The optimal control of nonlinear diffusion equations with rough initial data

We consider in this paper a problem consisting of the optimal boundary control of a nonlinear diffusion equation with a “rough” initial condition, that is, in which the initial condition is a (unit) mass at an (interior) point of its domain. Since the state space may need to be unbounded, it is necessary to study this problems by means of the techniques of nonstandard analysis. Firstly, a problem is transformed into a semi-infinite linear programming problem by embedding the spaces of admissible trajectory-control pairs into spaces of measures. Then this is mapped into an appropriate nonstandard structure, where a near-minimizer is found for the nonstandard optimization; this entity is mapped back as a minimizer for the original problem. An appendix is including introducing the basic concepts of nonstandard analysis.