Existence and uniform boundedness of optimal solutions of variational problems

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.