Arbitrarily Tight Bounds on a Singularly Perturbed Linear-Quadratic Optimal Control Problem

Abstract

We calculate arbitrarily tight upper and lower bounds on an unconstrained control, linear-quadratic, singularly perturbed optimal control problem whose exact solution is computationally intractable. It is well known that for the aforementioned problem, an approximate solution $\bar{V}^N(ε)$ can be constructed such that it is asymptotically equivalent in $ε$ to the solution $V(ε)$ of the singularly perturbed problem in the sense that $|V(ε)-\bar{V}^N(ε)| =O(ε^{N+1})$ for any integer $N\geq0$ as $ε\rightarrow 0$. For this approximation to be considered useful, the parameter $ε$ is typically restricted to be in some sufficiently small set; however, for values of $ε$ outside this set, a poor approximation can result. We improve on this approximation by incorporating a duality theory into the singularly perturbed optimal control problem and derive an upper bound $χ^N_u(ε)$ and a lower bound $χ^N_l(ε)$ of $V(ε)$ that hold for arbitrary $ε$ and, furthermore, satisfy the inequality $|χ^N_u(ε)-χ^N_l(ε)|=O(ε^{N+1})$ for any integer $N \geq 0$ as $ε\rightarrow 0$.