Infinite horizon optimal control for mean-field stochastic delay systems driven by Teugels martingales under partial information

In this article, we discuss an infinite horizon optimal control of the stochastic system with partial information, where the state is governed by a mean-field stochastic differential delay equation driven by Teugels martingales associated with Lévy processes and an independent Brownian motion. First, we show the existence and uniqueness theorem for an infinite horizon mean-field anticipated backward stochastic differential equation driven by Teugels martingales. Then applying different approaches for the underlying system, we establish two classes of stochastic maximum principles, which include two necessary conditions and two sufficient conditions for optimality, under a convex control domain. Moreover, compared with the finite horizon optimal control, we add the transversality conditions to the two kinds of stochastic maximum principles. Finally, using the stochastic maximum principle II, we settle an infinite horizon optimal consumption problem driven by Teugels martingales associated with Gamma processes.     

Stochastic optimal control problem for switching systems with constraints

This paper provides necessary conditions of optimality, in the form of a maximum principle, for optimal control problems of switching systems. Dynamics of the constituent processes take the form of stochastic differential equations with control terms in the drift and diffusion coefficients. The restrictions on the transitions or switches between operating modes are described by the collection of functional equalities. The main result is proved via an approximation functional and Ekeland's variational principle. Copyright © 2016 John Wiley & Sons, Ltd.     

Relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions

This paper is concerned with the relationship between maximum principle and dynamic programming principle for stochastic recursive optimal control problems of jump diffusions. Under the assumption that the value function is smooth, relations among the adjoint processes, the generalized Hamiltonian function, and the value function are given. A linear quadratic recursive utility portfolio optimization problem in the financial market is discussed to show the applications of the main result. Copyright © 2012 John Wiley & Sons, Ltd.     

Partially observed linear quadratic control problem with delay via backward separation method

In this paper, we study a partially observed linear quadratic optimal control problem derived by stochastic differential delay equations. Combining backward separation method with stochastic filtering, we obtain optimal feedback regulators in some special cases. Some filtering results for anticipated backward stochastic differential equations are also developed by expressing the solutions of the anticipated backward stochastic differential equations as some Itô's processes. Copyright © 2016 John Wiley & Sons, Ltd.     

Stochastic linear quadratic optimal control problem for systems driven by fractional Brownian motions

This paper is concerned with stochastic linear control systems driven by fractional Brownian motions (fBms) with Hurst parameter H∈(1/2,1) and the cost functional is quadratic with respect to the state and control variables. Here, the integrals with respect to fBms are the type of Stratonovich integrals. A stochastic maximum principle as a necessary condition of the optimal control is derived. The adjoint backward stochastic differential equation (BSDE) is driven by the fBms and its underlying standard Brownian motions. The existence and uniqueness of the solution of adjoint BSDE is proved. The explicit form of the unique optimal control is obtained.     

Maximum principle for optimal control problem of stochastic delay differential equations driven by fractional Brownian motions

In this paper, we consider the optimal control problem for delayed stochastic differential equations driven by fractional Brownian motions. Some necessary Pontryagin's type conditions are derived by considering the adjoint equations satisfying an anticipated backward stochastic differential equation driven by both fractional Brownian motions and the standard Brownian motions. Some new results on stochastic analysis about the control systems driven by fractional Brownian motions are presented. As an application, a linear quadratic problem is deduced, and a numerical example is shown to prove the effectiveness of our method. Copyright © 2014 John Wiley & Sons, Ltd.     

Fully coupled mean‐field FBSDEs with jumps and related optimal control problems

In this article, we study a type of fully coupled mean‐field forward‐backward stochastic differential equations with jumps under the monotonicity condition, including the existence and the uniqueness of the solution of our equations as well as the continuity property of the solutions with respect to the parameters. Then we establish the stochastic maximum principle for the corresponding optimal control problems and give the applications to the mean‐variance portfolio problem and linear‐quadratic problem, respectively.     

Maximum principle for optimal control of McKean‐Vlasov FBSDEs with Lévy process via the differentiability with respect to probability law

In this paper, we study stochastic optimal control problem for general McKean‐Vlasov–type forward‐backward differential equations driven by Teugels martingales, associated with some Lévy process having moments of all orders, and an independent Brownian motion. The coefficients of the system depend on the state of the solution process as well as of its probability law and the control variable. We establish a set of necessary conditions in the form of Pontryagin maximum principle for the optimal control. We also give additional conditions, under which the necessary optimality conditions turn out to be sufficient. The proof of our main result is based on the differentiability with respect to probability law and a corresponding Itô formula.     

The deterministic maximum principle for differential systems with a general cost functional

In this study, a deterministic optimal control problem is investigated in which the system is governed by an ordinary differential equation with a general cost functional. In the framework of Fréchet derivatives, we establish the maximum principle with the Hamilton systems for this optimal control problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal control of nonzero sum game mean‐field delayed Markov regime‐switching forward‐backward system with Lévy processes

This article investigates the optimal control problem of nonzero sum game mean‐field delayed Markov regime‐switching forward‐backward stochastic system with Lévy processes associated with Teugels martingales over the infinite time horizon. Based on the transversality conditions, assumption of convex control domain, infinite‐horizon version of stochastic maximum principle (Nash equilibrium), and necessary condition for optimality are established. Finally, the Nash equilibrium for the optimization problem in the financial market is considered to illustrate the observed theoretical results.     

Synthesis of optimal control in a mathematical model of tumour–immune dynamics

An optimal control problem for a mathematical model of tumour–immune dynamics under the influence of chemotherapy is considered. The toxicity effect of the chemotherapeutic agent on both tumour and immunocompetent cells is taken into account. A standard linear pharmacokinetic equation for the chemotherapeutic agent is added to the system. The aim is to find an optimal strategy of treatment to minimize the tumour volume while keeping the immune response not lower than a fixed permissible level as far as possible. Sufficient conditions for the existence of not more than one switching and not more than two switchings without singular regimes are obtained. The surfaces in the extended phase space, on which the last switching appears, are constructed analytically.Copyright © 2013 John Wiley & Sons, Ltd.     

On optimal singular control problem for general Mckean‐Vlasov differential equations: Necessary and sufficient optimality conditions

In this paper, we derive the necessary and sufficient conditions for optimal singular control for systems governed by general controlled McKean‐Vlasov differential equations, in which the coefficients depend on the state of the solution process as well as of its law and control. The control domain is assumed to be convex. The control variable has 2 components, ie, the first being absolutely continuous and the second being singular. The proof of our result is based on the derivative of the solution process with respect to the probability law and a corresponding Itô formula. Finally, an example is given to illustrate the theoretical results.     

A gradient algorithm for solution of the optimal control problem for hybrid switching systems

In this article, an algorithm is presented for solving the optimal control problem for the general form of a hybrid switching system. The cost function comprises terminal, running and switching costs. The controlled system is an autonomous hybrid switching system with jumps either at some switching times or some time varying switching manifolds. The proposed algorithm is an extension of the first-order gradient method for the conventional optimal control problem. The algorithm requires a low computational effort. The system's dynamical equations together with a set of algebraic equations are solved at each iteration in order to find the descent direction. The convergence of algorithm is proved and examples are provided to demonstrate the efficiency of the algorithm for different types of hybrid switching system optimal control problems.     

Determination of a controllable set for a class of non‐linear stochastic control systems

A controllable set of a class of non‐linear stochastic control systems to a given set is defined. A value function associated with an optimal control problem is introduced. Under some reasonable conditions, the controllable set is characterized by a level set of the viscosity solution of a Hamilton–Jacobi–Bellman equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Robust H∞ control for stochastic time‐delayed Markovian switching systems under partly known transition rates and actuator saturation via anti‐windup design

The paper deals with the problem of robust H_∞ control for stochastic time‐delayed Markovian switching systems under partly known transition rates and actuator saturation via anti‐windup design. The problem we address is the design of anti‐windup compensators, which guarantee that the resulting closed‐loop system is robustly stochastically stable with H_∞ performance. By employing local sector conditions and an appropriate Lyapunov–Krasovskii function, sufficient conditions for solving the problem are derived in the form of linear matrix inequalities. Finally, numerical examples are given to demonstrate the validity of the main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Inverse optimal noise‐to‐state stabilization of stochastic recurrent neural networks driven by noise of unknown covariance

In this paper, we extend our previous research results regarding the stabilization of recurrent neural networks from the concept of input‐to‐state stability to noise‐to‐state stability, and present a new approach to achieve noise‐to‐state stabilization in probability for stochastic recurrent neural networks driven by the noise of unknown covariance. This approach is developed by using the Lyapunov technique, inverse optimality, differential game theory, and the Hamilton–Jacobi–Isaacs equation. Numerical examples demonstrate the effectiveness of the proposed approach. Copyright © 2008 John Wiley & Sons, Ltd.     

Time optimal controls of the Cahn–Hilliard equation with internal control

This paper is concerned with the time optimal control problem governed by the internal controlled Cahn–Hilliard equation. We prove the existence of optimal controls. Moreover, we give necessary optimality conditions for an optimal control of our original problem by using the one of the approximate problems. Copyright © 2014 John Wiley & Sons, Ltd.     

Design of robust‐optimal controllers with low trajectory sensitivity for uncertain Takagi–Sugeno fuzzy model systems using differential evolution algorithm

By integrating the robust stabilizability condition, the orthogonal‐function approach (OFA) and the Taguchi‐sliding‐based differential evolution algorithm (TSBDEA), an integrative computational approach is presented in this paper to design the robust‐optimal fuzzy parallel‐distributed‐compensation (PDC) controller with low trajectory sensitivity such that (i) the Takagi–Sugeno (TS) fuzzy model system with parametric uncertainties can be robustly stabilized, and (ii) a quadratic finite‐horizon integral performance index for the nominal TS fuzzy model system can be minimized. In this paper, the robust stabilizability condition is proposed in terms of linear matrix inequalities (LMIs). Based on the OFA, an algebraic algorithm only involving the algebraic computation is derived for solving the nominal TS fuzzy feedback dynamic equations. By using the OFA and the LMI‐based robust stabilizability condition, the robust‐optimal fuzzy PDC control problem for the uncertain TS fuzzy dynamic systems is transformed into a static constrained‐optimization problem represented by the algebraic equations with constraint of LMI‐based robust stabilizability condition; thus, greatly simplifying the robust‐optimal PDC control design problem. Then, for the static constrained‐optimization problem, the TSBDEA has been employed to find the robust‐optimal PDC controllers with low trajectory sensitivity of the uncertain TS fuzzy model systems. A design example is given to demonstrate the applicability of the proposed new integrative approach. Copyright © 2011 John Wiley & Sons, Ltd.     

High‐order maximum principles for the stability analysis of positive bilinear control systems

We consider a continuous‐time positive bilinear control system, which is a bilinear control system with Metzler matrices. The positive orthant is an invariant set of such a system, and the corresponding transition matrix is entrywise nonnegative for all time. Motivated by the stability analysis of positive linear switched systems under arbitrary switching laws, we define a control as optimal if it maximizes the spectral radius of the transition matrix at a given final time. We derive high‐order necessary conditions for optimality for both singular and bang–bang controls. Our approach is based on combining results on the second‐order derivative of a simple eigenvalue with the generalized Legendre‐Clebsch condition and the Agrachev–Gamkrelidze second‐order optimality condition. Copyright © 2015 John Wiley & Sons, Ltd.     

L2‐L∞ filtering for a class of stochastic nonlinear systems with aperiodic sampling

This work investigates the problem of L₂‐L_∞ filtering for a class of stochastic nonlinear systems with nonuniform sampling. The sampled‐data filter developed in this paper is an impulsive differential system whose states change abruptly at every sampling instant. The resulting filtering error system is modeled as a stochastic nonlinear impulsive differential system. The goal is to propose a method for designing a target filter that ensures the stochastic asymptotic stability of the filtering error system and guarantees a prescribed L₂‐L_∞ performance. Based on a time‐varying Lyapunov functional, by virtue of a convex combination technique, a design method to achieve such a filter is formulated in the form of solving a set of linear matrix inequalities. The effectiveness of the proposed filtering strategy is shown via a numerical example of a stochastic Chua's circuit system.