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This paper mainly focuses on discrete‐time Markov jump linear stochastic systems (DMJLSSs) with external disturbances and multiplicative noise. Due to the multi‐player in the systems and multi‐objective cost function, the strategy design for these systems is more difficult than the traditional Markov jump stochastic systems. We combine H∞$$ {H}_{\infty } $$ theory with Pareto control and use the weighted sum method of the objective functional of each player to solve the problem of Pareto cooperative game. Four coupled difference matrix‐valued recursion equations (CDMREs) are derived by stochastic bounded real lemma (SBRL). Then, a sufficient and necessary condition for the existence of the Pareto optimal strategy is obtained, and the Pareto solution for every controller is derived under this strategy. A simulation example is given to verify the correctness of the theory. [ABSTRACT FROM AUTHOR]

This paper investigates the Pareto optimal strategy of discrete-time stochastic systems under H ∞ constraint, in which the weighting matrices of the weighted sum cost function can be indefinite. Combining the H ∞ control theory with the indefinite LQ control theory, the generalized difference Riccati equations (GDREs) are obtained. By means of the solution of the GDREs, the Pareto optimal strategy with H ∞ constraint is derived, and the necessary and sufficient conditions for the existence of the strategy are presented. Then the Pareto optimal solution under the worst-case disturbance is solved. Finally, the efficiency of the obtained results is illustrated by a numerical example. [ABSTRACT FROM AUTHOR]

This article presents results on designing the Pareto-optimal strategy under $H_{\infty }$ constraint for the linear mean-field stochastic systems disturbed by external disturbances. First, combining the stochastic $H_{\infty }$ control theory with the stochastic mean-field theory, we derive the stochastic bounded real lemma (SBRL) of our considered linear mean-field stochastic systems with the stochastic initial condition. Second, we use the mean-field forward–backward stochastic differential equation to solve the mean-field linear quadratic Pareto-optimal problem with indefinite cost functionals. It is proved that the existence of a closed-loop Pareto-optimal strategy is equivalent to the solvability of the coupled generalized differential Riccati equations when some conditions are satisfied. Finally, a necessary and sufficient condition for the Pareto-optimal strategy under the $H_{\infty }$ constraint is researched by four-coupled matrix-valued equations. Besides, we also obtain the Pareto frontier for the mean-field stochastic system with only state-dependent noise. A practical example is presented to show the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

In this paper, the issues about weighted H ∞ performance analysis and H ∞ control for the stochastic switched nonlinear systems (SSNSs) with multiplicative noise are investigated. The mode-dependent average dwell time (MDADT) method is used to deal with the switching in different modes. Firstly, we get a sufficient condition to show that the considered system with all stable subsystems achieves the exponential stability in mean square sense and a specified weighted H ∞ performance, which is extended to the case that both stable and unstable subsystems coexist in terms of second-order Hamilton-Jacobi inequalities (HJIs). Then, by using Takagi-Sugeno (T-S) fuzzy approach, the H ∞ controller for SSNSs is designed via solving a set of linear matrix inequalities (LMIs). Finally, an example is supplied to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

Summary: This paper mainly studies the locally/globally asymptotic stability and stabilization in probability for nonlinear discrete‐time stochastic systems. Firstly, for more general stochastic difference systems, two very useful results on locally and globally asymptotic stability in probability are obtained, which can be viewed as the discrete versions of continuous‐time Itô systems. Then, for a class of quasi‐linear discrete‐time stochastic control systems, both state‐ and output‐feedback asymptotic stabilization are studied, for which, sufficient conditions are presented in terms of linear matrix inequalities. Two simulation examples are given to illustrate the effectiveness of our main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with the necessary and sufficient conditions for the Pareto optimality in the finite horizon stochastic cooperative differential game. Based on the necessary and sufficient characterization of the Pareto optimality, the problem is transformed into a set of constrained stochastic optimal control problems with a special structure. Utilizing the stochastic Pontryagin minimum principle, the necessary conditions for the existence of the Pareto solutions are put forward. Under certain convex assumptions, it is shown that the necessary conditions are also sufficient ones. Next, we study the indefinite linear quadratic (LQ) case. It is pointed out that the solvability of the related generalized differential Riccati equation (GDRE) provides the sufficient condition under which all Pareto efficient strategies can be obtained by the weighted sum optimality method. Two examples shed light on the effectiveness of theoretical results. [ABSTRACT FROM AUTHOR]

This paper considers quadratic stabilizability and H ∞ feedback control for stochastic discrete-time uncertain systems with state- and control-dependent noise. Specifically, the uncertain parameters considered are norm-bounded and external disturbance is an l2-square summable stochastic process. Firstly, both quadratic stability and quadratic stabilization criteria are presented in the form of linear matrix inequalities (LMIs). Then we design the robust H ∞ state and output feedback H ∞ controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H ∞ controllable. Sufficient conditions for the existence of the desired robust H ∞ controllers are obtained via LMIs. Finally, some examples are supplied to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

This paper mainly discusses the robust quadratic stability and stabilization of linear discrete-time stochastic systems with state delay and uncertain parameters. By means of the linear matrix inequality (LMI) method, a sufficient condition is, respectively, obtained for the stability and stabilizability of the considered system. Moreover, we design the robust H∞ state feedback controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H∞ controllable. A sufficient condition for the existence of the desired robust H∞ controller is obtained. Finally, an example with simulations is given to verify the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

This paper studies robust stability, stabilization, and H∞ control for a class of nonlinear discrete time stochastic systems. Firstly, the easily testing criteria for stochastic stability and stochastic stabilizability are obtained via linear matrix inequalities (LMIs). Then a robust H∞ state feedback controller is designed such that the concerned system not only is internally stochastically stabilizable but also satisfies robust H∞ performance. Moreover, the previous results of the nonlinearly perturbed discrete stochastic system are generalized to the system with state, control, and external disturbance dependent noise simultaneously. Two numerical examples are given to illustrate the effectiveness of the proposed results. [ABSTRACT FROM AUTHOR]