On stability with respect to the part of variables of a non-autonomous system in a cylindrical phase space

The stability problem of a system of differential equations with a right-hand side periodic with respect to the phase (angular) coordinates is considered. It is convenient to consider such systems in a cylindrical phase space which allows a more complete qualitative analysis of their solutions. The authors propose to investigate the dynamic properties of solutions of a non-autonomous system with angular coordinates by constructing its topological dynamics in such a space. The corresponding quasi-invariance property of the positive limit set of the system’s bounded solution is derived. The stability problem with respect to part of the variables is investigated basing of the vector Lyapunov function with the comparison principle and also basing on the constructed topological dynamics. Theorem like a quasi-invariance principle is proved on the basis of a vector Lyapunov function for the class of systems under consideration. Two theorems on the asymptotic stability of the zero solution with respect to part of the variables (to be more precise, non-angular coordinates) are proved. The novelty of these theorems lies in the requirement only for the stability of the comparison system, in contrast to the classical results with the condition of the corresponding asymptotic stability property. The results obtained in this paper make it possible to expand the usage of the direct Lyapunov method in solving a number of applied problems.

Asymptotic Stabilization for a Class of Linear Fractional-Order Composite Systems

In this paper, we present the design for a decentralized control method comprising a series of local state feedback controllers for a class of linear fractional composite systems. In addition, the corresponding asymptotic stabilization criterion is derived. First, we design the local state feedback controllers for each subsystem of the linear fractional composite system. Then, based on the vector Lyapunov function, we combine these local state feedback controllers into a single decentralized controller through which the asymptotic stabilization criterion is proposed for the class of linear fractional composite system. Finally, numerical simulation of a class of linear fractional composite systems is used to verify the accuracy and effectiveness of the decentralized control method.

Stochastically Globally Exponential Stability of Stochastic Impulsive Differential Systems with Discrete and Infinite Distributed Delays Based on Vector Lyapunov Function

This paper deals with stochastically globally exponential stability (SGES) for stochastic impulsive differential systems (SIDSs) with discrete delays (DDs) and infinite distributed delays (IDDs). By using vector Lyapunov function (VLF) and average dwell-time (ADT) condition, we investigate the unstable impulsive dynamics and stable impulsive dynamics of the suggested system, and some novel stability criteria are obtained for SIDSs with DDs and IDDs. Moreover, our results allow the discrete delay term to be coupled with the nondelay term, and the infinite distributed delay term to be coupled with the nondelay term. Finally, two examples are given to verify the effectiveness of our theories.