Generalized flow invariance for differential inclusions
It is considered the differential inclusion \[ \dot{x}\in F(t,x),\quad x(t_{0}) = x_{0}, \tag{1} \] where \(F(t,x)\subset \mathbb{R}^{n} \) is a nonempty convex set for each \((t,x)\in \mathbb{R}_{+}\times \mathbb{R}^{n};F\) is upper semicontinuous. It is assumed that \(v\in F(t,x)\) implies that \(\| v\| \leq g(t,\| x\| )\), where \(g\in C(\mathbb{R}^{2}_{+},\mathbb{R}_{+}), g(t,w)\) is nondecreasing in \(w\), and the maximal solution \(r(t) = r(t,t_{0},w_{0})\), of the scalar differential equation \(\dot{w} = g(t,w),\, w(t_{0})= w_{0}\geq 0\) exists on \([t_{0},\infty).\) Let \(S(t), t\in [0,\infty)\) be a family of nonempty closed subsets of \(\mathbb{R}^{n}\). The system \(\{S(t), F): t\geq t_{0}\}\) is said to be weakly invariant if for all \( x_{0}\in S(t_{0})\), there exists an Euler solution \(x(t)\) of (1) on \([t_{0},\infty)\) such that \(x(t_{0}) = x_{0}\) and \(x(t)\in S(t)\), for \(t\geq t_{0}.\) A set of sufficient conditions for the weak and strong invariance are obtained in the paper.
scientific article; zbMATH DE number 5134571
scientific article