Reaching a target in a time-costly area using a two-stage optimal control method

Consider that an agent, which moves on the two-dimensional coordinate space and is modeled as a linear time-invariant system, must be steered from a given initial condition towards an elliptical target region. This article presents a methodology to design control policies that minimize a cost subject to the terminal constraint that the target region is reached. In contrast to existing work, we consider that a time-costly elliptical area encompasses the target region. More specifically, we adopt a cost that linearly combines a quadratic term and a function of the duration of the time interval that starts when the agent first enters the time-costly area and ends when it reaches the target region. We propose a solution method that breaks up the problem in two stages (before and after entering the time-costly area), each modeled as an optimal control subproblem. The overall solution is obtained by solving an augmented problem that consists of the first stage subproblem subject to a terminal penalty determined by the optimal second-stage cost. We recast this problem, which is nonconvex, as a quadratic program with two quadratic constraints. We obtain a solution by proving that strong duality holds when certain symmetry conditions are satisfied. A numerical example is provided that illustrates our method.