The Global Control of Shock Waves

We consider control problems associated with nonlinear wave equations, in which the slope of the admissible trajectories can be made to approach infinity by choosing parameters in an appropriate form. Thus, the solutions near shock waves, and we try to control these. A problem is first reformulated as one consisting of the minimization 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 infinite dimensional linear program and by well-proven approximation procedures a finite dimensional linear program is found by means of which nearly-optimal curves can be constructed for the original problem. A numerical example is given. Mathe..