带有随机跳跃干扰的线性二次随机最优控制问题

给出一类布朗运动和泊松过程混合驱动的正倒向随机微分方程解的存在唯一性结果,应用这一结果研究带有随机跳跃干扰的线性二次随机最优控制问题,并得到最优控制的显式形式,可以证明最优控制是唯一的.然后,引入和研究一类推广的黎卡提方程系统,讨论该方程系统的可解性并由该方程的解得到带有随机跳跃干扰的线性二次随机最优控制问题最优的线性反馈.     

Optimal control problems for stochastic delay evolution equations in Banach spaces

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given.     

Feasible solution to discrete-time linear quadratic stochastic Stackelberg difference game

This paper investigates the discrete-time linear quadratic (LQ) stochastic Stackelberg game, which has not been thoroughly addressed in previous literature. Firstly, we derive the maximum principle for the stochastic Stackelberg difference game using the variational method, and obtain the necessary and sufficient solvability conditions. However, due to the coupling between the two players and the presence of stochastic noise, obtaining explicit optimal leader and follower's strategies becomes challenging. Therefore, we present a feasible suboptimal control strategy instead. As a result, we derive a feasible suboptimal control strategy. To achieve this, we assume a linear homogeneous relationship to decouple the group of stochastic game forward-backward stochastic differential equations (SG-FBSDEs), which serves as a compromise for obtaining the optimal solution. With this approach, we derive a feasible solution to the stochastic Stackelberg difference game based on the solution to symmetric Riccati equations.     

Necessary and sufficient conditions of risk‐sensitive optimal control and differential games for stochastic differential delayed equations

In this paper, we consider risk‐sensitive optimal control and differential games for stochastic differential delayed equations driven by Brownian motion. The problems are related to robust stochastic optimization with delay due to the inherent feature of the risk‐sensitive objective functional. For both problems, by using the logarithmic transformation of the associated risk‐neutral problem, the necessary and sufficient conditions for the risk‐sensitive maximum principle are obtained. We show that these conditions are characterized in terms of the variational inequality and the coupled anticipated backward stochastic differential equations (ABSDEs). The coupled ABSDEs consist of the first‐order adjoint equation and an additional scalar ABSDE, where the latter is induced due to the nonsmooth nonlinear transformation of the adjoint process of the associated risk‐neutral problem. For applications, we consider the risk‐sensitive linear‐quadratic control and game problems with delay, and the optimal consumption and production game, for which we obtain explicit optimal solutions.     

Linear‐Quadratic Optimal Control Problem for Partially Observed Forward‐Backward Stochastic Differential Equations of Mean‐Field Type

This paper is concerned with the linear‐quadratic optimal control problem for partially observed forward‐backward stochastic differential equations (FBSDEs) of mean‐field type. Based on the classical spike variational method, backward separation approach as well as filtering technique, we first derive the necessary and sufficient conditions of the optimal control problem with the non‐convex domain. Nextly, by means of the decoupling technique, we obtain two Riccati equations, which are uniquely solvable under certain conditions. Also, the optimal cost functional is represented by the solutions of the Riccati equations for the special case.     

Infinite horizon linear quadratic differential games for discrete-time stochastic systems

This paper deals with the infinite horizon linear quadratic(LQ)differential games for discrete-time stochastic systems with both state and control dependent noise.The Popov-Belevitch-Hautus(PBH)criteria for exact observability and exact detectability of discrete-time stochastic systems are presented.By means of them,we give the optimal strategies (Nash equilibrium strategies)and the optimal cost values for infinite horizon stochastic differential games.It indicates that the infinite horizon LQ stochastic differential games are associated with four coupled matrix-valued equations.Furthermore, an iterative algorithm is proposed to solve the four coupled equations.Finally,an example is given to demonstrate our results.     

Optimal control of a stochastic delay heat equation with boundary-noise and boundary-control

In this paper, a controlled stochastic delay heat equation with Neumann boundary-noise and boundary-control is considered. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of the backward stochastic differential equations, which is applied to the optimal control problem.     

Finite Horizon Minimax Optimal Control of Stochastic Partially Observed Time Varying Uncertain Systems

We consider a linear-quadratic problem of minimax optimal control for stochastic uncertain control systems with output measurement. The uncertainty in the system satisfies a stochastic integral quadratic constraint. To convert the constrained optimization problem into an unconstrained one, a special S-procedure is applied. The resulting unconstrained game-type optimization problem is then converted into a risk-sensitive stochastic control problem with an exponential-of-integral cost functional. This is achieved via a certain duality relation between stochastic dynamic games and risk-sensitive stochastic control. The solution of the risk-sensitive stochastic control problem in terms of a pair of differential matrix Riccati equations is then used to establish a minimax optimal control law for the original uncertain system with uncertainty subject to the stochastic integral quadratic constraint. Date received: May 13, 1997. Date revised: March 18, 1998.     

Optimal Control Problem for Risk‐Sensitive Mean‐Field Stochastic Delay Differential Equation with Partial Information

This paper deals with the risk‐sensitive control problem for mean‐field stochastic delay differential equations (MF‐SDDEs) with partial information. Firstly, under the assumptions that the control domain is not convex and the value function is non‐smooth, we establish a stochastic maximum principle (SMP). Then, by means of Itô's formula and some continuous dependence, we prove the existence and uniqueness results for another type of MF‐SDDEs. Meanwhile, the verification theorem for the MF‐SDDEs is obtained by using a clever construction of the Hamiltonian function. Finally, based on our verification theorem, a linear‐quadratic system is investigated and the optimal control is also derived by the stochastic filtering technique.     

Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes

ABSTRACT

In this paper, we introduce a new class of backward doubly stochastic differential equations (in short BDSDE) called mean-field backward doubly stochastic differential equations (in short MFBDSDE) driven by Itô-Lévy processes and study the partial information optimal control problems for backward doubly stochastic systems driven by Itô-Lévy processes of mean-field type, in which the coefficients depend on not only the solution processes but also their expected values. First, using the method of contraction mapping, we prove the existence and uniqueness of the solutions to this kind of MFBDSDE. Then, by the method of convex variation and duality technique, we establish a sufficient and necessary stochastic maximum principle for the stochastic system. Finally, we illustrate our theoretical results by an application to a stochastic linear quadratic optimal control problem of a mean-field backward doubly stochastic system driven by Itô-Lévy processes.     

Necessary condition for near optimal control of linear forward–backward stochastic differential equations

This paper investigates the near optimal control for a kind of linear stochastic control systems governed by the forward–backward stochastic differential equations, where both the drift and diffusion terms are allowed to depend on controls and the control domain is not assumed to be convex. In the previous work (Theorem 3.1) of the second and third authors, some problem of near optimal control with the control dependent diffusion is addressed and our current paper can be viewed as some direct response to it. The necessary condition of the near-optimality is established within the framework of optimality variational principle developed by Yong and obtained by the convergence technique to treat the optimal control of FBSDEs in unbounded control domains by Wu. Some new estimates are given here to handle the near optimality. In addition, an illustrating example is discussed as well.     

Linear−quadratic optimal control and nonzero‐sum differential game of forward−backward stochastic system

An existence and uniqueness result for one kind of forward–backward stochastic differential equations with double dimensions was obtained under some monotonicity conditions. Then this result was applied to the linear‐quadratic stochastic optimal control and nonzero‐sum differential game of forward–backward stochastic system. The explicit forms of the optimal control and the Nash equilibrium point are obtained respectively. We note that our method is effective in studying the uniqueness of Nash equilibrium point. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

The optimal control related to Riemannian manifolds and the viscosity solutions to Hamilton–Jacobi–Bellman equations

In this paper we study the optimal stochastic control problem for stochastic differential equations on Riemannian manifolds. The cost functional is specified by controlled backward stochastic differential equations in Euclidean space. Under some suitable assumptions, we conclude that the value function is the unique viscosity solution to the associated Hamilton–Jacobi–Bellman equation which is a fully nonlinear parabolic partial differential equation on Riemannian manifolds.     

Maximum principle for the stochastic optimal control problem with delay and application

In this paper, we consider an optimal control problem for the stochastic system described by stochastic differential equations with delay. We obtain the maximum principle for the optimal control of this problem by virtue of the duality method and the anticipated backward stochastic differential equations. Our results can be applied to a production and consumption choice problem. The explicit optimal consumption rate is obtained.     

H2/H∞ control for stochastic systems with delay

This paper is concerned with the H₂/H_∞ control problem for stochastic linear systems with delay in state, control and external disturbance-dependent noise. A necessary and sufficient condition for the existence of a unique solution to the control problem is derived. The resulting solution is characterised by a kind of complex generalised forward–backward stochastic differential equations with stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the equivalent feedback solution via a new type of Riccati equations. To explain the theoretical results, we apply them to a population control problem.     

Stochastic linear quadratic optimal control problem with terminal state constraints

This paper addresses the stochastic linear quadratic (LQ) control problem with first- and second-order moment constraints on the terminal state. The problem is a modified version of the optimal covariance control problem, where the terminal state is steered to a given probability distribution. Studying a multiplicative-noise stochastic system rather than an additive-noise system is a salient feature. Unlike the existing ideas in the optimal steering, by using the Lagrange multipliers method and establishing the stochastic maximum principle, our problem is converted into solving forward–backward stochastic difference equations (FBSDEs), which is a special stochastic two-point boundary-value problem (TPBVP). We provide the optimal closed-loop controller and necessary and sufficient solvability conditions by developing a nonhomogeneous relationship between the optimal state and costate in FBSDEs. Finally, numerical examples are given to demonstrate our results.     

不确定拟哈密顿系统的随机最优控制

本文提出了不确定拟哈密顿系统、基于随机平均法、随机极大值原理和随机微分对策理论的一种随机极大极小最优控制策略.首先,运用拟哈密顿系统的随机平均法,将系统状态从速度和位移的快变量形式转化为能量的慢变量形式,得到部分平均的It随机微分方程;其次,给定控制性能指标,对于不确定拟哈密顿系统的随机最优控制,根据随机微分对策理论,将其转化为一个极小极大控制问题;再根据随机极大值原理,建立关于系统与伴随过程的前向-后向随机微分方程,随机最优控制表达为哈密顿控制函数的极大极小条件,由此得到最坏情形下的扰动参数与极大极小最优控制;然后,将最坏扰动参数与最优控制代入部分平均的It随机微分方程并完成平均,求解与完全平均的It随机微分方程相应的Fokker-Planck-Kolmogorov(FPK)方程,可得受控系统的响应量并计算控制效果;最后,将上述不确定拟哈密顿系统的随机最优控制策略应用于一个两自由度非线性系统,通过数值结果说明该随机极大极小控制策略的控制效果.     

有限时间二次型数值算法研究及其应用

为了实际需要和学术发展的要求,研究了以倒立摆为控制对象,通过闭环网络形成的反馈控制系统的随机传输时延的最优控制问题。在求解有限时间最优控制律过程中,通过矩阵Raccati方程的离散变换,利用Matlab中计算无限时间二次型最优控制器的LQR函数,从而求出有限时间LQR问题的数值解。通过仿真结果证明,研究的方法能够使倒立摆系统最终稳定,从而说明提出的算法对于求解有限时间LQR问题是有效的。     

正倒向随机微分方程与一类线性二次随机最优控制问题

讨论一类正倒向随机微分方程解的存在唯一性及其对应的一类线性二次随机最优控制问题,利用单调性方法证明了一类特殊的正倒向随机微分方程解的存在唯一性定理,利用该结果研究一类耦合了一个倒向随机微分方程的线性随机控制系统广义最优指标随机控制问题,得到由正倒向随机微分方程的解所表示的唯一最优控制的显式表达式,并得到精确的线性反馈及其对应的Riccati方程.     

Linear‐Quadratic Optimal Control Problems for Mean‐Field Stochastic Differential Equations with Jumps

In this paper, we study a linear‐quadratic optimal control problem for mean‐field stochastic differential equations driven by a Poisson random martingale measure and a one‐dimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean‐field forward‐backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying a decoupling technique, we establish the connection between two Riccati equations and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.