M. Rotkowitz, R. Cogill, and S. Lall
Convexity of Optimal Control over Networks with Delays and Arbitrary Topology
special issue on "Information Processing and Decision Making in Distributed Control Systems",
International Journal of Systems, Control, and Communications. vol. 2, no. 1/2/3, pp. 30-54, 2010.

Abstract

We consider the problem of multiple subsystems, each with its own controller, such that the dynamics of each subsystem may affect those of other subsystems with some propagation delays, and the controllers may communicate with each other with some transmission delays. We wish to synthesize controllers to minimize a closed-loop norm for the entire system. We show that if the transmission delays satisfy the triangle inequality, then the simple condition that the transmission delay between any two subsystems is less than the propagation delay between those subsystems allows for the optimal control problem to be recast as a convex optimization problem. This is shown to unify and broadly generalize the class of such systems amenable to convex synthesis.

We develop analagous results for spatio-temporal systems, showing that if transmission delays satisfy the triangle inequality, the the transmission delay between any two points being less than the propagation delay between those two points similarly allows the optimal control problem to be cast as a convex optimization problem. When considering the special case of spatially invariant systems, this is shown to yield a simple characterization of which of those problems are amenable to convex synthesis, which itself is a broad generalization of such previously characterized convex problems.