Journal of Applied Analysis EMIS ELibM Electronic Journals Journal of Applied Analysis
Vol. 3, No. 1, pp. 67-92 (2003)

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Optimality conditions for control problems governed by abstract semilinear differential equations in complex Banach spaces

U. Ledzewicz and A. Nowakowski

Urszula Ledzewicz
Department of Mathematics
and Statistics
Southern Illinois University
at Edwardsville
Edwardsville, Illinois 62026
Andrzej Nowakowski
Faculty of Mathematics
University of Lodz
ul. Banacha 22
90-238 Lodz, Poland

Abstract: We consider the problem to minimize an integral functional defined on the space of absolutely continuous functions and measurable controls with values in an infinite-dimensional complex Banach space. The states are governed by abstract first order semilinear differential equations and are subject to periodic or anti-periodic type boundary conditions. We derive necessary conditions for optimality and introduce the notion of a dual field of extremals to obtain sufficient conditions for optimality. Such a dual field of extremals is constructed and a dual optimal synthesis is proposed. The paper is an extension of an earlier paper written for real Banach spaces. This extension covers optimal control problems which are governed by equations like the Schrödinger equation and other equations arising in Quantum mechanics.

Keywords: Abstract optimal control, semilinear differential equations, necessary optimality conditions, sufficient optimality conditions, complex Banach spaces

Classification (MSC2000): 49J, 35B

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Electronic version published on: 12 Jun 2003. This page was last modified: 12 Jun 2003.

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