Controllability of some nonlinear systems in Hilbert spaces

Dauer J., Mahmudov N.

Journal of Optimization Theory and Applications, vol.123, no.2, pp.319-329, 2004 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 123 Issue: 2
  • Publication Date: 2004
  • Doi Number: 10.1007/s10957-004-5151-3
  • Journal Name: Journal of Optimization Theory and Applications
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.319-329
  • Keywords: controllability, fixed-point theorems, Infinite-dimensional systems, linear systems
  • Open Archive Collection: Article
  • Azerbaijan State University of Economics (UNEC) Affiliated: No

Abstract

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.