Finite-time stability and stabilization of nonlinear stochastic hybrid systems

This paper deals with the problem of finite-time stability and stabilization of nonlinear Markovian switching stochastic systems which exist impulses at the switching instants. Using multiple Lyapunov function theory, a sufficient condition is established for finite-time stability of the underlying systems. Furthermore, based on the state partition of continuous parts of systems, a feedback controller is designed such that the corresponding impulsive stochastic closed-loop systems are finite-time stochastically stable. A numerical example is presented to illustrate the effectiveness of the proposed method.     

A Notion of Stochastic Input-to-State Stability and Its Application to Stability of Cascaded Stochastic Nonlinear Systems

In this paper, the property of practical input-to-state stability and its application to stability of cascaded nonlinear systems are investigated in the stochastic framework. Firstly, the notion of （practical） stochastic input-to-state stability with respect to a stochastic input is introduced, and then by the method of changing supply functions, （a） an （practical） SISS-Lyapunov function for the overall system is obtained from the corresponding Lyapunov functions for cascaded （practical） SISS subsystems.     

A survey of results and perspectives on stabilization of switched nonlinear systems with unstable modes

This paper surveys the recent theoretical results on the stabilization of switched nonlinear systems with unstable modes. Two cases are considered. (1) Some modes are stable, and others may be unstable. The stabilization can be achieved via the trade-off among stable modes and unstable ones. (2) All modes may be unstable. The stabilization can also be achieved via the trade-off among the potentially stable parts of all modes, or with the help of the jump dynamics at switching instants. The practical usefulness is illustrated by several applications including supervisory control, fault tolerant control, multi-agent systems, and networked control systems. Some perspectives are also provided.     

On delay-dependent stability for a class of nonlinear stochastic systems with multiple state delays

Global asymptotic stability conditions for nonlinear stochastic systems with multiple state delays are obtained based on the convergence theorem for semimartingale inequalities, without assuming the Lipschitz conditions for nonlinear drift functions. The Lyapunov–Krasovskii and degenerate functionals techniques are used. The derived stability conditions are directly expressed in terms of the system coefficients. Nontrivial examples of nonlinear systems satisfying the obtained stability conditions are given.     

Global stabilization of a certain class of nonlinear dynamical systems using state detection

This note is concerned with the stabilization of control systems using an estimated state feedback. The global stabilization problem for a relatively broad class of nonlinear plants is discussed. Moreover, using the “input to state stability” property introduced by Sontag [1–4] and detectability condition, we show that the system can be globally asymptotically stable using a state detection.     

Orbital stability index for stochastic systems

The Known concepts of Lyapunov exponent, moment Lyapunov exponents, and stability index for stationary points of stochastic systems are carried over for invariant orbits with nonvanishing diffusion. The obtained geneal results are applied to investiating stochastic stability and stabilization of orbits on the plane. These questions are considered under small diffusion as well.     

Stabilization of a class of nonlinear composite stochastic systems

This paper deals with the stability for a class of nonlinear composite stochastic systems by feedback laws.Firstly,we give sufficient conditions for the existence of feedback laws which render the equilibrium solution of the stochastic system globally asymptotically stable in probability.Secondly,for stochastic systems of the same type,we prove that there exists a linear feedback law which exponentially stabilizes in mean square the closed–loop stochastic system at its equilibrium.     

Global fixed-time stabilization for a class of switched nonlinear systems with general powers and its application

This paper investigates the problem of global fixed-time stabilization for a class of uncertain switched nonlinear systems with the general powers, namely, the powers of the considered systems can be different odd rational numbers, even rational numbers or both odd and even rational numbers. By skillfully using the common Lyapunov function method and the adding a power integrator technique, a common state feedback control strategy is developed. It is proved that the proposed controller can guarantee that the state of the resulting closed-loop system converges to zero for any given fixed time under arbitrary switchings. Simulation results of the liquid-level system are provided to show the effectiveness of the proposed method.     

Stabilization of stochastic singular nonlinear hybrid systems

This paper deals with the control of the class of singular nonlinear stochastic hybrid systems. Under some appropriate assumptions, results on stochastic stability and stochastic stabilization are developed. Two state feedback controllers (linear and nonlinear) that stochastically stabilize the class of systems we are considering are designed. LMI sufficient conditions are developed to compute the gains of these controllers.     

Robust reliable stabilization of stochastic switched nonlinear systems under asynchronous switching

This paper is concerned with the problem of robust reliable control for a class of uncertain stochastic switched nonlinear systems under asynchronous switching, where the switching instants of the controller experience delays with respect to those of the system. A design scheme for the reliable controller is proposed to guarantee almost surely exponential stability for stochastic switched systems with actuator failures, and the dwell time approach is utilized for the stability analysis. Then the approach is extended to take into account stochastic switched system with Lipschitz nonlinearities and structured uncertainties. Finally, a numerical example is employed to verify the proposed method.     

Global finite-time stabilization of switched high-order rational power nonlinear systems

This works is concerned with the finite-time optimal stabilization problem for a class of switched non-strict-feedback nonlinear systems whose powers are possibly different positive odd rational numbers in the sense the powers of each subsystem might differ from others. It is well known that high-order nonlinear systems are intrinsically challenging as feedback linearization and backstepping method successfully developed for low-order systems fail to work. To this purpose, the nested saturation homogeneous controller is constructively devised to achieve global finite-time stability. Furthermore, the corresponding design parameters are optimized by minimizing a well-defined cost function, and thus an optimal controller being independent of switching signals is obtained. Simulation results are eventually provided to validate the effectiveness of the proposed control scheme.     

On stability of switched homogeneous nonlinear systems

In this paper, the problem of stability of switched homogeneous systems is addressed. First of all, if there is a quadratic Lyapunov function such that nonlinear homogeneous systems are asymptotically stable, a matrix Lyapunov-like equation is obtained for a stable nonlinear homogeneous system using semi-tensor product of matrices, and Lyapunov equation of linear system is just its particular case. Following the previous results, a sufficient condition is obtained for stability of switched nonlinear homogeneous systems, and a switching law is designed by partition of state space. In particular, a constructive approach is provided to avoid chattering phenomena which is caused by the switching rule. Then for planar switched homogeneous systems, an LMI approach to stability of planar switched homogeneous systems is presented. Similar to the condition for linear systems, the LMI-type condition is easily verifiable. An example is given to illustrate that candidate common Lyapunov function is a key point for design of switching law.     

On the uniform exponential stability of a wide class of linear time-delay systems

This paper deals with the global uniform exponential stability independent of delay of time-delay linear and time-invariant systems subject to point and distributed delays for the initial conditions being continuous real functions except possibly on a set of zero measure of bounded discontinuities. It is assumed that the delay-free system as well as an auxiliary one are globally uniformly exponentially stable and globally uniform exponential stability independent of delay, respectively. The auxiliary system is typically a part of the overall dynamics of the delayed system but not necessarily the isolated undelayed dynamics as usually assumed in the literature. Since there is a great freedom in setting such an auxiliary system, the obtained stability conditions are very useful in a wide class of practical applications.     

Global asymptotic stability of a stochastic Lotka-Volterra model with infinite delays

In this paper, sufficient criteria for global asymptotic stability of a general stochastic Lotka-Volterra system with infinite delays are established. Some simulation figures are introduced to support the analytical findings.     

On almost sure stability conditions of linear switching stochastic differential systems

In modeling practical systems, it can be efficient to apply Poisson process and Wiener process to represent the abrupt changes and the environmental noise, respectively. Therefore, we consider the systems affected by these random processes and investigate their joint effects on stability. In order to apply Lyapunov stability method, we formulate the action of the infinitesimal generator corresponding to such a system. Then, we derive the almost sure stability conditions by using some fundamental convergence theorem. To illustrate the theoretical results, we construct an example to show that it is possible to achieve stabilization by using random perturbations.     

Complete controllability of nonlinear stochastic impulsive functional systems

This paper is concerned with the controllability of a kind of nonlinear stochastic impulsive functional systems. Sufficient conditions for the complete controllability of the nonlinear stochastic impulsive functional systems are obtained by using Schauder fixed-point theorem. A numerical example is given to illustrate the effectiveness of the theoretical results.     

Uniform exponential stabilization of nonlinear systems in Banach spaces

This paper presents necessary and sufficient conditions for uniform exponential stabilization of a class of nonlinear systems in Banach state spaces. The stabilization assumptions are formulated in terms of integral estimates involving the control operator and the state of the uncontrolled version of the system at hand. An explicit estimate of the convergence speed is given. Applications to feedback stabilization of affine control systems are given. Illustrative examples are further provided.     

Improved results on robust stability of neutral systems with mixed time-varying delays and nonlinear perturbations

This paper considers the problem of robust stability of neutral systems with mixed time-varying delays and nonlinear perturbations. Two type uncertainties such as nonlinear time-varying parameter perturbations and norm-bounded uncertainties have been discussed. Based on the new Lyapunov–Krasovskii functional with triple integral terms, some integral inequalities and convex combination technique, a new delay-dependent stability criterion for the system is established in terms of linear matrix inequalities (LMIs). Finally, four numerical examples are given to illustrate the effectiveness and an improvement over some existing results in the literature with the proposed results.     

The stability and stabilization of heat equation in non-cylindrical domain

This paper is addressed to a study of the stability and stabilization of heat equation in non-cylindrical domain. Special solutions of the system are first given by the method of the undetermined function and the similarity variables, which indicate that the system is not exponentially stable. Then the stability and stabilization of the system are obtained by the energy estimate and the backstepping method.