Limit differential inclusions and the invariance principle for nonautonomous systems (Q483460)

Let \(x' \in F(t,x)\) be a differential inclusion, where \(F : \mathbb{R}^{n+1} \to \mathbb{R}^n\) is measurable in \(t\), upper semi-continuous in \(x\) and has compact values. Considering the translations \(F(t+\tau,x)\), the author introduces the concept of limit differential inclusion \(y' \in F^*(y)\), and the concept of limit differential inclusion with respect to a sequence \(\tau_k \to \infty\), \(y' \in F'(t,y)\). Sufficient conditions on \(F\) are given to insure the existence of solutions to those limit problems. Also, it is shown that from \(x\) a solution to the original problem, one can deduce the existence of a solution to the limit problem \(y'\in F'(t,y)\) such that \(y\) is in the \(\omega\)-limit set of \(x\). Finally, the invariance properties of the \(\omega\)-limit sets of solutions to the original problem are studied. Using Lyapunov functions with derivative of constant sign, the author proves an analog of LaSalle's invariance principle.