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Inverse optimal control of regime-switching jump diffusions

This paper studies the inverse optimal control using Legendre-Fenchel (in short, LF) translation method for regime-switching jump diffusions. Our approach is to first design inverse pre-optimal stabilization controllers and then obtain inverse optimal stabilizers, which avoids solving a Hamilton-Jacobi-Bellman equation. Finally, an application to stochastic Hamiltonian systems with Markov regime-switching is studied in detail for illustration.


Introduction.
Jump processes provide mathematical model in many research fields such as mathematical biology, financial market, population processes, control problems, and so on, see for example, [1,2,3,14,16]. Due to the random change of the environment concerning system lives, regime-switching processes have attracted much interest lately, progress towards these problems have so far involved a wide range of ideas surround as biological and ecological systems, risk theory, stabilization theory, we refer to readers [4,12,15,17,18,24] and the references therein. Inspired by problems of model complexity in random environments, in which both structural changes and sudden changes of parameters as well as large fluctuations coexist and are intertwined, this paper study regime-switching jump diffusion processes. The quantitative properties of these systems have greatly expanded over the previous few years, with most results making a connection between the control theory and the qualitative dynamics or long-time behavior of the associated dynamical system, see, for example, [5,17,18,19,20,21,22,23,25] and references therein.
As a well-defined dynamic system, there are few results on the optimal stabilization theory of regime-switching jump diffusion systems. It is the most intriguing topics to investigate the optimality of the regime-switching jump diffusion systems. It was shown that the dynamic programming method and the viscosity approach are two powerful tools to study stochastic optimal control problems by means of Hamilton-Jacobi-Bellman equation. However, those method have many limitations.
Because the optimal control problem is ultimately reduced to that of solving a partial differential equation (in short, PDE). Thus, a main drawback in the fields of stochastic optimal control problem is the fact that the solution of the PDE does not exist, or exists but is not unique. In other words, the premise of this method is to guarantee the existence of the regular solution to the Hamilton-Jacobi-Bellman equation. However, the existence and uniqueness of the solutions to the Hamilton-Jacobi-Bellman equation are available only for some special cases so far. This is not the case in general. This question is relaxed in the theory of inverse optimal control, which provides a powerful approach to solve optimal control problems, see [10]. Afterwards, Deng and Krstic [6,7] extended the previous results to stochastic case and the design scheme of inverse optimal control for nonlinear systems are presented in detail. The main approach to inverse optimal stabilization is to design pr-inverse optimal stabilizer then design inverse optimal controller with the aid of LF transform. For more details on this topic, the readers are referred to Do [8,9] and the references therein. The problem that we study in this paper is a continuation of [9,13]. Compared with the work of [9], this framework addresses the influence of Markov regime-switching. It is also worth pointing out that the current results are similar to those in pure jump diffusion work, some additional care is needed due to the presence of regime switching. The gain function is closely related to the LU (·, ·, ·). The potential difficulty is to compute LU (·, ·, ·) associated with the process (X, Λ). Consequently, the optimal stabilization problem is transferred to some existing results and techniques. There are several available work, see [5,25]. In [13], the state feedback control is used to stabilize a given unstable hybrid SDE. In this article, our main purpose is to provide an optimization scheme for the design of the controller for hybrid SDEs with the state feedback control. More precisely, these problem can be split in two parts: the sufficient condition of the stabilization of system is obtained by Lyapunov function and then the corresponding controller is optimized by means of LF transform. This paper is organized as following: In Section 2 we detail the additional notations used in this article. Section 3 contains the proof of our main results. Section 4 is devoted to example, some specific model to which our results apply. A summary of the key point is given in Section 5.
We consider a stochastic control model is governed by the following equation: where initial date X(t 0 ) = X 0 , Λ(t 0 ) = i 0 , f (·, ·, ·) : is the control process. The switching process Λ(·) satisfies: as ∆ ↓ 0, where q ij denotes the transition rate from i to j with q ij > 0 if i = j. Let Q = (q ij ) n×n be irreducible and conservative, i.e., q i = −q ii = Σ i =j q ij < ∞. In addition, we assume that the Brownian motion B(·), the Poisson random measure N (·, ·), and the Markov chain Λ(·) are mutually independent. To examine the wellposedness of global solutions to the SDE (1), we introduce some notations. For any U (·, ·, ·) ∈ C 1,2 (R + × R d × S; R + ), let us define the operator by where ∇U and ∇ 2 U denote the gradient and Hessian matrix of U (t, X, i), respectively. To examine the properties of system (1), we need the following assumptions: Then process (X(t), Λ(t)) is regular.
The results have been well studied, see Zhen et al. ( [5], Theorem 3.1) and Zong et al. [25]. [11]) Let us present Legendre-Fenchel (in short, LF) transform: Definition 2.5. There exist a positive definite function ϕ : , a symmetric positive definite matrix-valued function G(t, X, i) and feedback control u such that p-th moment inverse optimal exponential stabilization for (1) is solvable and the gain function is minimal, where the gain function is given as Denote by s,X,i the collection of all admissible control. The corresponding value function is described as An admissible control u * ∈ s,X,i is called optimal, if it holds Remark 2. Definition 2.5 indicates that value function includes ϕ(t, X, i), η(r), G(t, X, i) and feedback control u. When we search for different K ∞ function, this leads to different inverse optimal control law. Hence, the flexibility of the cost function is left to the designer.
Theorem 3.1. There exists a Lyapunov function U (t, X, i) satisfies condition (i) in Theorem 2.2. Let G(t, X, i) be a symmetric positive definite matrix-valued function. Let u := u(t, X, i) be pre-optimal controller. Suppose that u(t, X, i) ensures the p-th moment exponential stabilization of system (1). Let be optimal controller for all α ≥ 2. Then u * (t, X, i) ensures the p-th moment inverse optimal exponential stabilization of system (1) by minimizing the cost functional where Moreover, U (t, X, i) satisfies the family Hamilton-Jacobi-Bellman Proof. The proof is separated into three steps.
Step 2: This step is to show that the u * is also an admissible feedback control, which will be the desired optimal feedback control.
This means that u * is also an admissible feedback control and ensures system (1) p-th moment exponential stabilization. Since U (t, X, i) ∈ C 1,2 ([0, T )×R n ), we have for all (t, X) ∈ [0, T ) × R n , u * ∈ s,X,i , by Itô's formula Taking the expectation yields that Furthermore, invoking the equalities (7) and (12) and substituting them into (6) lead to Now using (4) yields that ∇U H(t, X, i)u(t, X, i) Directly computing, we obtain that Next, using (5) yields that Similarly, using (5) and (15), we have Substituting (14), (15) and (16) into (13) that Invoking Lemma 2.3, we have where "=" holds if and only if Since (19) holds for x ∈ R n , we have u = u * . In addition, because u = u * stabilizes the system (1), we obtain the desired limit with lim t→∞ EU (t, X, i) = 0. In view of (18), the optimal cost functional are arg min u J(t, X, i, u) = u * and min u J(t, X, i, u) = 2αEU (t 0 , X 0 , i 0 ).
Letū guarantee system (1) p-th moment inverse optimal exponential stabilization. Replacing u of (13) withū yields that where Invoking Young's inequality shows that .
Step 3: Proof of continuity We complete the proof.
Remark 3. The theorem above indicates that the gain function J(t, X, u, i) and Hamilton-Jacobi-Bellman equation (8) are generated after the feedback control u.

Remark 4.
In summary, the main procedure for design of an inverse optimal control for regime-switching jump diffusions include the following steps: (1) Set and It is obvious that A and A * are positive define matrices. Choosing an appropriate Lyapunov function is employed to guarantee for LU ≤ −µU ; (2) Searching for K ∞ function η(r), then the G(t, X, i) is determined by the matrix A.
(3) Finally, inverse optimal controller is obtained by (4). Now, we would like to present sufficient conditions of inverse optimal stabilizers. and where β 1 and β 2 are two positive constants. Then, inverse optimal stabilization is solvable.
To design inverse optimal control stabilizers u for the system (31), the corresponding controlled system becomes The method of the designing is very technical so we break it into three steps.