Theory and Applications of Bang-Bang and Singular Control Problems

  • Date: 05/18/2010
Helmut Maurer (Universitaet Muenster, Institut fuer Numerische und Angewandte Mathematik)

University of British Columbia


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.


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