Relaxations and approximations for mixed-integer optimal control (Q2873479)

The excellent PhD-thesis of Michael Jung deals with mixed-integer optimal control, it was written under the supervision of S. Sager and G. Reinelt. The thesis is influenced by the Heidelberg research team of optimization. The topic is also referred to as hybrid optimal control. An optimal control system is studied which is described by ODEs or DAEs, respectively, control-state inequalities and control restrictions, and a certain functional is to minimize.NEWLINENEWLINE Necessary optimality conditions and well-known numerical methods for classical controls are listed. The new problem under consideration has an integer feasibility requirement on a subset of the control functions. The author proposes the use of a direct approach to formulate numerical algorithms. Therefore, a discretization of the process is given. Then, an integral relaxation of different modes ODEs and constraints is considered. There are inner convexification and outer convexification. Different approaches are discussed in terms of numerical stability and general solvability.NEWLINENEWLINE The next step treats approximation of the controls in order to obtain integer control functions out of relaxed ones. The solution structure of a branch-and-bound algorithm with a Lagrangian relaxation as subproblems is investigated.NEWLINENEWLINE The last chapter gives numerical examples, it covers four problems: 1) a prototype model from the literature, 2) a Lotka-Volterra-model for fishing control, 3) a sewage network, where the overflow onto streets and environment has to be controlled, 4) an energy-optimal problem of a truck cruise.