Relaxation results for hybrid inclusions
The goal of the paper is to extend the Filippov-Ważevski relaxation theorem, established for differential inclusions, to the setting of hybrid inclusions. Hybrid inclusions can be written as \[ {\mathcal H}:\quad \left\{ \begin{matrix} \dot x \in F(x)&\quad x \in C \\ x^{+} \in G(x) &\quad x \in D\end{matrix} \right. \tag{1} \] That is, the state \(x\) evolution in time is described by the flow map \(F\) when \(x\) is in the flow set \(C\), and \(x\) can also jump according to the jump map \(G\) from the jump set \(D\). A relaxed hybrid inclusion \[ {\mathcal H}^{\text{con}}:\quad \left\{ \begin{matrix} \dot x \in \text{con}\; F(x)&\quad x \in C \\ x^{+} \in G(x) &\quad x \in D\end{matrix} \right. \tag{2} \] is obtained from the original \(\mathcal H\) by convexification of the flow map \(F\). The paper establishes conditions when solutions to \(\mathcal H\) are dense in the set of solutions to \({\mathcal H}^{\text{con}}\), with respect to the distance between their graphs. As an application of this relaxation result to stability theory, it is shown that uniform global asymptotic stability of a compact set for Eq.~1 implies that property for Eq.~2. The relaxation theorem leads to results on continuous dependence of the solution set to hybrid inclusions on initial conditions. The case of constrained differential inclusions and the case of systems with logic variables are considered also.
continuous dependence on initial conditions
scientific article; zbMATH DE number 5529267
scientific article