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

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 $\bar{V}^N(\epsilon)$ can be constructed such that it is asymptotically equivalent in $\epsilon$ to the solution $V(\epsilon)$ of the singularly perturbed problem in the sense that $|V(\epsilon)-\bar{V}^N(\epsilon)| =O(\epsilon^{N+1})$ for any integer $N\geq0$ as $\epsilon \rightarrow 0$. For this approximation to be considered useful, the parameter $\epsilon$ is typically restricted to be in some sufficiently small set; however, for values of $\epsilon$ 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 $\chi^N_u(\epsilon)$ and a lower bound $\chi^N_l(\epsilon)$ of $V(\epsilon)$ that hold for arbitrary $\epsilon$ and, furthermore, satisfy the inequality $|\chi^N_u(\epsilon)-\chi^N_l(\epsilon)|=O(\epsilon^{N+1})$ for any integer $N \geq 0$ as $\epsilon \rightarrow 0$.