Non-Linear-Quadratic Optimal Control Problem for a Unicycle: Maximin Solution

A finite-horizon optimal control problem for a nonlinear unicycle with constant linear velocity is considered. The cost functional consists of the squared norm of a final position and the integral penalty term for the control effort, i.e., both the miss distance and the control are soft-constrained. A finite horizon formulation arises, for instance, in coordinated guidance attack against a stationary target, in which all interceptors have to arrive at the target at the same time. The soft constraint on terminal position allows for tradeoff between the miss distance and control effort. Semi-analytical solution is derived by representing the squared norm as a maximum of a quadratic form and by changing the order of maximization and minimization. The inner minimization problem becomes a problem of calculus of variations, which Euler-Lagrange equation writes as a nonlinear pendulum equation. Based on the solution of this equation, a numerical scheme for constructing the suboptimal control is developed. As a by-product of the approach, the posterior control bounds are obtained.