In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for partial stability and partial-state stabilization of stochastic dynamical systems. Partial asymptotic stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation, and hence, guaranteeing both partial stability in probability and optimality. The overall framework provides the foundation for extending optimal linear-quadratic stochastic controller synthesis to nonlinear-nonquadratic optimal partial-state stochastic stabilization. Connections to optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic cost functionals are also provided. Finally, we also develop optimal feedback controllers for affine stochastic nonlinear systems using an inverse optimality framework tailored to the partial-state stochastic stabilization problem and use this result to address polynomial and multilinear forms in the performance criterion.
In Ref. , we extended the framework developed in Refs. [2,3] to address the problem of optimal partial-state stabilization, wherein stabilization with respect to a subset of the system state variables is desired. Partial-state stabilization arises in many engineering applications [4,5]. Specifically, in spacecraft stabilization via gimballed gyroscopes, asymptotic stability of an equilibrium position of the spacecraft is sought while requiring Lyapunov stability of the axis of the gyroscope relative to the spacecraft . Alternatively, in the control of rotating machinery with mass imbalance, spin stabilization about a nonprincipal axis of inertia requires motion stabilization with respect to a subspace instead of the origin . The most common application where partial stabilization is necessary is adaptive control, wherein asymptotic stability of the closed-loop plant states is guaranteed without necessarily achieving parameter error convergence.
In this paper, we extend the framework developed in Ref.  to address the problem of optimal partial-state stochastic stabilization. Specifically, we consider a notion of optimality that is directly related to a given Lyapunov function that is positive definite and decrescent with respect to part of the system state. In particular, an optimal partial-state stochastic stabilization control problem is stated, and sufficient Hamilton–Jacobi–Bellman conditions are used to characterize an optimal feedback controller. Another important application of partial stability and partial stabilization theory is the unification it provides between time-invariant stability theory and stability theory for time-varying systems [3,6]. We exploit this unification and specialize our results to address optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic cost functionals.
Our approach focuses on the role of the Lyapunov function guaranteeing stochastic stability of the closed-loop system and its connection to the steady-state solution of the stochastic Hamilton–Jacobi–Bellman equation characterizing the optimal nonlinear feedback controller. In order to avoid the complexity in solving the stochastic steady-state, Hamilton–Jacobi–Bellman equation, we do not attempt to minimize a given cost functional, but rather, we parameterize a family of stochastically stabilizing controllers that minimizes a derived cost functional that provides the flexibility in specifying the control law. This corresponds to addressing an inverse optimal stochastic control problem [7–13].
The inverse optimal control design approach provides a framework for constructing the Lyapunov function for the closed-loop system that serves as an optimal value function and, as shown in Refs. [11,12], achieves desired stability margins. Specifically, nonlinear inverse optimal controllers that minimize a meaningful (in the terminology of Refs. [11,12]) nonlinear-nonquadratic performance criterion involving a nonlinear-nonquadratic, non-negative-definite function of the state and a quadratic positive-definite function of the feedback control are shown to possess sector margin guarantees to component decoupled input nonlinearities in the conic sector .
The paper is organized follows. In Sec. 2, we establish notation, definitions, and present some key results on partial stability of nonlinear stochastic dynamical systems. In Sec. 3, we consider a stochastic nonlinear system with a performance functional evaluated over the infinite horizon. The performance functional is then evaluated in terms of a Lyapunov function that guarantees partial asymptotic stability in probability. We then state a stochastic optimal control problem and provide sufficient conditions for characterizing an optimal nonlinear feedback controller guaranteeing partial asymptotic stability in probability of the closed-loop system. These results are then used to address a stochastic optimal control problem for uniform asymptotic stabilization in probability of nonlinear time-varying stochastic dynamical systems.
In Sec. 4, we develop optimal feedback controllers for affine stochastic nonlinear systems using an inverse optimality framework tailored to the partial-state stochastic stabilization problem. This result is then used to derive time-varying extensions of the results in Refs. [14,15] involving nonlinear feedback controllers minimizing polynomial and multilinear performance criteria. In Sec. 5, we provide two illustrative numerical examples that highlight the optimal partial-state stochastic stabilization framework. In Sec. 6, we present conclusions and highlight some future research directions. Finally, we note that a preliminary version of this paper appeared in Ref. . The present paper considerably expands on Ref.  by providing detailed proofs of all the results along with examples and additional motivation.
Notation, Definitions, and Mathematical Preliminaries
In this section, we establish notation, definitions, and review some basic results on partial stability of nonlinear stochastic dynamical systems [17–22]. Specifically, denotes the set of real numbers, denotes the set of positive real numbers, denotes the set of non-negative numbers, denotes the set of positive integers, denotes the set of n × 1 real column vectors, denotes the set of n × m real matrices, denotes the set of n × n non-negative-definite matrices, and denotes the set of n × n positive-definite matrices. We write for the open ball centered at x with radius ε, for the Euclidean vector norm or an induced matrix norm (depending on context), for the Frobenius matrix norm, AT for the transpose of the matrix A, ⊗ for the Kronecker product, ⊕ for the Kronecker sum, and In or I for the n × n identity matrix. Furthermore, denotes the σ-algebra of Borel sets in , and denotes a σ-algebra generated on a set .
We define a complete probability space as , where Ω denotes the sample space, denotes a σ-algebra, and defines a probability measure on the σ-algebra ; that is, is a non-negative countably additive set function on such that . Furthermore, we assume that w(⋅) is a standard d-dimensional Wiener process defined by , where is the classical Wiener measure [22, p. 10], with a continuous-time filtration generated by the Wiener process w(t) up to time t. We denote a stochastic dynamical system by generating a filtration adapted stochastic process on satisfying , such that , for all Borel sets contained in the Borel σ-algebra . Here, we use the notation x(t) to represent the stochastic process x(t, ω) omitting its dependence on ω.
We denote the set of equivalence classes of measurable, integrable, and square-integrable or (depending on context) valued random processes on over the semi-infinite parameter space [0, ∞) by , and , respectively, where the equivalence relation is the one induced by -almost-sure equality. In particular, elements of take finite values -almost surely (a.s.). Hence, depending on the context, will denote either the set of n × 1 real variables or the subspace of comprising random processes that are constant almost surely. All inequalities and equalities involving random processes on are to be understood to hold -almost surely. Furthermore, and denote, respectively, the expectation with respect to the probability measure and with respect to the classical Wiener measure .
Finally, we write tr(⋅) for the trace operator, for the inverse operator, for the Fréchet derivative of V at x, for the Hessian of V at x, and for the Hilbert space of random vectors with finite average power, that is, . For an open set denotes the set of all the random vectors in induced by . Similarly, for every . Furthermore, C2 denotes the space of real-valued functions that are two-times continuously differentiable with respect to .
where, for every and are such that is a -measurable random state vector, is an open set with , w(t) is a d-dimensional independent standard Wiener process (i.e., Brownian motion) defined on a complete filtered probability space is independent of , and is such that, for every and f1(⋅, x2) is locally Lipschitz continuous in x1, and is such that, for every is locally Lipschitz continuous in x2. In addition, the function is continuous such that, for every , and is continuous.
where the integrals in Eq. (3) are Itô integrals. Note that for each fixed t ≥ t0, the random variable assigns a vector x(ω) to every outcome ω ∈ Ω of an experiment, and for each fixed ω ∈Ω, the mapping is the sample path of the stochastic process x(t), t ≥ t0. A pathwise solution of Eqs. (1) and (2) in is said to be right maximally defined if x cannot be extended (either uniquely or nonuniquely) forward in time. We assume that all right maximal pathwise solutions to Eqs. (1) and (2) in exist on [t0, ∞), and hence, we assume that Eqs. (1) and (2) are forward complete. Sufficient conditions for forward completeness or global solutions to Eqs. (1) and (2) are given by Corollary 6.3.5 of Ref. .
for some Lipschitz constant L > 0, and hence, since and x(t0) is independent of , it follows that there exists a unique solution of Eqs. (1) and (2) in the following sense. For every , there exists τx > 0 such that, if and are two solutions of Eqs. (1) and (2); that is, if , with continuous sample paths almost surely, solve Eqs. (1) and (2), then and . Sufficient conditions for forward existence and uniqueness in the absence of the uniform Lipschitz continuity condition and growth restriction condition can be found in Refs. [23,24].
A solution is said to be regular if and only if for all , where τe is the first stopping time of the solution to Eqs. (1) and (2) from every bounded domain in . Recall that regularity of solutions implies that solutions exist for t ≥ t0 almost surely. Here, we assume regularity of solutions to Eqs. (1) and (2), and hence, τx = ∞ [18, p. 75]. Moreover, the unique solution determines a -valued, time-homogeneous Feller continuous Markov process x(⋅), and hence, its stationary Feller transition probability function is given by (Refs. [18, Theorem 3.4] and [20, Theorem 9.2.8]) for all and t ≥ t0, and all Borel subsets of , where , denotes the probability of transition of the point at time instant s into the set at time instant t. Finally, recall that every continuous process with Feller transition probability function is also a strong Markov process [18, p. 101].
In the following definition, we introduce the notion of stochastic partial stability.
for all t ≥ 0 and all.
uniformly in x20 for all.
(iii)is globally asymptotically stable in probability with respect to x1 uniformly in x20 if is Lyapunov stable in probability with respect to x1 uniformly in x20 andholds uniformly in x20 for all.
Remark 2.1. It is important to note that there is a key difference between the stochastic partial stability definitions given in Definitions 2.2 and the definitions of stochastic partial stability given in Ref. . In particular, the stochastic partial stability definitions given in Ref.  require that both the initial conditions x10 and x20 lie in a neighborhood of origin, whereas in Definition 2.2 x20 can be arbitrary. As will be seen below, this difference allows us to unify autonomous stochastic partial stability theory with time-varying stochastic stability theory. An additional difference between our formulation of the stochastic partial stability problem and the stochastic partial stability problem considered in Ref.  is in the treatment of the equilibrium of Eqs. (1) and (2). Specifically, in our formulation, we require the weaker partial equilibrium condition f1(0, x2) = 0 and D1(0, x2) = 0 for every , whereas in Ref.  the author requires the stronger equilibrium condition , and D2(0, 0) = 0.
Remark 2.2. A more general stochastic stability notion can also be introduced here involving stochastic stability and convergence to an invariant (stationary) distribution. In this case, state convergence is not to an equilibrium point but rather to a stationary distribution. This framework can relax the vanishing perturbation assumption , and requires a more involved analysis and synthesis framework showing stability of the underlying Markov semigroup .
Note that Eqs. (11) and (12) are in the same form as the system given by Eqs. (1) and (2), and Definition 2.2 applied to Eqs. (11) and (12) specializes to the definitions of uniform Lyapunov stability in probability, uniform asymptotic stability in probability, and global uniform asymptotic stability in probability of Eq. (10); for details, see Refs.  and .
Next, we provide sufficient conditions for partial stability of the nonlinear stochastic dynamical system given by Eqs. (1) and (2). For the statement of this result, recall the definitions of a class and functions given in Ref. [3, p. 162].
(ii) If there exist a two-times continuously differentiable function, classfunctions α(⋅) and β(⋅), and a classfunction γ(⋅) satisfying Eqs.(13)and(14), then the nonlinear dynamical system given by Eqs.(1)and(2)is globally asymptotically stable in probability with respect to x1 uniformly in x20.
which proves partial Lyapunov stability in probability with respect to x1 uniformly in x20.
Furthermore, it follows from partial Lyapunov stability in probability that is an invariant set with respect to the solutions of Eqs. (1) and (2) as ε → 0, and hence, using Corollary 4.2 of Ref.  with it follows that . Furthermore, using the properties of the class functions and γ(⋅), it follows that , which yields . Hence, as x10 → 0, which proves partial asymptotic stability in probability with respect to x1 uniformly in x20.
(ii) Finally, for , globally asymptotically stable in probability with respect to x1 uniformly in x20 is direct consequence of the radially unbounded condition on V(⋅, ⋅) using standard arguments and the fact that α(⋅) and β(⋅) are class functions. ▪
Stochastic Optimal Partial-State Stabilization
where is jointly continuous in x1 and x2, and x1(t) and x2(t), t ≥ 0, satisfy Eqs. (1) and (2), can be evaluated in a convenient form so long as Eqs. (1) and (2) are related to an underlying Lyapunov function that is positive definite and decrescent with respect to x1 and proves asymptotic stability in probability of Eqs. (1) and (2) with respect to x1 uniformly in x20.
Proof. Let x1(t) and x2(t), t ≥ t0, satisfy Eqs. (1) and (2). Then, Eqs. (16) and (17) are a restatement of Eqs. (13) and (14), and hence, it follows from Theorem 2.1 that the system is globally asymptotically stable in probability with respect to x1 uniformly in x20. Consequently, holds for all initial conditions .
where is jointly continuous in t and x, and x(t), t ≥ t0, satisfies Eq. (10).
Then, the stochastic nonlinear dynamical system (10) is globally uniformly asymptotically stable in probability andfor all.
Proof. The result is a direct consequence of Theorem 3.1 with , and . ▪
where, for every , and F1(0, x2, 0) = 0 and D1(0, x2, 0) = 0 for every .
Here, we assume that u(⋅) satisfies sufficient regularity conditions such that Eqs. (32) and (33) have a unique solution forward in time. Specifically, we assume that the control process u(⋅) in Eqs. (32) and (33) is restricted to the class of admissible controls consisting of measurable functions u(⋅) adapted to the filtration such that , and, for all is independent of , and , and hence, u(⋅) is nonanticipative. Furthermore, we assume u(⋅) takes values in a compact, metrizable set U and the uniform Lipschitz continuity and growth conditions (4) and (5) hold for the controlled drift and diffusion terms and uniformly in u. In this case, it follows from Theorem 2.2.4 of Ref.  that there exists a pathwise unique solution to Eqs. (32) and (33) in .
Note that restricting our minimization problem to , that is, inputs corresponding to partial-state null convergent in probability solutions, can be interpreted as incorporating a partial-state system detectability condition through the cost.
Proof. Global asymptotic stability in probability with respect to x1 uniformly in x20 is a direct consequence of Eqs. (37) and (38) by applying Theorem 2.1 to the closed-loop system given by Eqs. (34) and (35). Furthermore, using Eq. (40), condition (43) is a restatement of Eq. (19) as applied to the closed-loop system.
Now, noting that for all , define the random variable . In this case, the sequence of -measurable random variables on Ω, where , satisfies .
In this case, Eq. (37) implies that V(⋅) is positive definite with respect to x, and the conditions of Theorem 3.2 reduce to the conditions given in Chap. 4 of Ref.  characterizing the classical stochastic optimal control problem for time-invariant systems on an infinite interval.
Proof. The result is a consequence of Theorem 3.2 with and . ▪
Proof. The proof is a direct consequence of Theorem 3.2 with , and . ▪
for time-varying stochastic systems on a finite or infinite interval.
Inverse Optimal Stochastic Control
In this section, we construct state feedback controllers for nonlinear affine in the control stochastic dynamical systems that are predicated on an inverse optimal control problem [7–13]. In particular, as noted in the Introduction, to avoid the complexity in solving the steady-state, stochastic Hamilton–Jacobi–Bellman equation (62), we do not attempt to minimize a given cost functional, but rather, we parameterize a family of stabilizing controllers that minimize some derived cost functional that provides flexibility in specifying the control law. The performance integrand is shown to explicitly depend on the nonlinear system dynamics, the Lyapunov function of the closed-loop system, and the stabilizing feedback control law, wherein the coupling is introduced via the stochastic Hamilton–Jacobi–Bellman equation. Hence, by varying the parameters in the Lyapunov function and the performance integrand, the proposed framework can be used to characterize a class of globally partial-state stabilizing (in probability) controllers that can meet closed-loop system response constraints.
Proof. The proof is identical to the proof of Corollary 3.2. ▪
which verifies Eq. (76).
and hence, Eq. (77) holds.
Finally, we specialize Theorem 4.1 to linear time-varying stochastic systems controlled by nonlinear controllers that minimize a multilinear cost functional. For the following result, define and , with x and A appearing k times, where k is a positive integer. Furthermore, define and let , where r is a positive integer, and be continuous and uniformly bounded, , and , for some μ > 0 and for all t ≥ t0.
which verifies Eq. (76).
Illustrative Numerical Examples
In this section, we provide two illustrative numerical examples to highlight the optimal and inverse optimal partial-state asymptotic stabilization framework developed in the paper.
Optimal Partial Stabilization of a Rigid Spacecraft.
Let , , , and . Figure 1 shows the sample average along with the standard deviation of the controlled system state versus time for 20 sample paths for . Note that a.s. as t → ∞, whereas does not converge to zero. Figure 2 shows the sample average along with the standard deviation of the corresponding control signal versus time. Finally, .
Thermoacoustic Combustion Model.
In this example, we consider control of thermoacoustic instabilities in combustion processes. Engineering applications involving steam and gas turbines and jet and ramjet engines for power generation and propulsion technology involve combustion processes. Due to the inherent coupling between several intricate physical phenomena in these processes involving acoustics, thermodynamics, fluid mechanics, and chemical kinetics, the dynamic behavior of combustion systems is characterized by highly complex nonlinear models [32,33]. The unstable dynamic coupling between heat release in combustion processes generated by reacting mixtures releasing chemical energy and unsteady motions in the combustor develop acoustic pressure and velocity oscillations that can severely affect operating conditions and system performance.
representing a time-averaged, two-mode thermoacoustic combustion model with state-dependent stochastic disturbances, where α1 > 0 and α2 > 0 represent decay constants, θ1 and represent frequency shift constants, , where γ denotes the ratio of specific heats and ω1 is the frequency of the fundamental mode, σ1, σ2, and σ3 are such that and and represent augmentation factors of the variance of the state-dependent stochastic disturbance, and u is the control input signal. As shown in Refs. [32,33], only the first two states q1 and q2 representing the modal amplitudes of a two-mode thermoacoustic combustion model are relevant in characterizing system instabilities, since the third state q3 represents the phase difference between the two modes . Hence, we require asymptotic stability of , and , which necessitates partial stabilization.
which implies that Furthermore, since which is positive definite with respect to x1, and hence, Eq. (59) holds.
Let , and q30 = 10. Figure 3 shows the sample average along with the standard deviation of the controlled system state versus time, whereas Fig. 4 shows the sample average along with the standard deviation of the corresponding control signal versus time for 20 sample paths. Note that as t → ∞, whereas is unstable. Finally, .
In this paper, an optimal control problem for partial-state stochastic stabilization is stated, and sufficient conditions are derived to characterize an optimal nonlinear feedback controller that guarantees asymptotic stability in probability of part of the closed-loop system state. Specifically, we utilized a steady-state stochastic Hamilton–Jacobi–Bellman framework to characterize optimal nonlinear feedback controllers with a notion of optimality that is directly related to a given Lyapunov function that is positive definite and decrescent with respect to part of the system state. This result was then used to address optimal linear and nonlinear regulation for linear and nonlinear time-varying stochastic systems with quadratic and nonlinear-nonquadratic performance measures. In addition, we developed inverse optimal feedback controllers for affine nonlinear systems and linear time-varying stochastic systems with polynomial and multilinear performance criteria. Extensions of this framework for addressing discrete-time systems with computation constraints as well as optimal adaptive controllers for stochastic dynamical systems are currently under development.
This work was supported in part by the Air Force Office of Scientific Research under Grant No. FA9550-16-1-0100.