Convex synthesis of distributed controllers for spatio-temporal systems with subadditive support functions

We consider the problem of spatio-temporal systems, with sensor measurements and control actions at each point in a spatial domain, such that the dynamics at each point may effect those at any other point with some propagation delay, and the controllers may communicate between any points with some transmission delays. We wish to synthesize such a controller to minimize a closed-loop norm for the entire system. We show that, similar to results for finite subsystems, if the transmission delays satisfy the triangle inequality, then the simple condition that the transmission delay between any two points is less than the propagation delay between those points allows for the optimal control problem to be recast as a convex optimization problem. We can then view spatially invariant systems as a special case, and a broad generalisation of the class of such systems amenable to convex synthesis quickly falls out. In particular, the class of systems determined by funnel causal propagation functions in one dimension can be generalised to those determined by subadditive propagation functions in arbitrary dimensions.