Existence and Uniform Boundedness of Optimal Solutions of Variational Problems

Given an x0 ∈ Rn we study the infinite horizon problem of minimizing the expression ∫ T 0 f(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 . Introduction The study of variational and optimal control problems defined on infinte intervals has recently been a rapidly growing area of research. These problems arise in engineering (see Anderson and Moore [1], Artstein and Leizarowitz [2]), in models of economic growth (see Rockafellar [14], Zaslavski [20]), in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals which are under discussion in Aubry and Le Daeron [3], Zaslavski [16] and in the theory of thermodynamical equilibrium of materials (see Leizarowitz and Mizel [12], Coleman, Marcus and Mizel [7], Zaslavski [17,18]). We consider the infinite horizon problem of minimizing the expression ∫ T 0 f(t, x(t), x′(t))dt as T grows to infinity where a function x : [0,∞) → K is absolutely continuous (a.c.) and satisfies the initial condition x(0) = x0, K ⊂ R is a 1991 Mathematics Subject Classification. 49J99, 58F99.