Existence theorems for general control problems of Bolza and Lagrange

The existence of solutions is established for a very general class of problems in the calculus of variations and optimal control involving ordinary differential equations or contingent equations. The theorems, while relatively simple to state, cover, besides the more classical cases, problems with considerably weaker assumptions of continuity or boundedness. For example, the cost functional may only be lower semicontinuous in the control and may approach + ∞ as one nears certain boundary points of the control region; both endpoints in the problem may be “free”. Earlier results of Cesari, Olech and the author are thereby extended.The development is based on the theory of convex integral functionals and their conjugates. The first step is to show that, for purposes of existence theory, the problem can be reduced to a simpler model where control variables are not present as such. This model, resembling a classical problem of Bolza in the calculus of variations, but where the functions are extended-real-valued, is then investigated using, above all, the conjugacy correspondence between generalized Lagrangians and Hamiltonians.