奇摄动线性二次最优控制问题的严格紧界
Arbitrarily Tight Bounds on a Singularly Perturbed Linear-Quadratic Optimal Control Problem
摘要：
我们计算任意紧的上限和下限的无约束控制，线性二次，奇摄动最优控制问题的精确解是计算困难的。众所周知，对于上述问题，可以构造近似解$\bar{V}^N(ε)$，使得它在$ε$中渐近等价于奇异摄动问题的解$V(ε)$，在这个意义上，对于任何整数$N\geq0$，$|V(ε)-\bar{V}^N(ε)| =O(ε^{N+1})$为$ε\rightarrow 0$。为了使这种近似被认为是有用的，参数$ε$通常被限制在某个足够小的集合中;然而，对于该集合之外的$ε$的值，可能导致差的近似。通过将对偶理论引入到奇摄动最优控制问题中，我们改进了这种近似，并得到了$V(ε)$的上界$χ^N_u(ε)$和下界$χ^N_l(ε)$，它们对任意的$ε$都成立，而且对于任何整数$N \geq 0$都满足不等式$|χ^N_u(ε)-χ^N_l(ε)|=O(ε^{N+1})$作为$ε\rightarrow 0$ .
英文摘要：
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 $ε\rightarrow 0$.
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