Controllability results for evolution inclusions with non-local conditions
type: Item
instance of: scholarly article
title: Controllability results for evolution inclusions with non-local conditions
zbMATH Open document ID: 1052.34073
DOI: 10.4171/ZAA/1153
author: Mouffak Benchohra
author: Efrosini P. Gatsori
author: Sotiris K. Ntouyas
author: Lech Górniewicz
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-09-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 34H05
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 47H04
Mathematics Subject Classification ID: 47H10
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 93B05
zbMATH DE Number: 2096454
zbMATH Keywords: control problem
zbMATH Keywords: evolution inclusion
zbMATH Keywords: nonlocal condition
zbMATH Keywords: fixed-point theorem
reviewed by: Daniel C. Biles
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.4171/zaa/1153
OpenAlex ID: W1975566841
description: scientific article; zbMATH DE number 2096454