Regularity of Minimizers

This chapter provides conditions under which minimizers for dynamic optimization problems possess regularity properties of interest, such as Lipschitz continuity or boundedness of higher derivatives. We discuss the significance of regularity analysis, regarding the derivation of necessary conditions of optimality, the selection of numerical schemes for the computation of optimal strategies and the implementation of such strategies. Tonelli identified conditions (Tonelli existence hypotheses), under which the basic problem in the calculus of variations has a minimizer in the class of absolutely continuous functions that satisfy specified end-point conditions. Tonelli’s existence hypotheses do not guarantee, however, that minimizers satisfy the Euler Lagrange condition; this is a set-back to solution techniques based on seeking a minimizer among arcs satisfying standard necessary conditions of optimality. For problems with smooth Lagrangians and in which the arcs are scalar valued, Tonelli showed however that, under his existence hypotheses, the velocity of the minimizing arcs is locally essentially bounded at all times in an open set of full Lebesgue measure. This regularity property can be exploited, to show that minimizers satisfy the Euler Lagrange condition under additional hypotheses. In sections of this chapter dealing with Tonelli regularity, we cover subsequent extensions of Tonelli’s original discoveries: the velocities of minimizers are essentially locally bounded on a relatively open subset of full measure of the reference time interval, under Tonelli’s existence hypotheses, even when we allow vector valued arcs and nonsmooth Lagrangians. We show that this extended Tonelli regularity property leads to improved criteria for validity of the Euler Lagrange condition. The proof of these extensions, which is based on the construction of auxiliary Lagrangians, is a showcase for application of nonsmooth methods, which introduce a new flexibility into the use of Tonelli’s methods, by allowing us to construct nonsmooth auxiliary Lagrangians. These aspects of Tonelli regularity theory relate to calculus of variations problems. We briefly consider extensions of the theory gives conditions for the essential boundedness of the minimizing control, for dynamic optimization problems associated with a linear control system. The chapter also includes material on another approach to establishing regularity of minimizers, in circumstances when standard necessary conditions are not valid and therefore cannot be (directly) used in the regularity analysis. This is based on reparameterization of the minimizer by a change of independent variable. Applications of this approach, when first introduced, were restricted to autonomous problems satisfying the superlinear growth conditions in the Tonelli existence hypotheses. We take account of recent extensions of reparameterization methods, which can be used to establish regularity properties of minimizers, specifically Lipchitz continuity, in some situations where the problem concerned is non-autonomous and the superlinear growth hypothesis is replaced by a less restrictive, slow growth hypothesis. In the final section of this chapter, we restrict attention to certain dynamic optimization problems with dynamic constraint a controlled differential equation, where minimizers are known to satisfy the maximum principle. Here we show how these necessary conditions can be applied directly to establish Lipschitz continuity of state trajectories.