On the Existence and Computation of Minimum Attention Optimal Control Laws

One means of capturing the cost of control implementation of a general nonlinear control system is via Brockett’s minimum attention criterion, defined as a multidimensional integral of the rate of change of the control with respect to state and time. Although shown to be important in human motor control and robotics applications, a practical difficulty with this criterion is that the existence of solutions is not always assured; even when they exist, obtaining local solutions numerically is difficult. In this article, we prove that, for the class of controls consisting of the sum of a time-varying feedforward term and a time-varying feedback term linear in the state, existence of a suboptimal solution can be guaranteed. We also derive a provably convergent gradient descent algorithm for obtaining a local solution, by appealing to the Liouville equation representation of a nonlinear control system and adapting iterative methods originally developed for boundary flow control. Our methodology is illustrated with a two degree-of-freedom planar robot example.