On stability and dissipativity of stochastic nonlinear systems

Input-to-state stability of nonlinear control system is described in several different manners, and has been a central concept since the equivalences among them were verified. In this paper, a framework of stability and dissipativity for stochastic control systems is constructed on the maximal existence interval of behaviors (states and external inputs), by the aid of stochastic Barbalat lemma and stochastic dissipativity. The main work consists of three aspects. First, input-to-state stability and robust stability are extended to the stochastic case, and several criteria are established. Second, two forms of dissipativity and their criteria are presented. Third, the key relations among the definitions of stability and dissipativity are verified. Compared with the existing results on stochastic input-to-state stability, our methods allow for non-globally Lipschitz condition, dynamic inputs without knowledge about boundedness and non-smoothness of storage functions, which are more effective and convenient to be used in practice.