Maximum Process Problems in Optimal Control Theory

where c > 0 and the supremum is taken over all admissible controls v satisfying vt 2 [ 0; 1] for all t up to = inf f t > 0 j Xt = 2 (`0; `1) g with 0 < 0 < 1 and `0 < 0 < `1 given and fixed. The following control v is proved to be optimal: ’Pull as hard as possible’ that is v t = 0 if Xt < g (St) , and ’push as hard as possible’ that is v t = 1 if Xt > g (St) , where s 7! g (s) is a switching curve that is determined explicitly (as the unique solution to a nonlinear differential equation). The solution found demonstrates that the problem formulations based on a maximum functional can be successfully included in optimal control theory (calculus of variations) in addition to the classic problem formulations due to Lagrange, Mayer and Bolza.