Strong (In)stability of Differential Inclusions and Lyapunov Characterizations

In the preceding chapter we have discussed strong stability and strong instability results for differential inclusions where the term strong refers to all solutions of a differential inclusion. In this chapter we discuss weak stability and weak instability results, where properties need to be satisfied for at least one solution instead of for all solutions. While the results from the last chapter allowed us to draw conclusions in terms of robustness, the results in this chapter guarantee stabilizability or destabilizability of the origin. In particular, in the context of a control system ( 2.8), weak stability guarantees the existence of an input \(u(\cdot )\) such that the origin can be reached asymptotically and weak instability describes the fact that it is possible to steer away from the origin by an appropriate selection of \(u(\cdot )\). In contrast to the last chapter where it turned out that strong stability in forward time is equivalent to strong instability in backward time, the same equivalence is not true for weak stability/instability results.