Incremental stability analysis of stochastic hybrid systems

In this paper, we study the incremental stability of stochastic hybrid systems, based on the contraction theory, and derive sufficient conditions of global stability for such systems. As a special case, the conditions to ensure the second moment exponential stability which is also called exponential stability in the mean square of stochastic hybrid systems are obtained. The theoretical results in this paper extend previous works from deterministic or stochastic systems to general stochastic hybrid systems, which can be applied to qualitative and quantitative analysis of many physical and biological phenomena. An illustrative example is given to show the effectiveness of our results.     

On the stabilization of interconnected stochastic systems

The aim of this work is to prove that if the equilibrium solution of a nonlinear control stochastic system is locally asymptotically stable in probability by means of a continuous state feedback law, then the resulting stochastic system obtained by adding an integrator is also locally asymptotically stable in probability by means of a smooth, except possibly at the equilibrium solution, state feedback law. This result extends to the stabilization of stochastic systems a result proved by Tsinias [9] for deterministic systems. In our proof, we make use of the stochastic version of Artstein's theorem established in [4]     

State constrained reachability for stochastic hybrid systems

Many control problems can be formulated as driving a system to reach some target states while avoiding some unwanted states. We study this problem for systems with regime change operating in uncertain environments. Nowadays, it is a common practice to model such systems in the framework of stochastic hybrid system models. In this casting, the problem is formalized as a mathematical problem named state constrained stochastic reachability analysis. In the state constrained stochastic reachability analysis, this probability is computed by imposing a constraint on the system to avoid the unwanted states. The scope of this paper is twofold. First we define and investigate the state constrained reachability analysis in an abstract mathematical setting. We define the problem for a general model of stochastic hybrid systems, and we show that the reach probabilities can be computed as solutions of an elliptic integro-differential equation. Moreover, we extend the problem by considering randomized targets. We approach this extension using stochastic dynamic programming. The second scope is to define a developmental setting in which the state constrained reachability analysis becomes more tractable. This framework is based on multilayer modelling of a stochastic system using hierarchical viewpoints. Viewpoints represent a method originated from software engineering, where a system is described by multiple models created from different perspectives. Using viewpoints, the reach probabilities can be easily computed, or even symbolically calculated. The reach probabilities computed in one viewpoint can be used in another viewpoint for improving the system control. We illustrate this technique for trajectory design.     

Solving approximately a prediction problem for stochastic jump-diffusion systems

In this paper, a new approach to solving a prediction problem for nonlinear stochastic differential systems with a Poisson component is discussed. In this approach, the prediction problem is reduced to an analysis of stochastic jump-diffusion systems with terminating and branching paths. The prediction problem can be approximately solved by using numerical methods for stochastic differential equations and methods for modeling inhomogeneous Poisson flows.     

Approximate boundary controllability of Sobolev-type stochastic differential systems

The objective of this paper is to investigate the approximate boundary controllability of Sobolev-type stochastic differential systems in Hilbert spaces. The control function for this system is suitably constructed by using the infinite dimensional controllability operator. Sufficient conditions for approximate boundary controllability of the proposed problem in Hilbert space is established by using contraction mapping principle and stochastic analysis techniques. The obtained results are extended to stochastic differential systems with Poisson jumps. Finally, an example is provided which illustrates the main results.     

Finite dimensional Markov process approximation for stochastic time-delayed dynamical systems

This paper presents a method of finite dimensional Markov process (FDMP) approximation for stochastic dynamical systems with time delay. The FDMP method preserves the standard state space format of the system, and allows us to apply all the existing methods and theories for analysis and control of stochastic dynamical systems. The paper presents the theoretical framework for stochastic dynamical systems with time delay based on the FDMP method, including the FPK equation, backward Kolmogorov equation, and reliability formulation. A simple one-dimensional stochastic system is used to demonstrate the method and the theory. The work of this paper opens a door to various studies of stochastic dynamical systems with time delay.     

On nonsmooth and discontinuous problems of stochastic systems optimization

A class of stochastic optimization problems is analyzed that cannot be solved by deterministic and standard stochastic approximation methods. We consider risk-control problems, optimization of stochastic networks and discrete event systems, screening irreversible changes, and pollution control. The results of Ermoliev et al. are extended to the case of stochastic systems and general constraints. It is shown that the concept of stochastic mollifier gradient leads to easily implementable computational procedures for systems with Lipschitz and discontinuous objective functions. New optimality conditions are formulated for designing stochastic search procedures for constrained optimization of discontinuous systems.     

Global stability and stabilization of more general stochastic nonlinear systems

This paper considers the global stability and stabilization of more general stochastic nonlinear systems. Due to the absence of the conventional assumptions (e.g., Lipschitz condition), the stochastic nonlinear systems under investigation may have more than one weak solution. However, the most associated results are only applicable to the stochastic systems having a unique strong solution, and therefore, it is meaningful to refine and extend the relevant concepts and methods to the more general case. In this paper, the concepts of stochastic stability in the more general sense are first introduced to cover the stochastic nonlinear systems having more than one weak solution. Then, the generalized stochastic Barbashin–Krasovskii theorem and LaSalle theorem are established, which present the criterions of stochastic stability for more general stochastic nonlinear systems. As one of the main contributions in this paper, we rigorously prove the generalized stochastic Barbashin–Krasovskii theorem. Moreover, based on the generalized theorems, the output-feedback and state-feedback stabilization are accomplished respectively for two classes of high-order stochastic nonlinear systems under rather weaker assumptions comparing to the existing literature.     

Constrained stochastic controllability of infinite-dimensional linear systems

The main purpose of this article is to investigate the problem of (ε, δ)-stochastic controllability for linear systems of evolution type in infinite-dimensional spaces, wherein the controls are subjected to norm-bounded constrained sets. Some basic prerequisites of infinite-dimensional measures, in particular, Gaussian distributed type, are discussed. Corresponding to this measure, various properties of (ε, δ)-stochastic attainable sets in Hilbert spaces are studied. Necessary and sufficient conditions for (ε, δ)-stochastic controllability with respect to Hilbert space valued linear systems are obtained. Relationships with the deterministic counterpoint are noted. Pursuit game problems are also considered. Examples on systems governed by stochastic linear partial differential equations and stochastic differential delay equations are given for illustration.     

Controllability of stochastic Volterra integrodifferential systems

In this paper, sufficient conditions for the controllability of stochastic integrodifferential systems in Banach spaces are established. The results are obtained by using a fixed point theorem. An example is provided to illustrate the theory.     

Almost sure exponential stability of stochastic reaction diffusion systems

The Lyapunov direct method, as the most effective measure of studying stability theory for ordinary differential systems and stochastic ordinary differential systems, has not been generalized to research concerning stochastic partial differential systems owing to the emptiness of the corresponding Ito differential formula. The goal of this paper is just employing the Lyapunov direct method to investigate the stability of Ito stochastic reaction diffusion systems, including asymptotical stability in probability and almost sure exponential stability. The obtained results extend the conclusions of [X.X. Liao, X.R. Mao, Exponential stability and instability of stochastic neural networks, Stochastic Analysis and Applications 14 (2) (1996) 165-185; X.X. Liao, S.Z. Yang, S.J. Cheng, Y.L. Fu, Stability of general neural networks with reaction-diffusion, Science in China (F) 44 (5) (2001) 389-395].     

Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks

This paper is concerned with the stability of n-dimensional stochastic differential delay systems with nonlinear impulsive effects. First, the equivalent relation between the solution of the n-dimensional stochastic differential delay system with nonlinear impulsive effects and that of a corresponding n-dimensional stochastic differential delay system without impulsive effects is established. Then, some stability criteria for the n-dimensional stochastic differential delay systems with nonlinear impulsive effects are obtained. Finally, the stability criteria are applied to uncertain impulsive stochastic neural networks with time-varying delay. The results show that, this convenient and efficient method will provide a new approach to study the stability of impulsive stochastic neural networks. Some examples are also discussed to illustrate the effectiveness of our theoretical results.     

Stability analysis and optimal control of stochastic singular systems

In this paper, problems of stability and optimal control for a class of stochastic singular systems are studied. Firstly, under some appropriate assumptions, some new results about mean-square admissibility are developed and the corresponding LMI sufficient condition is given. Secondly, finite-time horizon and infinite-time horizon linear quadratic (LQ) control problems for the stochastic singular system are investigated, in which the coefficients are allowed to be random in control input and quadratic criterion. Some results involving new stochastic generalized Riccati equation are discussed as well. Finally, the proposed LQ control model for stochastic singular systems provides an appropriate and effective framework to study the portfolio selection problem in light of the recent development on general stochastic LQ problems.     

On unified concepts of detectability and observability for continuous-time stochastic systems

This paper is concerned with detectability and observability of continuous-time stochastic linear systems. By adopting the idea used in defining these two concepts for time-varying systems and Markovian jump linear systems that have been studied in the literature, corresponding definitions for continuous-time stochastic linear systems are proposed. These two definitions are not only able to unify some recent definitions on these two concept reported in the literature, but also allow us to propose an efficient rank criterion to test observability of continuous-time stochastic linear systems. It seems that this rank criterion is quite analogous to the rank criterion for deterministic linear systems. With the help of these two concepts and the new criteria, the stochastic Lyapunov equation is revisited and some recent published work on this equation are generalized. Numerical examples are given to illustrate the effectiveness of the proposed approach.     

Online stochastic reservation systems

This paper considers online stochastic reservation problems, where requests come online and must be dynamically allocated to limited resources in order to maximize profit. Multi-knapsack problems with or without overbooking are examples of such online stochastic reservations. The paper studies how to adapt the online stochastic framework and the consensus and regret algorithms proposed earlier to online stochastic reservation systems. On the theoretical side, it presents a constant sub-optimality approximation of multi-knapsack problems, leading to a regret algorithm that evaluates each scenario with a single mathematical programming optimization followed by a small number of dynamic programs for one-dimensional knapsacks. It also proposes several integer programming models for handling cancellations and proves their equivalence. On the experimental side, the paper demonstrates the effectiveness of the regret algorithm on multi-knapsack problems (with and without overbooking) based on the benchmarks proposed earlier.     

Controllability of fractional neutral stochastic functional differential systems

In this paper, we study a class of fractional neutral stochastic functional differential systems. We obtain the controllability of the stochastic functional differential systems by the Sadovskii’s fixed point theorem under some suitable assumptions. An example is given to illustrate the theory.     

Qualitative analysis of stochastic ratio-dependent predator-prey systems

In this paper, two stochastic ratio-dependent predator-prey systems are considered. One is just with white noise, and the other one is taken into both white noise and color noise account. Sufficient criteria for extinction and persistence in time average are established. The critical value between persistence and extinction is obtained. Moreover, we show that there is stationary distribution for the stochastic system with regime-switching. Finally, examples and simulations are carried on to verify these results.     

A NARMAX model-based state-space self-tuning control for nonlinear stochastic hybrid systems

A novel state-space self-tuning control methodology for a nonlinear stochastic hybrid system with stochastic noise/disturbances is proposed in this paper. via the optimal linearization approach, an adjustable NARMAX-based noise model with estimated states can be constructed for the state-space self-tuning control in nonlinear continuous-time stochastic systems. Then, a corresponding adaptive digital control scheme is proposed for continuous-time multivariable nonlinear stochastic systems, which have unknown system parameters, measurement noise/external disturbances, and inaccessible system states. The proposed method enables the development of a digitally implementable advanced control algorithm for nonlinear stochastic hybrid systems.     

Razumikhin method and exponential stability of hybrid stochastic delay interval systems

This paper deals with the exponential stability of hybrid stochastic delay interval systems (also known as stochastic delay interval systems with Markovian switching). The known results in this area (see, e.g., [X., Mao, Exponential stability of stochastic delay interval systems with Markovian switching, IEEE Trans. Automat. Control 47 (10) (2002) 1604-1612]) require the time delay to be a constant or a differentiable function and the main reason for such a restriction is due to the analysis of mathematics. The main aim of this paper is to remove this restriction to allow the time delay to be a bounded variable only. The Razumikhin method is developed to cope with the difficulty arisen from the nondifferentiability of the time delay.