23. Existence and regularity

To this point, our discussion of optimal control has focused on the issue of necessary conditions. Before turning to the attendant issues of existence and regularity, we digress somewhat to take note of an entirely new consideration, one that is called relaxation. Just as an unstable equilibrium in mechanics is considered somewhat meaningless, one may consider that certain optimal control problems are not well posed, since their solution lacks stability, in a certain sense. It is the goal of relaxation to reformulate the problem so as to avoid this phenomenon. We then proceed to obtain three existence theorems for optimal control. Roughly speaking, these theorems require that velocity sets be convex. When there is a running cost, its convexity relative to the control variable is the functional counterpart of that property. In addition, some growth restriction is required. This is most easily supplied by taking the control set to be compact; otherwise, when the controls are unbounded, coercivity of the running cost can be postulated. We close the chapter by identifying criteria under which the optimal control must be continuous.