AI 中文总结
该研究针对热扩散模型，精确了拟线性抛物型方程最优控制问题的线性近似，证明小数据下拟线性与线性问题的最优控制及轨迹差为高阶无穷小，内点情形近似阶最优。
AI 中文摘要
受热扩散模型的启发，当初始数据、目标及控制作用足够小时，受控拟线性抛物型方程可形式上替换为其线性近似。本文在最优控制问题层面精确化该近似：首先，给出区分最优控制是容许控制集的内点还是边界点的验证准则；接着证明，拟线性与线性抛物型方程对应的最优控制及最优轨迹之差，是关于成本泛函中小初始数据和目标的高阶无穷小量。因此，当数据较小时，由拟线性方程支配的最优控制问题可由对应的线性问题近似，在内点情形下，所得近似阶是最优的。
英文摘要
Motivated by thermal diffusion models, a controlled quasi-linear parabolic equation can be formally replaced by its linear approximation, when the initial datum, target and control actions are sufficiently small. This paper makes this approximation precise at the level of optimal control problems. First, we provide verification criteria which distinguish whether an optimal control is an interior point or a boundary point of the admissible control set. We then prove that the differences between the optimal controls and the corresponding optimal trajectories for the quasi-linear and linear parabolic equations are higher-order infinitesimals with respect to the small initial datum and target appearing in the cost functional. Consequently, for small data, the optimal control problem governed by the quasi-linear equation can be approximated by the corresponding linear problem. In the interior-point case, the resulting approximation order is sharp.
Comments24 pp
