On smoothness properties and approximability of optimal control functions

The theory of discretization methods to control problems and their convergence under strong stable optimality conditions in recent years has been thoroughly investigated by several authors. A particularly interesting question is to ask for a ‘natural’ smoothness category for the optimal controls as functions of time. In several papers, Hager and Dontchev considered Riemann integrable controls. This smoothness class is characterized by global, averaged criteria. In contrast, we consider strictly local properties of the solution function. As a first step, we introduce tools for the analysis of L∞ elements ‘at a point’. Using afterwards Robinson's strong regularity theory, under appropriate first and second order optimality conditions we obtain structural as well as certain pseudo-Lipschitz properties with respect to the time variable for the control. Consequences for the behavior of discrete solution approximations are discussed in the concluding section with respect to L∞ as well as L2 topologies.