The global control of shock waves

We consider control problems associated with nonlinear wave equa-tions, in which the slope of the admissible trajectories can be made to approach innity by choosing parameters in an appropriate form. Thus, the solutions near shock waves, and we try to control these. A problem is rst reformulated as one consisting of the minimiza-tion of an integral in a space of functions satisfying a set of integral equalities; this is then transfered to a nonstandard framework, in which Loeb measures take the place of the functions and a near-minimizer can always be found. This is mapped back to the standard world by means of the standard part map; its image is a minimizer, so that the optimization is global. The minimizer is shown to be the solution of an in nite dimensional linear program and by well-proven approximation procedures a nite dimensional linear program is found by means of which nearly-optimal curves can be constructed for the original prob-lem. A numerical example is given. Mathematics Subject Classication: 49J27, 03H1