Controllability results for evolution inclusions with non-local conditions
Controllability is studied for the evolution inclusion with nonlocal condition \[ y'- A(t,y)y \in F(t,y) + (Bu)(t),\quad t\in [0, b],\quad y(0)+\sum_{k=1}^p c_ky(t_k)=y_0, \] in a real separable Banach space \(E\), in which \(F\) is set-valued. This problem is said to be non-locally controllable if for every \(x_1\in E\), there exists a control \(u\in L^2([0,b])\) such that the mild solution \(y\) satisfies \(y(b)+\sum^p_{k=1} c_ky(t_k)=x_1\). In Theorem 3.1, nonlocal controllability is proven in the case in which \(F\) is convex-valued using a fixed-point theorem for condensing maps due to Martelli. Theorems 3.2 and 3.3 give sufficient conditions for nonlocal controllability in the case in which \(F\) is not assumed to be convex-valued. Theorem 3.2 uses the Covitz-Nadler fixed-point theorem for contractive multi-valued maps, while Theorem 3.3 is proven using Schaefer's fixed-point theorem.
scientific article; zbMATH DE number 2096454
scientific article