We study optimal proportional feedback controllers for spatially invariant systems when the controller has access to delayed state measurements received from different spatial locations. We analyze how delays affect the spatial locality of the optimal feedback gain leveraging the problem decoupling in the spatial frequency domain. For the cases of expensive control and small delay, we provide exact expressions of the optimal controllers in the limit for infinite control weight and vanishing delay, respectively. In the expensive control regime, the optimal feedback control law decomposes into a delay-aware filtering of the delayed state and the optimal controller in the delay-free setting. Under small delays, the optimal controller is a perturbation of the delay-free one which depends linearly on the delay. We illustrate our analytical findings with a reaction-diffusion process over the real line and a multi-agent system coupled through circulant matrices, showing that delays reduce the effectiveness of optimal feedback control and may require each subsystem within a distributed implementation to communicate with farther-away locations.

In this paper we investigate the design of optimal spatially distributed controllers for a linear and spatially invariant reaction-diffusion process over the real line. The controller receives state measurements from different spatial locations with non-negligible delays. In this set-up and for the class of proportional spatially invariant state feedback controllers, the optimal control synthesis problem is equivalent to a feedback gain optimization for a spatially distributed delay system. We show that the spatial locality of optimal feedback gains is affected not only by diffusion and reaction coefficients, but also by the parameter representing communication time-delay that causes a sharp flattening of the control gains. In the expensive control regime, the optimal controller is solved analytically, yielding some practical design guidelines.