Discrete-time Pontryagin maximum principle under rate constraints: Necessary conditions for optimality

Limited bandwidth and limited saturation in actuators are practical concerns in control systems. Mathematically, these limitations manifest as constraints imposed on the control actions, their rates of change, and, more generally, the global behavior of their paths. While the problem of actuator saturation has been studied extensively, little attention has been devoted to the problem of actuators having limited bandwidth. While attempts have been made in the direction of incorporating frequency constraints on state-action trajectories before, rate constraints on the control at the design stage have not been studied extensively in the discrete-time regime. This article contributes toward filling this lacuna: We establish a new discrete-time Pontryagin maximum principle with rate constraints imposed on the control trajectories and derive first-order necessary conditions for optimality. A brief discussion on the existence of optimal control is also included.