Prescribed-time stability in switching systems with resets: A hybrid dynamical systems approach

Abstract

We consider the problem of achieving prescribed-time stability (PT-S) in a class of hybrid dynamical systems that incorporate switching nonlinear dynamics, exogenous inputs, and resets. By “prescribed-time stability”, we refer to the property of having the main state of the system converge to a particular compact set of interest before a given time defined a priori by the user. We focus on hybrid systems that achieve this property via time-varying gains. For continuous-time systems, this approach has received significant attention in recent years, with various applications in control, optimization, and estimation problems. However, its extensions beyond continuous-time systems have been limited. This gap motivates this paper, which introduces a novel class of switching conditions for switching systems with resets that incorporate time-varying gains, ensuring the PT-S property even in the presence of unstable modes. The analysis leverages tools from hybrid dynamical system’s theory, and a contraction–dilation property that is established for the hybrid time domains of the solutions of the system. We present the model and main results in a general framework, and subsequently apply them to two different problems: (a) PT control of dynamic plants with uncertainty and intermittent feedback; and (b) PT decision-making in non-cooperative switching games using algorithms that incorporate momentum, resets, and dynamic gains. Numerical results are presented to illustrate all our results.