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Summary: LaSalle theorem (also known as the LaSalle invariance principle) plays an essential role in the systems and control theory. Recently, it has been extensively studied and developed for various types of one‐dimensional (1‐D) systems including deterministic and stochastic 1‐D systems in discrete‐ and continuous‐time domains. For two‐dimensional (2‐D) systems, such studies have received considerably less attention. In this article, based on discrete martingale theory, a LaSalle‐type theorem is first developed for a class of discrete‐time nonlinear stochastic 2‐D systems described by a Roesser model. The proposed result can be regarded as an extension of stochastic Lyapunov‐like theorem, which guarantees the convergence almost surely of system state trajectories. Extensions to the problem of optimal guaranteed cost control of nonlinear stochastic 2‐D systems are also presented. The proposed schemes are then utilized to derive tractable synthesis conditions of a suboptimal state‐feedback controller for uncertain 2‐D systems with multiplicative stochastic noises. The effectiveness of the obtained results is illustrated by given numerical examples and simulations. [ABSTRACT FROM AUTHOR]

This paper is concerned with the problems of almost sure ultimate bound estimation and controller design for Markov jump linear systems with bounded stochastic disturbances. By utilising a Lyapunov-based scheme proposed in this paper, an almost sure estimation of ellipsoidal ultimate bound (EUB) of the system is obtained through tractable matrix inequalities. On the basis of the estimation results, the problem of designing mode-dependent state feedback controllers that make the closed-loop system admit a prescribed ellipsoid as an EUB is considered. The obtained results on estimation and synthesis are then extended to the case of systems with deficient mode information. Finally, a practical example in DC motor devices is presented to demonstrate the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, the problem of reachable set estimation of two-dimensional (2-D) discrete-time systems described by the Roesser model with interval time-varying delays is considered for the first time. New 2-D weighted summation inequalities, which provide a tighter lower bound than the commonly used Jensen summation inequality, are proposed. Based on the Lyapunov-Krasovskii functional approach, and by using the 2-D weighted summation inequalities presented in this paper, new delay-dependent conditions are derived to ensure the existence of an ellipsoid that bounds the system states in the presence of bounded disturbances. The derived conditions are expressed in terms of linear matrix inequalities, which can be solved by various computational tools to determine a smallest possible ellipsoidal bound. Applications to exponential stability analysis of 2-D systems with delays are also presented. The effectiveness of the obtained results are illustrated by numerical examples. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, the problem of finite-time stability of linear nonautonomous systems with time-varying delays is considered. Using a novel approach based on some techniques developed for linear positive systems, we derive new explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. These conditions are shown to be relaxed for the Lyapunov asymptotic stability. A numerical example is given to illustrate the effectiveness of the obtained result. [Copyright &y& Elsevier]

This article presents new exponential stability and stabilisation conditions for Takagi-Sugeno fuzzy time-varying delay systems. Based on an improved Lyapunov-Krasovskii functional, the exponential stability and stabilisation conditions are derived in terms of linear matrix inequalities, which allows one to compute simultaneously the two bounds that characterise the exponential stability rate of the solution. Finally, some numerical examples are given to illustrate the effectiveness of the proposed conditions. [ABSTRACT FROM AUTHOR]