization control law; (iii) formulation of the two fluids and rigid body as a global fluid; (iv) derivation of work and power of the two fluids, (v) derivation of a Lyapunov function, including stabilization errors, energy of the two fluids and surface tension. Future work is to consider an unbounded domain.

Funding Data

Australian Research Council (Grant No. DP230100390; Funder ID: 10.13039/501100000923).

Data Availability Statement

The authors attest that all data for this study are included in the paper.

References

1.

Fossen

,

T. I.

,

2002

,

Marine Control Systems

,

Marine Cybernetics

,

Trondheim, Norway

.

2.

Do

,

K. D.

, and

Pan

,

J.

,

2009

,

Control of Ships and Underwater Vehicles

,

Springer

, Berlin, Germany.

Google Scholar

Crossref

3.

Falcao

,

A.

,

2010

, “

Wave Energy Utilization: A Review of the Technologies

,”

Renewable Sustainable Energy Rev.

,

14

(

3

), pp.

899

–

918

.10.1016/j.rser.2009.11.003

Google Scholar

Crossref

4.

Do

,

K. D.

,

2022

, “

Nonlinear Control With Wave Observer to Maximize Harvested Power for Point Absorber Wave Energy Converters

,”

Asian J. Control

,

24

(

1

), pp.

16

–

45

.10.1002/asjc.2461

Google Scholar

Crossref

5.

Lackner

,

M. A.

, and

Rotea

,

M. A.

,

2011

, “

Structural Control of Floating Wind Turbines

,”

Mechatronics

,

21

(

4

), pp.

704

–

719

.10.1016/j.mechatronics.2010.11.007

Google Scholar

Crossref

6.

Bai

,

Y.

, and

Jin

,

W. L.

,

2015

,

Marine Structural Design

, 2nd ed.,

Elsevier

, Amsterdam, The Netherlands.

7.

Paidoussis

,

M. P.

,

1998

,

Fluid-Structure Interactions

, Vol.

1

,

Academic Press

,

San Diego, CA

.

8.

Faltinsen

,

O.

,

1993

,

Sea Loads on Ships and Offshore Structures

,

Cambridge University Press

, Cambridge, UK.

9.

Koo

,

W. C.

, and

Kim

,

M. H.

,

2007

, “

Fully Nonlinear Wave–Body Interactions With Surfacepiercing Bodies

,”

Ocean Eng.

,

34

(

7

), pp.

1000

–

1012

.10.1016/j.oceaneng.2006.04.009

Google Scholar

Crossref

10.

Koo

,

W. C.

, and

Kim

,

M. H.

,

2007

, “

Current Effects on Nonlinear Wave–Body Interactions by a 2D Fully Nonlinear Numerical Wave Tank

,”

J. Waterway, Port, Coastal, Ocean Eng.

,

133

(

2

), pp.

136

–

146

.10.1061/(ASCE)0733-950X(2007)133:2(136)

Google Scholar

Crossref

11.

Li

,

Y.

, and

Lin

,

M.

,

2010

, “

Wave–Body Interactions for a Surface-Piercing Body in Water of Finite Depth

,”

J. Hydrodyn., Ser. B

,

22

(

6

), pp.

745

–

752

.10.1016/S1001-6058(09)60112-8

Google Scholar

Crossref

12.

Li

,

Y.

, and

Lin

,

M.

,

2012

, “

Regular and Irregular Wave Impacts on Floating Body

,”

Ocean Eng.

,

42

, pp.

93

–

101

.10.1016/j.oceaneng.2012.01.019

Google Scholar

Crossref

13.

Desjardins

,

B.

, and

Esteban

,

M. J.

,

1999

, “

Existence of Weak Solutions for the Motion of Rigid Bodies in a Viscous Fluid

,”

Arch. Ration. Mech. Anal.

,

146

(

1

), pp.

59

–

71

.10.1007/s002050050136

Google Scholar

Crossref

14.

Feireisl

,

E.

,

2003

, “

On the Motion of Rigid Bodies in a Viscous Incompressible Fluid

,”

J. Evol. Equations

,

3

(

3

), pp.

419

–

441

.10.1007/s00028-003-0110-1

Google Scholar

Crossref

15.

Desjardins

,

B.

,

Esteban

,

M. J.

,

Grandmont

,

C.

, and

Tallec

,

P. L.

,

2001

, “

Weak Solutions for a Fluid-Structure Interaction Model

,”

Rev. Mat. Complut.

,

14

, pp.

523

–

538

.10.5209/rev_REMA.2001.v14.n2.17030

Google Scholar

Crossref

16.

Desjardins

,

B.

, and

Esteban

,

M. J.

,

1999

, “

On Weak Solutions for Fluid-Rigid Structure Interaction: Compressible and Incompressible Models

,”

Commun. Partial Differ. Equations

,

25

(

7–8

), pp.

263

–

285

.10.1080/03605300008821553

17.

Conca

,

C.

,

Martin

,

J. H. S.

, and

Tucsnak

,

M.

,

2000

, “

Existence of Solutions for the Equations Modelling the Motion of a Rigid Body in a Viscous Fluid

,”

Commun. Partial Differ. Equations

,

25

(

5–6

), pp.

99

–

110

.10.1080/03605300008821540

18.

Grandmont

,

C.

, and

Maday

,

Y.

,

2000

, “

Existence for an Unsteady Fluid-Structure Interaction Problem

,”

ESAIM: Math. Modell. Numer. Anal.

,

34

(

3

), pp.

609

–

636

.10.1051/m2an:2000159

Google Scholar

Crossref

19.

Gunzburger

,

M. D.

,

Lee

,

H. C.

, and

Seregin

,

G. A.

,

2000

, “

Global Existence of Weak Solutions for Viscous Incompressible Flows Around a Moving Rigid Body in Three Dimensions

,”

J. Math. Fluid Mech.

,

2

(

3

), pp.

219

–

266

.10.1007/PL00000954

Google Scholar

Crossref

20.

Takahashi

,

T.

,

2003

, “

Existence of Strong Solutions for the Equations Modelling the Rigid Motion of a Rigid-Fluid System in a Bounded Domain

,”

Adv. Differ. Equations

,

8

(

12

), pp.

1499

–

1532

.

21.

Takahashi

,

T.

, and

Tucsnak

,

M.

,

2004

, “

Global Strong Solutions for the Two Dimensional Motion of a Rigid Body in a Viscous Fluid

,”

J. Math. Fluid Mech.

,

6

(

1

), pp.

53

–

77

.10.1007/s00021-003-0083-4

Google Scholar

Crossref

22.

Do

,

K. D.

,

2023

, “

Stabilization of a Rigid Body in a Viscous Incompressible Fluid

,”

Ocean Eng.

,

269

, p.

113481

.10.1016/j.oceaneng.2022.113481

Google Scholar

Crossref

23.

Do

,

K. D.

,

2023

, “

Almost Global Practical Exponential Stabilisation of a Floating Rigid Body

,”

Int. J. Control

,

96

(

7

), pp.

1817

–

1833

.10.1080/00207179.2022.2070777

Google Scholar

Crossref

24.

Temam

,

R.

,

2001

,

Navier-Stokes Equations: Theory and Numerical Analysis

,

AMS Chelsea Publishing

,

New York

.

25.

Robinson

,

J. C.

,

Rodrigo

,

J. L.

, and

Sadowski

,

W.

,

2016

,

The Three-Dimensional Navier–Stokes Equations

,

Cambridge University Press

,

Cambridge, UK

.

Google Scholar

Crossref

26.

Fefferman

,

C. L.

,

2000

,

The Millennium Prize Problems: Existence and Smoothness of the Navier-Stokes Equation

,

Clay Mathematics Institute

,

Cambridge, MA

, pp.

57

–

67

.

27.

Takahashi

,

T.

,

Tucsnak

,

M.

, and

Weiss

,

G.

,

2015

, “

Stabilization of a Fluid-Rigid Body System

,”

J. Differ. Equations

,

259

(

11

), pp.

6459

–

6493

.10.1016/j.jde.2015.07.024

Google Scholar

Crossref

28.

Badra

,

M.

,

Takahashi

,

T.

, and

CNRS/Univ Pau & Pays Adour, Laboratoire de Mathématiques et, de leurs Applications de Pau -Fédération IPRA, UMR5142, 64000, Pau, France

,

2017

, “

Feedback Boundary Stabilization of 2D Fluid-Structure Interaction Systems

,”

Discrete Contin. Dyn. Syst.

,

37

(

5

), pp.

2315

–

2373

.10.3934/dcds.2017102

Google Scholar

Crossref

29.

Badra

,

M.

, and

Takahashi

,

T.

,

2014

, “

Feedback Stabilization of a Fluid-Rigid Body Interaction System

,”

Adv. Differ. Equations

,

19

, pp.

1137

–

1184

.10.57262/ade/1408367290

30.

Badra

,

M.

, and

Takahashi

,

T.

,

2014

, “

Feedback Stabilization of a Simplified 1D Fluid-Particle System

,”

Annales de l'Institut Henri Poincaré, Analyse Non Linéaire

, Vol.

31

, No.

2

, Elsevier, Amsterdam, The Netherlands, pp.

369

–

389

.

Google Scholar

Crossref

31.

Do

,

K. D.

,

2019

, “

Forwarding Control Design for Strict-Feedforward Systems

,”

ASME J. Dyn. Syst., Meas., Control

,

141

(

8

), p.

081010

.10.1115/1.4042881

Google Scholar

Crossref

32.

Bost

,

C.

,

Cottet

,

G. H.

, and

Maitre

,

E.

,

2010

, “

Convergence Analysis of a Penalization Method for the Three-Dimensional Motion of a Rigid Body in an Incompressible Viscous Fluid

,”

SIAM J. Numer. Anal.

,

48

(

4

), pp.

1313

–

1337

.10.1137/090767856

Google Scholar

Crossref

33.

Krstic

,

M.

,

Kanellakopoulos

,

I.

, and

Kokotovic

,

P.

,

1995

,

Nonlinear and Adaptive Control Design

,

Wiley

,

New York

.

34.

Do

,

K. D.

,

2019

, “

Inverse Optimal Gain Assignment Control of Evolution Systems and Its Application to Boundary Control of Marine Risers

,”

Automatica

,

106

, pp.

242

–

256

.10.1016/j.automatica.2019.05.020

Google Scholar

Crossref

35.

Adams

,

R. A.

, and

Fournier

,

J. J. F.

,

2003

,

Sobolev Spaces

, 2nd ed.,

Academic Press

,

Oxford, UK

.

36.

Tsiotras

,

P.

,

1996

, “

Stabilization and Optimality Results for the Attitude Control Problem

,”

J. Guid., Control, Dyn.

,

19

(

4

), pp.

772

–

779

.10.2514/3.21698

Google Scholar

Crossref

37.

San Martin

,

J. A.

,

Starovoitov

,

V.

, and

Tucsnak

,

M.

,

2002

, “

Global Weak Solutions for the Two Dimensional Motion of Several Rigid Bodies in an Incompressible Viscous Fluid

,”

Arch. Ration. Mech. Anal.

,

161

(

2

), pp.

113

–

147

.10.1007/s002050100172

38.

Nouri

,

A.

, and

Poupaud

,

F.

,

1995

, “

An Existence Theorem for the Multifluid Navier-Stokes Problem

,”

J. Differ. Equations

,

122

(

1

), pp.

71

–

88

.10.1006/jdeq.1995.1139

Google Scholar

Crossref

39.

Abels

,

H.

,

2007

, “

On Generalized Solutions of Two-Phase Flows for Viscous Incompressible Fluids

,”

Interfaces Free Boundaries

,

9

(

1

), pp.

31

–

65

.10.4171/ifb/155

Google Scholar

Crossref

40.

Fischer

,

J.

, and

Hensel

,

S.

,

2020

, “

Weak–Strong Uniqueness for the Navier–Stokes Equation for Two Fluids With Surface Tension

,”

Arch. Ration. Mech. Anal.

,

236

(

2

), pp.

967

–

1087

.10.1007/s00205-019-01486-2

Google Scholar

Crossref

41.

Ambrosio

,

L.

,

Fusco

,

N.

, and

Pallara

,

D.

,

2000

,

Functions of Bounded Variation and Free Discontinuity Problems

,

Clarendon Press

,

Oxford, UK

.

Google Scholar

Crossref

42.

Gérard-Varet

,

D.

, and

Hillairet

,

M.

,

2014

, “

Existence of Weak Solutions Up to Collision for Viscous Fluid-Solid Systems With Slip

,”

Commun. Pure Appl. Math.

,

67

(

12

), pp.

2022

–

2076

.10.1002/cpa.21523

Google Scholar

Crossref

43.

Bello

,

J. A.

,

1996

, “

L^r Regularity for the Stokes and Navier-Stokes Problems

,”

Annali di Matematica Pura ed Applicata

, Vol. CLXX, Springer, Berlin, pp.

187

–

206

.10.1007/BF01758988

44.

Evans

,

L.

,

2000

,

Partial Differential Equations

,

American Mathematical Society

,

Providence, RI

.

45.

Galdi

,

G. P.

,

1989

,

An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems

,

Springer

,

New York

.

Stabilization of a Rigid Body in Two Fluids With Surface Tension

Abstract

1 Introduction

1.1 Notation of Functional Spaces.

2 Problem Statement

3 Control Design

3.1 Step 1.

3.2 Step 2.

4 Existence of a Weak Solution of the Closed-Loop System

4.1 Formulation of a Weak Solution.

4.2 Penalized System.

4.3 Existence of a Weak Solution.

5 Stability and Convergence of the Closed-Loop System

5.1 The Term ϖ.

5.1.1 Detail of A₁.

5.1.2 Detail of A₂ and A₃.

5.1.3 Detail of ϖ.

5.2 Convergence of the Closed-Loop System.

6 Simulations

7 Conclusions

Funding Data

Data Availability Statement

References

Get Email Alerts

Cited By

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

Stabilization of a Rigid Body in Two Fluids With Surface Tension

Abstract

1 Introduction

1.1 Notation of Functional Spaces.

2 Problem Statement

3 Control Design

3.1 Step 1.

3.2 Step 2.

4 Existence of a Weak Solution of the Closed-Loop System

4.1 Formulation of a Weak Solution.

4.2 Penalized System.

4.3 Existence of a Weak Solution.

5 Stability and Convergence of the Closed-Loop System

5.1 The Term ϖ.

5.1.1 Detail of A1.

5.1.2 Detail of A2 and A3.

5.1.3 Detail of ϖ.

5.2 Convergence of the Closed-Loop System.

6 Simulations

7 Conclusions

Funding Data

Data Availability Statement

References

Get Email Alerts

Cited By

Related Articles

Related Proceedings Papers

Related Chapters

ASME Journals

ASME Conference Proceedings

ASME eBooks

Resources

Opportunities

This Feature Is Available To Subscribers Only

5.1.1 Detail of A₁.

5.1.2 Detail of A₂ and A₃.