Abstract and Applied Analysis
Volume 3 (1998), Issue 3-4, Pages 265-282
doi:10.1155/S1085337598000566

Existence and uniform boundedness of optimal solutions of variational problems

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.