Criteria of asymptotic stability of differential inclusions and periodic motions of time-varying nonlinear control systems

New constructive criteria of asymptotic stability of selector-linear differential inclusions are established. The well-known absolute stability problem is also considered as a particular case of the above problem. Asymptotically stable inclusions are very similar in properties to linear stable time-invariant systems. This similarity concerns the wide range of dynamic properties. In particular, asymptotic stability of selector-linear differential inclusions has an exponential type and a response of the system to a bounded action is bounded. It turns out that there exist periodic motions at the boundary of asymptotic stability region for two- and three-dimensional systems. In the general case of n-dimensional systems the periodic motions proved to exist out of the closure of the asymptotic stability region. This property is the basis for the new criteria having the form of algebraic conditions. It is necessary to note that differential inclusions are particularly attractive for adequate description of dynamic systems with incomplete information.