On weak topology for optimal control of switched nonlinear systems

Optimal control of switched systems is challenging in general due to the discrete nature of the switching control input. One of the most well-known approaches is the embedding based method which addresses this challenge by solving a relaxed optimization problem with only continuous inputs and then projecting the relaxed solution back to obtain a solution to the original problem. In this paper, we present a unified topology based framework for analyzing and designing various embedding based switched optimal control algorithms. The proposed framework views the embedding based approaches from a novel topological perspective as a change of topology over the optimization space. A general procedure of constructing different switched optimal control algorithms with guaranteed convergence to a stationary point is described. Numerical examples are also provided to illustrate the effectiveness of the proposed framework.