New Existence Results for Optimal Controls in the Absence of Convexity: the Importance of Extremality

A new approach to "existence without convexity" is presented. Instead of applying a Weierstrass--Tonelli-type existence result to a convexified (i.e., relaxed) form of the control problem, which is standard, it uses a new extremum principle. This principle guarantees the existence of an optimal relaxed control function that is a convex combination of relaxations of ordinary control functions. The same linear relationship is maintained for the corresponding trajectories, and this is of much benefit in the final phase, which consists of a deconvexification by a global subgradient argument and Lyapunov's theorem. The extremum principle is based on extreme point features of the relaxed control problem. It can be stated in an abstract form and extends the classical extremum principle of Bauer. The new approach applies both to optimal control problems of suitable type (dynamics linear in the state variable) and to abstract variational problems without dynamics, well-known in economics. With a different interpretation, it can also give bang-bang existence results. Several new applications serve to illustrate the power of the approach.