On the Existence and Computation of Minimum Attention Optimal Control Laws

Brockett's minimum attention functional citeBrockett has been proposed as one means of capturing the cost of control implementation--regarded here as the rate of change of the control with respect to both state and time--for general nonlinear control systems, with applications ranging from human motor control to robotics. The main challenge in forging the minimum attention paradigm into a practical control design methodology is that the existence of solutions is not always assured, and finding numerical solutions is also difficult. In this paper we prove that, under the assumption of a control that is the sum of a time-varying feedforward term and a time-varying feedback term linear in the state, existence of a solution can be guaranteed. Under these assumptions we appeal to the Liouville equation representation of a nonlinear control system and derive the associated first-order optimality conditions. The one-shot method is then used to prove the existence of a solution and also to iteratively compute a solution. Our methodology is illustrated with an example involving a two degree-of-freedom robot arm.