Strong (In)stability of Differential Inclusions and Lyapunov Characterizations

Abstract

In this chapter we discuss extensions and generalizations of the stability and instability results of Sect. 2.2 for differential inclusions. In Sect. 3.1 $\mathcal {KL}$-stability and equivalent Lyapunov-characterizations are discussed. In Sect. 3.2 the results are mirrored to $\mathcal {K}_{\infty }\mathcal {K}_{\infty }$-instability. Connections between stability and instability are discussed in Sect. 3.3. In the last section of this chapter, Sect. 3.4, we briefly discuss $\mathcal {KL}$-stability with respect to two measures, which combines the concepts of stability and instability. The results discussed in this chapter correspond to strong stability notions, i.e., properties which are satisfied for all solutions of the differential inclusion (2.1). Thus, the results of this chapter describe robust stability properties of control systems as motivated for the dynamics (2.8). Weak stability results are discussed in Chap. 4.