Solvability and Optimal Controls of Fractional Impulsive Stochastic Evolution Equations with Nonlocal Conditions

This paper deals with the solvability and optimal controls of a class of impulsive fractional stochastic evolution equations with nonlocal initial conditions in a Hilbert space. Firstly, the existence and uniqueness of mild solutions for the considered system are investigated. Then, we derive the existence conditions of optimal pairs to the control systems. In the end, an example is presented to illustrate the effectiveness of our abstract results.     

Optimal Solutions of Fractional Nonlinear Impulsive Neutral Stochastic Functional Integro-Differential Equations

Abstract

In this article, we consider a new class of fractional impulsive neutral stochastic functional integro-differential equations with infinite delay in Hilbert spaces. First, by using stochastic analysis, fractional calculus, analytic α-resolvent operator and suitable fixed point theorems, we prove the existence of mild solutions and optimal mild solutions for these equations. Second, the existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. The results are obtained under weaker conditions in the sense of the fractional power arguments. Finally, an example is given for demonstration.     

On a fractional impulsive partial stochastic integro-differential equation with state-dependent delay and optimal controls

This paper is mainly concerned with a new class of fractional impulsive partial stochastic integro-differential equations with state-dependent delay and optimal controls in Hilbert spaces. Firstly, a more appropriate concept for mild solutions is introduced. Secondly, existence and uniqueness of mild solutions are proved by means of stochastic analysis theory, fractional calculus and the fixed point technique combined with solution operator. The existence of optimal pairs of system governed by fractional impulsive partial stochastic integro-differential equations is also presented. Finally, an example is given for demonstration.     

Stepanov-like pseudo almost periodic solutions for impulsive perturbed partial stochastic differential equations and its optimal control

This paper is mainly concerned with the Stepanov-like pseudo almost periodicity to a class of impulsive perturbed partial stochastic differential equations. Firstly, we prove the existence of $p$-mean piecewise Stepanov-like pseudo almost periodic mild solutions for the impulsive stochastic dynamical system in a Hilbert space under non-Lipschitz conditions. The results are obtained by using the fixed point techniques with fractional power arguments. Then the existence of optimal pairs of system governed by impulsive partial stochastic differential equations is also obtained. Finally, an example is provided to illustrate the developed theory.     

Solvability and Optimal Controls of a Fractional Impulsive Stochastic Partial Integro-Differential Equation with State-Dependent Delay

In this paper, a new class of fractional impulsive stochastic partial integro-differential control systems with state-dependent delay and their optimal controls in a Hilbert space is studied. We firstly prove an existence result of mild solutions for the control systems by using stochastic analysis, analytic \(\alpha \)-resolvent operator, fractional powers of closed operators and suitable fixed point theorems. Then we derive the existence conditions of optimal pairs to the fractional impulsive stochastic control systems. Finally, an example is given to illustrate the effectiveness of our main results.     

Pseudo almost periodic in distribution solutions and optimal solutions to impulsive partial stochastic differential equations with infinite delay

In this paper, we study a general class of impulsive partial stochastic differential equations with infinite delay and pseudo almost periodic coefficients in Hilbert spaces. Firstly, a more appropriate concept of pseudo almost periodic in distribution for stochastic processes of infinite class is introduced. Secondly, the existence of pseudo almost periodic in distribution mild solutions is investigated by utilizing the interpolation theory, the stochastic analysis techniques and fixed point theorem. The existence of optimal mild solutions of the systems is also proved. Finally, an example is provided to show the effectiveness of the theoretical results.     

An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations

In this paper, we will study an indefinite stochastic linear quadratic optimal control problem, where the controlled system is described by a stochastic differential equation with delay. By introducing the relaxed compensator as a novel method, we obtain the well-posedness of this linear quadratic problem for indefinite case. And then, we discuss the uniqueness and existence of the solutions for a kind of anticipated forward–backward stochastic differential delayed equations. Based on this, we derive the solvability of the corresponding stochastic Hamiltonian systems, and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form. The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.     

Generalized Directional Gradients, Backward Stochastic Differential Equations and Mild Solutions of Semilinear Parabolic Equations

We study a forward-backward system of stochastic differential equations in an infinite-dimensional framework and its relationships with a semilinear parabolic differential equation on a Hilbert space, in the spirit of the approach of Pardoux-Peng. We prove that the stochastic system allows us to construct a unique solution of the parabolic equation in a suitable class of locally Lipschitz real functions. The parabolic equation is understood in a mild sense which requires the notion of a generalized directional gradient, that we introduce by a probabilistic approach and prove to exist for locally Lipschitz functions. The use of the generalized directional gradient allows us to cover various applications to option pricing problems and to optimal stochastic control problems (including control of delay equations and reaction--diffusion equations), where the lack of differentiability of the coefficients precludes differentiability of solutions to the associated parabolic equations of Black--Scholes or Hamilton-Jacobi-Bellman type.     

Maximum Principle for Partially-Observed Optimal Control of Fully-Coupled Forward-Backward Stochastic Systems

This paper is concerned with partially-observed optimal control problems for fully-coupled forward-backward stochastic systems. The maximum principle is obtained on the assumption that the forward diffusion coefficient does not contain the control variable and the control domain is not necessarily convex. By a classical spike variational method and a filtering technique, the related adjoint processes are characterized as solutions to forward-backward stochastic differential equations in finite-dimensional spaces. Then, our theoretical result is applied to study a partially-observed linear-quadratic optimal control problem for a fully-coupled forward-backward stochastic system and an explicit observable control variable is given.     

Optimal control of backward stochastic heat equation with Neumann boundary control and noise

This paper considers a stochastic control problem in which the dynamic system is a controlled backward stochastic heat equation with Neumann boundary control and boundary noise and the state must coincide with a given random vector at terminal time. Through defining a proper form of the mild solution for the state equation, the existence and uniqueness of the mild solution is given. As a main result, a global maximum principle for our control problem is presented. The main result is also applied to a backward linear-quadratic control problem in which an optimal control is obtained explicitly as a feedback of the solution to a forward–backward stochastic partial differential equation.     

Infinite Horizon Stochastic Optimal Control Problems with Degenerate Noise and Elliptic Equations in Hilbert Spaces

Semilinear elliptic partial differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. These results are applied to a stochastic optimal control problem with infinite horizon. Applications to controlled stochastic heat and wave equations are given.     

Viscosity Solutions for a System of Integro-PDEs and Connections to Optimal Switching and Control of Jump-Diffusion Processes

We develop a viscosity solution theory for a system of nonlinear degenerate parabolic integro-partial differential equations (IPDEs) related to stochastic optimal switching and control problems or stochastic games. In the case of stochastic optimal switching and control, we prove via dynamic programming methods that the value function is a viscosity solution of the IPDEs. In our setting the value functions or the solutions of the IPDEs are not smooth, so classical verification theorems do not apply.     

A stochastic approach for the optimization of open-loop engine control systems

The operation of sensors and actuators in engine control systems is always affected by errors, which are stochastic in nature. In this paper it is shown that, because of the non-linear interactions between engine performance and control laws in an open-loop engine control system, these errors can give rise to unexpected deviations of control variables, fuel consumption and emissions from the optimal values, which are not predictable in an elementary way.A model for vehicle performance evaluation on a driving cycle is presented, which provides the expected values of fuel consumption and emissions in the case of stochastic errors in sensors and actuators, utilizing only steady-state engine data.The stochastic model is utilized to obtain the optimal control laws; the resultant non-linear constrained minimization problem is solved by an Augmented Lagrangian approach, using a Quasi-Newton technique. The results of the stochastic optimization analysis indicate that significant reductions in performance degradation may be achieved with respect to the solutions provided by the classical deterministic approach.     

A Criterion for Stationary Solutions of Retarded Linear Equations with Additive Noise

In this work, we shall consider stationary (mild) solutions for a class of retarded functional linear differential equations with additive noise in Hilbert spaces. We first introduce a family of Green operators for the stochastic systems and establish stability results which will play an important role in the investigation of stationary solutions. A criterion imposed on the Green operators is presented to identify a unique stationary solution for the systems considered. Under strong quasi-Feller property, it is shown that this criterion is a sufficient and necessary condition to guarantee a unique stationary solution, based on a method having its origins in optimal control theory.     

Optimization of Nonlinear Systems of Stochastic Difference Equations

We present new results concerning the synthesis of optimal control for systems of difference equations that depend on a semi-Markov or Markov stochastic process. We obtain necessary conditions for the optimality of solutions that generalize known conditions for the optimality of deterministic systems of control. These necessary optimality conditions are obtained in the form convenient for the synthesis of optimal control. On the basis of Lyapunov stochastic functions, we obtain matrix difference equations of the Riccati type, the integration of which enables one to synthesize an optimal control. The results obtained generalize results obtained earlier for deterministic systems of difference equations.     

Large deviations for stochastic PDE with Lévy noise

We prove a large deviation principle result for solutions of abstract stochastic evolution equations perturbed by small Lévy noise. We use general large deviations theorems of Varadhan and Bryc coupled with the techniques of Feng and Kurtz (2006) [15], viscosity solutions of integro-partial differential equations in Hilbert spaces, and deterministic optimal control methods. The Laplace limit is identified as a viscosity solution of a Hamilton-Jacobi-Bellman equation of an associated control problem. We also establish exponential moment estimates for solutions of stochastic evolution equations driven by Lévy noise. General results are applied to stochastic hyperbolic equations perturbed by subordinated Wiener process.     

On stability for a class of semilinear stochastic evolution equations

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and using a limiting argument to pass on some properties of strong solution to our purposes. Several examples are studied to illustrate our theory. In particular, by means of the derived results we lose conditions of certain stochastic evolution systems from Haussmann (1978) to obtain the pathwise stability for mild solution with probability one.     

Semilinear Kolmogorov Equations and Applications to Stochastic Optimal Control

Semilinear parabolic differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. Applications to stochastic optimal control problems are studied by solving the associated Hamilton–Jacobi–Bellman equation. These results are applied to some controlled stochastic partial differential equations.     

The existence and uniqueness of the solution for nonlinear Kolmogorov equations

By means of backward stochastic differential equations, the existence and uniqueness of the mild solution are obtained for the nonlinear Kolmogorov equations associated with stochastic delay evolution equations. Applications to optimal control are also given.     

L\'evy过程驱动的正倒向随机系统的随机最大值原理

研究了由Teugels鞅和与之独立的多维Brown运动共同驱动的正倒向随机控制系统的最优控制问题. 这里Teugels鞅是一列与L\'{e}vy 过程相关的两两强正交的正态鞅 (见Nualart, Schoutens 在2000年的结果). 在允许控制值域为一非空凸闭集假设下, 采用凸变分法和对偶技术获得了最优控制存在所满足的充分和必要条件. 作为应用, 系统研究了线性正倒向随机系统的二次最优控制问题(简记为FBLQ问题), 通过相应的随机哈密顿系统对最优控制 进行了对偶刻画. 这里的随机哈密顿系统是由Teugels鞅和多维Brown运动共同驱动的线性正倒向随机微分方程, 其由状态方程、伴随方程和最优控制的对偶表示共同来构成.