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1997 | 17 | 1-2 | 5-27
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Measure solutions for semilinear evolution equations with polynomial growth and their optimal control

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In this paper we introduce a new concept of generalized solutions generalizing the notion of relaxed solutions recently introduced by Fattorini. We present some results on the question of existence of generalized or measure valued solutions for semilinear evolution equations on Banach spaces with polynomial nonlinearities. The results are illustrated by two examples one of which arises in nonlinear quantum mechanics. The results are then applied to some control problems.
  • Department of Electrical Engineering, Department of Mathematics, University of Ottawa, Canada
  • [1] N.U.Ahmed, Semigroup Theory with Applications to Systems and Control, Pitman Res. Notes in Math. Ser., 246, Longman Scientific and Technical and John Wiley, London, New York 1991.
  • [2] N.U.Ahmed, Optimal control of infinite dimensional systems governed by functional differential inclusions, Discussiones Mathematicae-Differential Inclusions 15 (1995), 75-94.
  • [3] N.U.Ahmed, Optimal relaxed controls for nonlinear infinite dimensional stochastic differential inclusions, Optimal Control of Differential Equations, (Ed. N.H.Pavel), Lect. Notes in Pure and Applied Mathematics, MarcelDekker, Inc. 160 (1994), 1-19.
  • [4] H.O. Fattorini, A remark on existence of solutions of infinite dimensional noncompact optimal control problems, SIAM Journal on Control and Optimization 35 (4) (1997), 1422-1433.
  • [5] H.O. Fattorini, Relaxation theorems, differential inclusions, and Filippov's theorem for relaxed controls in semilinear infinite dimensional systems,Journal of Differential Equations 112 (1994), 131-153.
  • [6] R. Larsen, Functional Analysis: An Introduction, Marcel Dekker Inc., New York 1973.
  • [7] N. Dunford and J.T. Schwartz, Linear Operators, Part 1, Interscience Publishers Inc., New York 1958.
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