Theory and Applications of Bang-Bang and Singular Control Problems

We consider optimal control problems with control appearing linearly. The evaluation of the Pontryagin Minimum Principle shows that optimal controls are composed of bang-bang and singular arcs. Values of bang-bang controls switch discontinuously between their upper and lower bounds, whereas singular controls can take values in the interior of the control region. The optimal control problem induces a finite-dimensional optimization problem with respect to the switching times between bang-bang and singular arcs. The arc-parametrization is an efficient method for solving the induced optimization problems and allows to check second-order sufficient conditions (SSC). It can be shown that SSC for the induced optimization problem and a regularity property imply SSC for bang-bang control problem. SSC for singular control problems require stronger conditions which are currently under investigation. SSC constitute the basis for a parametric sensitivity analysis and the development of real-time control techniques. Several examples illustrate theory and numerics: optimal control of (1) a Van-der-Pol oscillator, (2) a semiconductor laser, (3) a batch fermentation process and (4) chemotherapy of HIV.