Optimal Control on Disconnected Sets Using Extreme Point Relaxations and Normality Approximations

This paper presents new mathematical results for exact and approximate relaxations of non-convex optimal control problems defined on disconnected control sets. The exact relaxations are based upon extreme point relaxations wherein the relaxed control set is constructed so that its extreme points belong to the original set. A system property required for exactness is normality. In the absence of normality, control set approximations and system dynamic perturbations are introduced so that normality holds. The results hold for linear dynamical systems and nonlinear systems affine in the control. It is shown that the convex relaxations enable the efficient solution of problems in aerospace engineering, sparsity-promoting optimization, over-actuated systems, and multiplexing systems. The computational guarantees associated with convex optimization make the relaxation techniques suitable for real-time control.