Abstract and Applied Analysis
Volume 3 (1998), Issue 3-4, Pages 265-292
doi: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 Alexander J. Zaslavski. 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 x0∈Rn 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 E⊂Rn 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.