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Abstract and Applied Analysis
Volume 3, Issue 3-4, Pages 265-292
http://dx.doi.org/10.1155/S1085337598000566

Existence and uniform boundedness of optimal solutions of variational problems

Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel

Received 9 December 1996

Copyright © 1998 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Given an x0Rn we study the infinite horizon problem of minimizing the expression 0Tf(t,x(t),x(t))dt as T grows to infinity where x:[0,)Rn satisfies the initial condition x(0)=x0. We analyse the existence and the properties of approximate solutions for every prescribed initial value x0. We also establish that for every bounded set ERn the C([0,T]) norms of approximate solutions x:[0,T]Rn for the minimization problem on an interval [0,T] with x(0),x(T)E are bounded by some constant which does not depend on T.