Optimal control of a 1D diffusion process with a team of mobile actuators under jointly optimal guidance

This paper describes an optimization framework to control a distributed parameter system (DPS) using a team of mobile actuators. The optimization simultaneously seeks efficient guidance of the mobile actuators and effective control of the DPS such that an integrated cost function associated with both the mobile actuators and the DPS is minimized. Since the optimization does not have a constraint restricting the actuators to the domain of the DPS, the actuators may actuate outside the domain with no contribution towards regulating the DPS. We show that, under certain conditions, any guidance that steers the mobile actuators out of the spatial domain is nonoptimal. This result implies that optimal guidance is guaranteed to restrict the actuators to the domain even without explicit constraints. A gradient-descent method solves the integrated optimization problem numerically using its finite-dimensional approximation. We also synthesize the optimal feedback control of the DPS given jointly optimal guidance of the mobile actuators. A numerical example illustrates the optimization framework and the solution method.