Lyapunov‐Krasovskii stability analysis of delayed Filippov system: Applications to neural networks with switching control

Abstract

Summary This paper gives the stability analysis of Filippov system with delay. Under the Filippov‐framework, we introduce a general type of Lyapunov‐Krasovskii functional (LKF) to derive the stability of delayed differential inclusions (DDIs), where the indefiniteness or positive definiteness of the derivative of LKF holds for almost everywhere along the trajectories of state solution. The proposed LKF of this paper generalizes the classic LKF whose derivative possesses negative definiteness or seminegative definiteness for everywhere. As a result, the stability, uniform stability, uniform asymptotic stability, and global exponential stability criteria of the trivial solution for DDI are established. Moreover, the developed LKF method is applied to solve the stabilization control issue of delayed neural networks, possessing discontinuous input‐output activation.

Theℒ1controller synthesis for piecewise continuous nonlinear systems via set invariance principles

“…For a more rigorous argument on the treatment of nonlinear systems with non-unique solutions, there have been a number of studies on piecewise continuous nonlinear systems. [24][25][26][27][28][29][30][31] This is a class of discontinuous dynamical systems, in which the types of linear systems, 8,9 continuous nonlinear systems, 11 and switched nonlinear systems 32 are involved as described shown in Figure 1. Here, solutions to piecewise continuous nonlinear systems should be described based on the arguments on Filippov's differential equations (inclusions) 33 because they cannot be defined on the classical Caratheodory-based arguments.…”

“…For a more rigorous argument on the treatment of nonlinear systems with non‐unique solutions, there have been a number of studies on piecewise continuous nonlinear systems 24‐31 . This is a class of discontinuous dynamical systems, in which the types of linear systems, 8,9 continuous nonlinear systems, 11 and switched nonlinear systems 32 are involved as described shown in Figure 1.…”

“…Here, solutions to piecewise continuous nonlinear systems should be described based on the arguments on Filippov's differential equations (inclusions) 33 because they cannot be defined on the classical Caratheodory‐based arguments. In a theoretical aspect for dealing with piecewise continuous nonlinear systems, various studies on stability and $$$ {\mathcal{L}}_2 $$$‐induced norms have been introduced recently in References 27‐31, respectively, but no argument on the $$$ {\mathcal{L}}_1 $$$ controller synthesis is discussed in those studies. With respect to a practical effectiveness, furthermore, there have been also various studies on employing piecewise continuous nonlinear systems to describe real systems such as frictional effects, 24 neural networks with discontinuous activation functions, 25 sliding mode control systems 26 and so on.…”

Stability for a Non-Smooth Filippov Ratio-Dependent Predator-Prey System through a Smooth Lyapunov Function

“…erefore, the classical theory of differential equations has been believed to be ineffective for dealing with the solutions of differential equations with discontinuous right-hand sides. Although the solution of discontinuous differential equations could be converted into a solution of differential inclusion by constructing the Filippov multimap [16], it is still difficult to prove the stability of the system's equilibrium point. e most recent extension of Lyapunov's second method to nonsmooth dynamic systems was stated by Shevitz and Paden [17], in which piecewise smooth Lyapunov functions were put forward.…”

Cooperation‐Based Modeling of Sustainable Development: An Approach from Filippov’s Systems

“…On the applications of Filippov systems, the works have been mainly oriented to friction oscillators [31,[37][38][39][40][41], neural networks activated by discontinuous functions [42][43][44][45][46], memristor-based neural networks [47][48][49][50][51][52][53], neural networks with switching control using the Filippov system with delay [54][55][56][57], and electronic converters [58]. On issues related to Sustainable Development, the number of papers is much more limited, with approaches from the analysis of communities [59], from the analysis of companies [60] and others that touch on close issues such as energy systems [61][62][63][64], pest or disease control [65][66][67], HIV behavior [68,69], behavior longterm communities [70], or communications security [71].…”