Practical Stability and Finite-Time Stability of Discontinuous Systems

The trajectory bounds of discontinuous systems are treated within a stability framework. In doing so, stability is defined in terms of prespecified subsets of the state space over an infinite time interval (practical stability) and over a finite time interval (finite-time stability). The discontinuous systems considered are those which are described by ordinary discontinuous differential equations which may be autonomous or nonautonomous, linear or nonlinear. In all cases it is assumed that the differential equation possesses solutions in the sense of Filippov. The results obtained yield sufficient conditions for practical stability and finite-time stability. In order to demonstrate application of the methods advanced, specific examples are considered.