Speaker
Description
Optimal control problems with the control variable appearing linearly occur in many applications. However, this setting provides several challenges, from the analytical as well as from the numerical point of view. The optimal control can be of bang-bang or singular type, or even a combination of both is possible.
In the case of ordinary differential equations (ODE), the situation is thoroughly investigated. Specifically, for pure bang-bang controls, second order sufficient optimality conditions have been developed which are based on the so-called induced optimization problem. This method of optimizing the switching times gives the opportunity to prove optimality in a finite-dimensional manner and, furthermore, provides a convenient numerical solution method. For controls concatenated by bang-bang and singular arcs, however, there are still certain gaps concerning the theory and numerical verification of second order sufficient conditions in this context.
For constraints of partial differential equations (PDE), the situation is much more complicated and there are still many open questions. Some results on sufficient conditions for bang-bang controls have been published. Also numerical examples of bang-bang controls can be found in the literature, but only in very few cases, optimality could be shown. Recently, an example with a bang-singular control has been published where optimality has been conjectured by numerical arguments.
In this talk, we consider optimal control problems with the control variable appearing linearly subject to partial differential equations, particularly for boundary controls. We are interested in numerical methods as well as in optimality conditions. We will extend the induced optimization problem technique including the numerical method of arc parameterization from the ODE to this PDE setting. Necessary as well as sufficient optimality conditions therein are developed and compared to known conditions for the control problem. Connections to the ODE theory will be discussed by semi-discretization of the PDE in space. The results will be illustrated with numerical examples.
