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. [923]).

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. [1321], 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 H01(Ω), see “Notation of functional spaces” below for a definition of space H01(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 [2830] on stabilization of a rigid ball in one-dimensional, two-dimensional, and three-dimensional with the initial value of the fluid velocity in space H01(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 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 H01, 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 κH,φf 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 B2=κΓ*(t)Hn·XsdH2(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 3, and a time constant T >0. Lp(Ω), where 1p<, denotes the standard Lebesgue space of measurable p-integrable functions; L(Ω) denotes the space of essentially bounded functions; H1(Ω) is the usual Sobolev space of order 1, see Ref. [35]; H01(Ω) denotes H1(Ω) with compact support; (a,b)Ω=Ωa·bdx with a·b=i(aibi) and ||a||Ω2=(a,a)Ω; Lp(0,T;X), where 1p< and X is a Banach space with the norm denoted by ||·||X, denotes a Bochner space with the norm ||u||Lp(0,T;X)=(0T||u||Xpdt)1/p. Let Y=X be the dual space of a Banach space X. The space Lω(Ω;Y) is defined as all weak-* measurable functions ν:ΩY, i.e., x(νx,F(x,·))=(νx,F(x,·))X,X, where νx,xΩ denotes a family of probability measures, is measurable for each FL1(Ω;X), such that ||ν||LΩ(Ω;Y):=esssupxΩ||νx||Y<. The space of k-times differentiable functions on a set Ω is denoted by Ck(Ω). We also use ||·||E to denote the Euclidean (point-wise) norm, i.e., ||a||E=a·a. For a scalar, we use |·|E to denote the absolute value.

2 Problem Statement

Let Ω3 be a C1 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 Ω1(t) and Ω2(t) at time t0, see Fig. 1. These fluid are separated by the interface Γ(t) and surround a rigid body represented by Ωs(t), which is a bounded open connected subdomain of Ω. We have Ω=Ω1Ω2Ωs. The domain Ωs*(t) will be defined and used in Sec. 5. The time-dependence of the domains Ωi,i=1,2 and Ωs is due to motion of the rigid body. We assume that Ωs(0)Ω. We will drop the argument t of Ωi,i=1,2, Ωs, and Γ when there is no confusion.

Fig. 1
Domain definition
For the fluids, we denote by ρi>0,μi>0,ui,pi,fi 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
(1)
where Qi=(0,T)×Ωi with the time constant T<, and D(Qi) denotes the space of distribution on Qi. We assume that
(2)
Denoting by Ωs and Ωi the boundary of Ωs and Ωi, respectively, we impose homogeneous Dirichlet boundary conditions on the boundaries
(3)
At the interface Γ between Ω1 and Ω2, 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
(4)
where V and H denote the normal velocity and mean curvature, respectively, of Γ taken with respect to the exterior normal n of Ω2, κ is the surface tension constant, σi is the fluid stress tensor of the fluid i given by
(5)
with D(ui)=12(ui+(ui)T) being the rate tensor of the fluid, T denotes the transpose of (this should not be confused with the time constant T), and I3 being the 3 × 3 identity matrix. On the interface Γ, the tangential divergence of a function ϕC1(Γ) is defined as
(6)

where Pτ is the orthogonal projection onto the tangential space, which is defined point-wisely on Γ, the symbol ⊗ denotes the outer product operator, i.e., ab=abT with bT being the transpose of b for a,b3, and A:B=Tr(AB) for 3 × 3 symmetric matrices A, B with Tr being the trace of matrices. The mean curvature is defined as H=divΓ(ϕ).

Remark 2.1. Due to the surface tension term κHn 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 κHn 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 ms, the density ρs>0, the vector of the center of gravity xcs(t) (measured with respect to the origin, which coincides with the center of Γ(0)) and its velocity vector ucs(t), the modified Rodrigues parameter vector ηs(t) representing the rigid body orientation, see Ref. [36], (this vector is related to the principal axis e and the principal angle γ through ηs(t)=etan(γ4), which is well-defined for all eigenaxis rotations in the range [0,2π), and should not be confusing with almost global stability in this paper), angular velocity vector ωs(t), the inertia matrix Js, the transformation matrix Rs, and the velocity vector field us(t,x) by
(7)
where rs=xxcs, the symbol × denotes the cross vector product operator, I3 denotes the 3 × 3 identity matrix, |Ωs|E denotes the volume of Ωs, G(ηs) denotes the 3 × 3 skew-symmetric matrix of ηs, it holds that
(8)
for all ηs,ωs3, 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
(9)
where Ωis is the part of Ωs submerged in Ωi, and hence Ωs=i=12Ωis, and Ωis is the interface between Ωi and Ωis; ni is the normal unit vector pointing outside of Ωis; Fk is the control force. The control force vector Fs and the control moment vector Ms are expressed in terms of actuator forces as follows:
(10)

where rFk=xFkxcs with xFk being the position vector of actuator forces Fk such that Fs3 and Ms3 (i.e., the rigid body is fully actuated).

At the interface between Ωis and Ωi, the transmission condition is expressed as continuity of the velocity
(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., σini=σsni on Ωis with σs being the Cauchy stress tensor, i.e., σsni 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)×Ω, and introduce the following function spaces:
(12)

where Ωf=i=12Ωi. Note that the elements of K(t) are given by the rigid body velocity in Ωs. 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
(13)

In this paper, we address the following control objective.

Control Objective 2.1. Under the initial data
(14)

whereΩ¯ΩandDηs=(,)3, design the control force vectorFsand control moment vectorMsto 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 xcs(0)Ω¯ was imposed to be compatible with Ωs(0)Ω, and ηs(0)Dηs was made due to the domain of the modified Rodrigues parameter vector ηs.

In the rest of this section, we write Eqs. (1)(4) as single NSEs for a global fluid, and rewrite Eq. (9) accordingly. We define an indicator function and global variables
(15)
We will drop the arguments (t,x) of all the above variables when there is no confusion. Clearly, we have
(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 χf, ρf, and μf based on [38] as
(17)

where Qf=[0,T]×Ωf. Here, χf is a finite Radon measure and ||χf||M(Ωf)=H2(Γ(t)), see Ref. [39]. There are no boundary conditions for χf, ρf, and μf. 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 Hχf 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 L2(0,T;H1(Ω)).

The rigid body dynamics Eq. (9) are now rewritten as
(18)
where ns is the normal unit vector pointing outside of Ωs, and the transmission condition (11) is rewritten as
(19)

3 Control Design

We extend the backstepping method in Ref. [33] and the forwarding method [31] to design the controls Fs and Ms 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
(20)
where αucs and αωs are virtual controls of ucs and ωs, respectively. At this step, we consider the following Lyapunov function candidate to design αucs and αωs:
(21)
Differentiating Eq. (21) and using Eqs. (18) and (20) yields
(22)
From Eq. (22), we design the virtual controls αucs and αωs as
(23)
where K11 and K12 are positive definite matrices. Substituting Eq. (23) into Eq. (22) and using Eq. (8) gives
(24)
Multiplying the first equation of Eq. (20) by m and the second equation of Eq. (20) by Js, then differentiating the results and using Eqs. (23) and (18) gives
(25)

3.2 Step 2.

To design the actual controls Fs and Ms, we consider the following Lyapunov function candidate:
(26)
Differentiating Eq. (26) and using Eqs. (25) and (24) gives
(27)
where
(28)
(29)
and
(30)
with
(31)

ε01 and ε02 being positive constants to be chosen.

Using Eqs. (7) and (8), and Young's inequality, we can bound terms in ϖ0 as follows:
(32)
where ε0i,i=5,,7 are positive constants to be chosen, and λM() denotes the maximum eigenvalue of . Using Eqs. (32) and (30), we can bound the sum ϖ0+Ξ as follows:
(33)
where
(34)
Substituting Eq. (33) into Eq. (27) gives
(35)
From Eq. (35), we design Fs and Ms as follows:
(36)

where K21 and K22 are positive definite matrices to be chosen. Note that the nonlinear term (1+||ηs||E2) is due to the nonlinear terms Rsωs and ωs×(Jsωs) in the rigid body dynamics Eq. (18). From Eqs. (36) and (10), we can solve for the actuator forces Fk. The controls Eq. (36) do not cancel any system functions. This is important for robustness.

Substituting Eq. (36) into Eq. (35) gives
(37)
where we have chosen the control gain matrices (K11,K21,K22), and positive constants ε0i,i=1,,7 such that
(38)

It can be easily shown that there always exist sufficiently large control gain matrices K11,K12,K21, and K22 such that all the inequalities in Eq. (38) hold for given positive constants ε0i,i=1,,7.

From Eqs. (21) and (26), we can write Eq. (37) as follows:
(39)
where
(40)
Remark 3.1. Let us define the state vector Yse=col(xcs,ucse,ηs,ωse). Then, we can write Eq. (36) as
(41)

where W is a positive definite matrix. This form means that the controls Fs and Ms 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 Ωsσfnsdτ and the fluid moment Ωsrs×(σfns)dτ, 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 Fk 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 Fk 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. (17) by a test function φIT and integrating over Ωf, and then integrating by parts gives
(42)
where
(43)

with H2 being the two-dimensional Hausdorff measure.

Next, we derive from Eq. (18) with controls Fk obtained from Eqs. (10) and (36) for φIT that
(44)
where δFk denotes the diagonal matrix of Dirac delta function of xxFk with xFk being the position of actuator forces Fk, i.e., xFk=xcs+rFk. In derivation of Eq. (44), we have used φs=vφs+ωφs×rs and
(45)

due to ωφs·(rs×δFkFk)=(ωφs×rs)·δFkFk.

Adding Eqs. (42) and (44) yields
(46)
Equation (46) is a global weak formulation of the momentum equations in Eqs. (17) and (18) with actuator forces Fk obtained from Eqs. (10) and (36), where the surface tension is taken into account. This goes together with the conservation of mass, which represents the transport of Ωs by the rigid vector field us, i.e., the indicator function
(47)
satisfies
(48)
Now, it is seen that Eqs. (46) and (48) involve with the fields uf,us,φf, and φs defined over Ω such that
(49)
Setting φf=uf and φs=us in Eq. (46) yields
(50)
where we have used [39]
(51)
Hence, the distributional gradient χf is a finite Radon measure and ||χf||M(Ωf)=H2(Γ), where M(Ωf) denotes the space of finite Randon measures.
We consider the “energy” E defined by
(52)
Differentiating Eq. (52) along the solutions of Eq. (52) and using Eqs. (8) and (36) and the Hölder inequality, we obtain formally
(53)
Using Ωsρs||us||E2dx=ms||ucs||E2+ωsTJsωs and the Poincaré inequality, we have
(54)
where
(55)
cp is the Poincaré constant, and ε1i,i=0,1,2 are positive constants chosen such that
(56)
Then, we have from Eq. (54) that
(57)
Integrating Eq. (57) from 0 to t gives
(58)
Due to Eq. (2), which ensures that 0tΞ20ds is bounded, we expect from Eqs. (57), (52), (51), (48), and (16), and the first three equations in Eq. (17) that
(59)

where χfL(0,T;BV(Ωf;{0,1})) means that χfL(0,T;BV(Ωf)) and χf{0,1} with BV(Ωf)={vL1(Ωf):vM(Ωf) being the space of functions with bounded variation [41]; ρfL(0,T;BV(Ωf;{ρ1,ρ2})) means that ρfL(0,T;BV(Ωf)) and ρf{ρ1,ρ2}; μfL(0,T;BV(Ωf;{μ1,μ2})) means that μfL(0,T;BV(Ωf)) and μf{μ1,μ2}; and χsL(0,T;(Ω;{0,1}) means χsL(0,T;Ω) and χs{0,1}.

The above derivations motivate the following weak solution definition.

Definition 4.1. Under the initial data Eq. (14), the tuple(xcs,ηs, χf, ρf, μf,uf,us,χs,V,p)is a weak solution of the closed-loop system consisting of Eqs. (1), (18), and (36)if they satisfy Eqs. (59) and (46), the first three equations in Eq. (17)inD(Qf)and Eq. (48)inD(Q), Eq. (53), and the compatibility condition
(60)

for allψC03(Ωf).

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 (xcs,ηs, χf, ρf, μf, uf,us,χs,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 1n>0 be a penalized parameter. Given the initial data
(61)
we wish to find the tuple (xcsn,ηsn, χfn, ρfn, μfn, un,χsn,Vn,pn) such that they satisfy
(62)
and are a solution to the penalized system
(63)
where
(64)
with
(65)
In addition, we impose the homogeneous Dirichlet boundary condition
(66)

and define Ωsn={xΩs,χsn(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 Fs and Ms given by Eq. (36) and nonhomogeneous viscosity and surface tension are taken into account.

It is clear that |Ωsn|E=Ωχsndx=|Ωs(0)|E because un,s is divergence free and hsn vanishes on Ω as we assume there is no collision between the rigid body and Ω. We also have
(67)

Since ρnmin(ρ1,ρ2,ρs)>0,msn is positive, and the matrix Jsn is symmetric and positive definite, see Eq. (7), and hence is invertible.

In the first equation of Eq. (63), the term un,s defined in Eq. (64) is the projection of un onto the velocity fields which are rigid on Ωsn because one can prove that, see Ref. [32]
(68)

where ζ is a rigid velocity field, i.e., there exist (vζ,ωζ)3×3 such that ζ=vζ+ωζ×rs(t,x), and un,s is given by Eq. (64). Hence, the penalized term nρnχsn(unun,s) in the first equation of Eq. (63) is the difference between un and its projection onto velocity fields rigid in the rigid body domain, i.e., un,s. Moreover, the viscosity term μn vanishes asymptotically in the solid part. Hence, there will be no uniform bound in V for un. This does not conflict with the penalized term nρnχsn(unun,s) because of the space K.

The density, viscosity, and fluid character function χfn are transported with the velocity field un. 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 (xcsn,ηsn, χfn, ρfn, μfn, un,χsn,Vn,pn) to the penalized system Eq. (63) that satisfies Eq. (62) for all t[0,T], where T is such that Ωsn(t)Ω and ηsn(t)Dηs for all t[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 En as in Eq. (52) with (xcs,ηs,u) being substituted by (xcsn,ηsn,un). This is due to inclusion of (xcs,ηs) and controls (Fs,Ms) 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(xcsn,ηsn, χfn, ρfn, μfn, un,χsn,Vn,pn)be a weak solution to the penalized system Eq. (63). Then, under the initial data Eq. (14), there exists a subsequence of(xcsn,ηsn, χfn, ρfn, μfn, un,χsn,Vn)such thatxcsnxcsstrongly inL(Q);ηsnηsstrongly inL((0,T)×Dηs);χfnχfinL(Q);ρfnρfinL(Q);μfnμfinL(Q);χsnχsstrongly inC(0,T;Lq(Ω)),q1;unustrongly inL2(Q)and weakly inL(0,T;H)L2(0,T;V); andVnVinL(0,T;H(Ωf×S2)), where>52, whenn. The tuple(xcs,ηs, χf, ρf, μf,uf,us,χs,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 thatΩs(t)Ωandηs(t)Dηsfor allt[0,T]. Moreover, there ispW1,(0,T;L2(Ω))such that
(69)

inD((0,T)×Ωf), whereσf=2μfD(uf)pI3.

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 VnV in L(0,T;H(Ωf×S2)), where >52. We consider the sequence Vn(t),t[0,) associated with Γn(t), i.e.,
(70)
where C0 denotes the closure of the usual space of continuous functions compactly supported on Ωf, and nn(x))=χfn|χfn|. Now, we set
(71)
Then, since VnLω(0,T;M(Ωf×S2)), see Sec. 1 for notation of Lω(0,T;M(Ωf×S2)), is uniformly bounded and M(Ωf×S2)H(Ωf×S2), where >2×312=52, we have
(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. (20) and (23) into Eq. (29) yields
(73)
where
(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
(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 uf 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 Ωs for K11xcs and (K12ηs)×rs. 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 Ωs*, see Fig. 1, to detail the terms A2 and A3.

5.1.1 Detail of A1.

Multiplying the fourth equation in Eq. (17) by uf and integrating over Ωf yields
(76)
where we have used the transmission condition (19). Substituting Eq. (76) into the first equation of Eq. (74) gives
(77)
Using Hölder's inequality results in
(78)

where ε20 is a positive constant to be chosen.

5.1.2 Detail of A2 and A3.

We define the domain Ωs*, which is an extension of the domain Ωs in the normal direction, such that ΩsΩs*Ω, see Fig. 1. We cite the following results for use in detailing A2 and A3.

Lemma 5.1. Consider the Stokes problem
(79)
Then, for anyw1Wk+21/r,r(Ωs), wherek0is an integer and1<r<, the Stokes problem Eq. (79)has a unique pair(w˜1,q˜1)withq˜1being q1 different from a constant, that satisfies
(80)

wherec(Ωs,k,r)is a constant depending only on Ωs, k, r.

Proof . See Ref. [43].▪

Lemma 5.2. Suppose1r. Assume that Ωs is bounded indandΩsisC1. Select a bounded open setΩs*such thatΩsΩs*. Then, there exists a bounded linear operator
(81)

such that for eachw˜1W1,p(Ωs): (i)Ew˜1=w˜1a.e. in Ωs, (ii)Ew˜1has support inΩs*, and iii)||w˜1||W1,r(d)c(r,Ωs,Ωs*)||w˜1||W1,r(Ωs), wherec(r,Ωs,Ωs*)is a constant depending on only r, Ωs, andΩs*).

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

In what follows, we denote:
(82)
We will derive a general formula for detailing Ak,k=2,3 with w1 playing the role of either K11xcs,t(K11xcs)=K11ucs,(K12ηs)×rs, or t((K12ηs)×rs)=(K12Rsωs)×rs(K12ηs)×ucs. This makes sense because on the boundary Ωs,w1 can take the value of either
Denote
(83)
Now, multiplying the fourth equation in Eq. (17) by Xs and integrating over Ωs* yields
(84)
Using integration by parts, the boundary condition Xs=0 on Ωs*, and the transmission condition (19), and noting that Ωs is time-dependent, we have
(85)
Substituting Eq. (85) into Eq. (84), and using Eq. (83) gives
(86)
where
(87)
where H,Xs* is H,Xs, which is restricted to Γ* with Γ* being the interface between Ω1 and Ω2 belonging to Ωs*. Using the Hölder inequality, we obtain
(88)
where ε2i,i=1,2,3 are positive constants to be chosen, and we have used
(89)

due to the embedding V(L6(Ωs*))3(L4(Ωs*))3 because Ωs* is a bounded domain, where c is a positive embedding constant.

Next, we calculate the bound of B2. By definition of H,Xs, see Eq. (43), we have
(90)
where we have partitioned Γ*(t) to N infinitesimal regions such that in each region H does not change its sign (Hi and Xis denote H and Xs in the i region, respectively, ni denotes the normal unit vector to Γi*), and ε24 is a positive constant to be chosen. Substituting Eqs. (88) and (90) into Eq. (86) yields
(91)
Now, setting w1=K11xcs and Xsxcs=Ew˜1 with tw1=K11ucs in Eq. (91) yields
(92)
On the other hand, setting w1=(K12ηs)×rs and noting that tw1=(K12Rsωs)×rs(K12ηs)×ucs in Eq. (91) yields
(93)

where Xsηs=Ew˜1 and |Ωs*|E denotes the size of Ωs*.

5.1.3 Detail of ϖ.

Substituting Eqs. (78), (92), and (93) into Eq. (73) gives
(94)
where
(95)

5.2 Convergence of the Closed-Loop System.

With ϖ detailed by Eq. (94), we consider the following Lyapunov function candidate for the closed-loop system:
(96)
where U1 is given by Eq. (26), E is given by Eq. (52), and ε0 is a positive constant to be chosen, and
(97)
Using Hölder's inequality, we can find the bound of U2 as
(98)
where ε31 and ε32 are positive constants chosen such that
(99)
and
(100)
Therefore, we can find the bound for U as
(101)

where ε¯0 and ε¯0 are positive constants by choosing a sufficiently large ε0.

Now, differentiating Eq. (96) and using Eqs. (39), (39), and (94) yield
(102)
where ε24 is a positive constant, and
(103)
Since we already proved the estimates in Eq. (59), by choosing a sufficiently large ε0, where we note that ϖ1* is given by Eq. (95), there exist a positive constant k¯ and a non-negative constant k¯0 (depending on the initial data due to H2(Γ) in Eq. (103), which is bounded because of Eqs. (58) and (52), such that
(104)

Next, we need the following lemma.

Lemma 5.3. Leta,g:[0,T]be given continuous functions. Then the solution of the nonhomogeneous ordinary differential equation
(105)
with the initial valuex(0)=x0, is defined fort[0,T]and is given by
(106)

wherea¯(t)=0ta(s)ds.

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

Applying Lemma 5.3 to Eq. (104) yields
(107)
which means that U(t) (hence ||u||Ω,||xcs||E,||ηs||E) exponentially converges to k¯00t||f(s)||Ω2ds, which is bounded because fi satisfy Eq. (2) and f is defined in Eq. (15). We summarize the main results in the following theorem.▪
Theorem 5.1. Under the initial data Eq. (14), the controlsFk, which are obtained from Eqs. (36) and (10), solves Control Objective 2.1 for allt[0,T], where T is such thatΩs(t)Ωandηs(t)Dηsfor allt[0,T]. In particular, the closed-loop system consisting of Eqs. (1), (9), and (36)has at least one weak solution, which is defined in Definition 4.1 for allt[0,T]. Moreover, the closed-loop system is almost globally practically exponentially stable at the origin, i.e.,
(108)

whereϒ(t)=||xcs(t)||E2+||ηs(t)||E2+||ucs||E2+||ωs||E2+Ωf||uf(t,x)||E2dx, 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: π10m×π10m×π10m and the mass: ms = 1 kg, which give Js=diag(0.01645,0.01645,0.01645)kgm2. For the air with μ1=1.81×105kg/ms and ρ1=1.225kg/m3 and water with μ2=1.793×103kg/ms and ρ2=980kg/m3, the domains Ω1 and Ω2 are also taken as cubes subtracting the rigid body. The dimensions of Ω1 and Ω2 are respectively taken as ([2π,2π]m×[2π,2π]m×[0,2π]m)Ω1s, where Ω1s is the part of the rigid body submerged in the air, and ([2π,2π]m×[2π,2π]m×[2π,0]m)Ω2s, where Ω2s is the part of the rigid body submerged in the water. The surface tension constant is κ=7.7×103kg/m. Thus, the domain Ω is a cube with dimensions L1×L2×L3=[2π,2π]m×[2π,2π]m×[2π,2π]m. We approximate all the sharp corners of Ω and Ωs by rounding them off to make Ω and Ωf Lipschitz, where recalling that Ωf=i=12Ωi. 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 f1=10211+t[sin(2t),cos(1.5t),sin(t)]Tkg/m2s2 and f2=101e0.1t[sin(1.5t),cos(2t),sin(1.5t)]Tkg/m2s2, which satisfy the condition (2).

We will use the semi-Galerkin method to the penalized system Eq. (63) to obtain a numerical weak solution, where we approximate
(109)

where cln(t) are scalar functions of time, al(x) are eigenfunctions of the Stokes operator. We substitute Eq. (109) into the first equation of Eq. (63) and multiply it by ξ=Spann{al(x);l=1,,n} to obtain a system of ODEs for cln(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 in3with Lipschitz boundary, then H coincides with the space of divergence free functions inL2(Ω)such thatu·n=0onΩ, wherenis the normal unit vector toΩ.

With this lemma, eigenfunctions of the Stokes problem are equivalent to those of the Laplace operator with the condition u·n=0 on Ω as we consider a weak solution in H. Hence, we look for al such that
(110)
A nontrivial calculation gives al (we neglect the round off corners) as follows:
(111)

where (l1,l2,l3)Z3{0} such that l2=l12+l22+l32, which are taken into account to have summing combination in calculating Eq. (109), and a¯l=12π3A with A=l12+l22+l32+l1l2l1l3+l2l3. It is easy to show that A >0 for all (l1,l2,l3)Z3{0}, and that al satisfies all the properties listed in Eq. (110) with λl=l12+l22+l32.

For the initial values of the fluid velocity, we take cln(0) to be random number in 1n2[1,1] if xΩ1(0)Ω1s(0) and in 1n2[2,2] if xΩ2(0)Ω2s(0). The initial value of the interface between the air and water is neutral, i.e., Ω1(0)=([2π,2π]m×[2π,2π]m×[0,2π]m)Ω1s(0) and Ω2(0)=([2π,2π]m×[2π,2π]m×[2π,0]m)Ω2s(0). This results in μ(0)=μ1 if xΩ1(0),μ(0)=μ2 if xΩ2(0),μ(0)=0 if xΩs(0); and ρ(0)=ρ1 if xΩ1(0),ρ(0)=ρ2 if xΩ2(0),ρ(0)=ρs if xΩs(0). The initial values of the rigid body are taken as xcs(0)=[0.5,0.5,0.2]Tm, ηs(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=4.9665rad. The initial values of the velocities ucs(0) and ωs(0) of the rigid body are determined via Eqs. (7) and (61), where u(0,x) is substituted by uδn(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 Fs=0 and Ms=0. The position vector xcs=[x1cs,x2cs,x3cs]T, orientation vector ηs=[η1s,η2s,η3s]T, linear velocity vector ucs=[u1cs,u2cs,u3cs]T, and angular velocity vector ωs=[ω1s,ω2s,ω3s]T are plotted in Fig. 2. The H-norm of the global velocity ||u||Ω, the control force vector Fs=[F1s,F2s,F3s]T, and control moment vector Ms=[M1s,M2s,M3s]T are plotted in Fig. 3. It is seen from these figures that x1cs,x2cs,η3s tend to drift away from the origin due to forces and moments from the fluids; x3cs,η1s,η2s tend to the origin due to (restoring) forces and moments from the fluids; ucs,||u||Ω, and ωs decay slowly to zero (decay is due to resistance of the fluids); and indeed the controls Fs and Ms are zero.

Fig. 2
Open-loop states: xcs, ηs, ucs, ωs
Fig. 2
Open-loop states: xcs, ηs, ucs, ωs
Close modal
Fig. 3
Open-loop velocity norm:||u||Ω=(∫Ω||u||E2dx)12 and open-loop controls: Fs and Ms
Fig. 3
Open-loop velocity norm:||u||Ω=(∫Ω||u||E2dx)12 and open-loop controls: Fs and Ms
Close modal

In the closed-loop simulations, we choose the control gains as follows: K11=5I3,K12=5I3,K21=15I3,K22=15I3. It can be checked that the conditions in Eq. (38) hold. The position vector xcs=[x1cs,x2cs,x3cs]T, orientation vector ηs=[η1s,η2s,η3s]T, linear velocity vector ucs=[u1cs,u2cs,u3cs]T, and angular velocity vector ωs=[ω1s,ω2s,ω3s]T are plotted in Fig. 4. The H-norm of the global velocity ||u||Ω, the control force vector Fs=[F1s,F2s,F3s]T, and control moment vector Ms=[M1s,M2s,M3s]T are plotted in Fig. 5. It is seen from these figures that all the states xcs,ηs,ucs,ωs, and the velocity norm ||u||Ω=(Ω||u||E2dx)12; and the controls Fs and Ms converge to the origin (note that the body forces f1 and f2 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
Closed-loop states: xcs, ηs, ucs, ωs
Fig. 4
Closed-loop states: xcs, ηs, ucs, ωs
Close modal
Fig. 5
Closed-loop velocity norm:||u||Ω=(∫Ω||u||E2dx)12 and closed-loop controls: Fs and Ms
Fig. 5
Closed-loop velocity norm:||u||Ω=(∫Ω||u||E2dx)12 and closed-loop controls: Fs and Ms
Close modal

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.
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
.
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
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
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
.
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
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
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
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
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
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
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
41.
Ambrosio
,
L.
,
Fusco
,
N.
, and
Pallara
,
D.
,
2000
,
Functions of Bounded Variation and Free Discontinuity Problems
,
Clarendon Press
,
Oxford, UK
.
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
43.
Bello
,
J. A.
,
1996
, “
Lr 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
.