Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control

In model-based nonlinear optimal control switching decisions that can be optimized often play an important role. Prominent examples of such hybrid systems are gear switches for transport vehicles or on/off valves in chemical engineering. Optimization algorithms need to take the discrete nature of the variables that model these switching decisions into account. Unnecessarily, for many applications still an equidistant time discretization and either rounding or standard mixed-integer solvers are used. In this article we survey recent progress in theoretical bounds, reformulations, and algorithms for this problem class and show how process control can benefit from them. We propose a comprehensive algorithm based on the solution of a sequence of purely continuous problems and simulations, and provide a new and more compact proof for its well-posedness. Instead of focusing on a particular application, we classify different solution behaviors in the applications section. We provide references to respective case studies with prototype character and cite newly emerging benchmark libraries. We conclude by pointing out future challenges for process control with switching decisions.