Review of the Latest Progress in Controllability of Stochastic Linear Systems and Stochastic GE-Evolution Operator

Null controllability results for stochastic delay systems with delayed perturbation of matrices

Several fundamental results from the theory of linear state-space systems in finite-dimensional space are extended to encompass a class of linear state-space systems in infinite-dimensional space. The results treated are those pertaining to the relationship between input-output and internal stability, the problem of dynamic output feedback stabilization, and the concept of joint stabilizability/detectability. A complete structural characterization of jointly stabilizable/detectable systems is obtained. The generalized theory applies to a large class of linear state-space systems, assuming only that: (i) the evolution of the state is governed by a strongly continuous semigroup of bounded linear operators; (ii) the state space is Hilbert space; (iii) the input and output spaces are finite-dimensional; and (iv) the sensing and control operators are bounded. General conclusions regarding the fundamental structure of control-theoretic problems in infinite-dimensional space can be drawn from these results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In this study, time-delayed stochastic dynamical systems of linear and nonlinear equations are discussed. The existence and uniqueness of the stochastic semilinear time-delay system in finite dimensional space is investigated. Introducing the delay Gramian matrix, we establish some sufficient and necessary conditions for the relative approximate controllability of time-delayed linear stochastic dynamical systems. In addition, by applying the Banach fixed point theorem, we establish some sufficient relative approximate controllability conditions for semilinear time-delayed stochastic differential systems. Finally, concrete examples are given to illustrate the main results.

On a fixed point index method for the analysis of the asymptotic behavior and boundary value problems of infinite dimensional dynamical systems and processes

In this paper we study linear discrete-time control systems in infinite-dimensional spaces with periodically time-varying bounded input and output operators. For such systems we consider a problem of arbitrary assignability of the upper Bohl exponent by non-stationary linear static state feedback and by non-stationary linear dynamic output feedback. This is a generalization of the stabilization problem, which has been well studied for systems of various types. The main results present sufficient conditions for considered types of assignability.

The classical theory of controllability and observability for deterministic systems is extended to linear stochastic time-varying systems defined on infinite dimensional Hilbert spaces. Two types of stochastic controllability (observability) are studied: approximate and complete controllability (observability). Tests for complete and approximate controllability (observability) are proved, and the relation between the controllability (observability) of linear stochastic systems and the controllability (observability) of the corresponding deterministic systems is studied.

SynopsisThe following theorem is proved: Let S(t), t≧0 be a dynamical system in an infinite dimensional Banach space X such that S(t) = S1(t)+S2(t) for t≧0, where (1) uniformly in bounded sets of x in X, and (2) S2(t) is compact for t sufficiently large. Then, if the orbit {S(t)x: t ≧0} of x ∈ X is bounded in X, it is precompact in X. Applications are made to an age dependent population model, a non-linear functional differential equation on an infinite interval, and a non-linear Volterra integrodifferential equation.

Observability and Controllability of Fractional Linear Dynamical Systems

In this paper, an optimal filter for a stochastic linear system with previous stage noise correlation is designed. Based on this result, together with the decomposition techniques of the stochastic singular linear system, the design of an optimal filter for a stochastic singular linear system is given.

In this paper, the notation of complete controllability for semilinear stochastic impulsive systems in infinite dimensional spaces is introduced. Sufficiently conditions ensuring the complete controllability of the systems are established. The results are obtained by using the Banach fixed point theorem.

The main objective of this paper is to present sufficient conditions for controllability of Hilfer fractional nonlinear stochastic systems in finite dimensional space. The main results are obtained by using the Nussbaum fixed point theorem, stochastic analysis approach and generalized fractional calculus (Hilfer fractional derivative) which is universality of Riemann-Liouville and Caputo fractional derivative. Finally, a numerical example is provided to show the effectiveness of the obtained theoretical result. The obtained result is more generalized one than the existing results on fractional stochastic system in finite dimensional space.

This paper is concerned with the controllability of linear and nonlinear fractional dynamical systems in finite dimensional spaces. Sufficient conditions for controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function. Examples are given to illustrate the effectiveness of the theory.

For the deterministic case, a linear controlled system is alwayspth order stable as long as we use the control obtained as the solution of the so-called LQ-problem. For the stochastic case, however, a linear controlled system with multiplicative noise is not alwayspth mean stable for largep, even if we use the LQ-optimal control. Hence, it is meaningful to solve the LP-optimal control problem (i.e., linear system,pth order cost functional) for eachp. In this paper, we define the LP-optimal control problem and completely solve it for the scalar case. For the multidimensional case, we get some results, but the general solution of this problem seems to be impossible. So, we consider thepth mean stabilization problem more intensively and give a sufficient condition for the existence of apth mean stabilizing control by using the contraction mapping method in a Hilbert space. Some examples are also given.

Optimal filtering and smoothing for discrete-time stochastic singular systems

We study classes of mappings between finite and infinite dimensional Banach spaces that are monotone and mappings which are differences of monotone mappings (DM). We prove a Radó–Reichelderfer estimate for monotone mappings in finite dimensional spaces that remains valid for DM mappings. This provides an alternative proof of the Fréchet differentiability a.e. of DM mappings. We establish a Morrey-type estimate for the distributional derivative of monotone mappings. We prove that a locally DM mapping between finite dimensional spaces is also globally DM. We introduce and study a new class of the so-called UDM mappings between Banach spaces, which generalizes the concept of curves of finite variation.