Controllability of nonlinear stochastic systems with multiple time-varying delays in control

Abstract

This paper is concerned with the problem of controllability of semi-linear stochastic systems with time varying multiple delays in control in finite dimensional spaces. Sufficient conditions are established for the relative controllability of semilinear stochastic systems by using the Banach fixed point theorem. A numerical example is given to illustrate the application of the theoretical results. Some important comments are also presented on existing results for the stochastic controllability of fractional dynamical systems.

Relative Controllability of Linear Difference Equations

“…Concerning the controllability problem, due to the infinite-dimensional nature of the dynamics of neutral functional differential equations and difference equations, several different notions of controllability can be used, such as exact, approximate, spectral, or relative controllability [5,30]. Relative controllability has been originally introduced in the study of control systems with delays in the control input [5,20,27], but this notion has later been extended and used to study also systems with delays in the state [13,29] and in more general frameworks, such as for stochastic control systems [19] or fractional integro-differential systems [2]. The main idea of relative controllability is that, instead of controlling the state x t : [−r, 0] → C d of (1.1), defined by x t (s) = x(t + s), in a certain function space such as C k ([−r, 0], C d ) or L p ((−r, 0), C d ), where r ≥ max j∈ 1,N Λ j , one controls only the final state x(t) = x t (0).…”

Boundary controllability of nonlocal Hilfer fractional stochastic differential systems with fractional Brownian motion and Poisson jumps

“…Stochastic differential equations driven by fractional Brownian motion have been considered extensively by research community in various aspects due to the salient features for real world problems (see [5][6][7][8][9][10][11][12]). In addition, controllability problems for different kinds of dynamical systems have been studied by several authors (see [13][14][15][16][17][18][19][20][21]), and the references therein. Few authors studied the controllability for linear and nonlinear systems when the control is on the boundary (see [22][23][24][25][26][27][28][29][30]).…”

A practical test for assessing the reachability of discrete-time Takagi–Sugeno fuzzy systems

“…The controllability of infinite dimensional systems has been studied in [9,10]. Further investigation has addressed the problem for stochastic [11,12,13,14], delayed [15,16], fractional [17,18,19] and switched [20,21,22,23] systems.…”