Solution Bounds, Stability and Attractors Estimates of Some Nonlinear Time-Varying Systems

TL;DR

This paper introduces a new method for estimating solution bounds, stability criteria, and trapping regions in nonlinear time-varying systems using differential inequalities and Lipschitz extensions, validated through simulations.

Contribution

It develops a novel approach based on scalar differential inequalities and nonlinear Lipschitz extensions to bound solutions and estimate stability regions for nonlinear systems.

Findings

Derived scalar differential inequalities for solution norms

Introduced nonlinear Lipschitz extension to improve bounds

Validated theoretical results with simulations

Abstract

Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous applied and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for a broad class of the corresponding systems. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and estimates the stability basins and trapping regions for the corresponding systems. Finally, we conform the theoretical results in representative simulations.