论文标题
通过最佳控制，提高动力学福克 - 普兰克方程的收敛速率
Improving the Convergence Rates for the Kinetic Fokker-Planck Equation by Optimal Control
论文作者
论文摘要
在过去的几年中，长期行为和详细的趋同分析受到了越来越多的关注。困难是由于缺乏基础动力学福克斯 - 普兰克操作员缺乏强制性，通常称为低调性，这是二阶随机微分方程的部分确定性的结果。在本手稿中，研究了控制限制电位的效果，而无需更改原始不变措施。这导致了一个抽象的双线性控制系统，具有无限但无限的可允许的控制操作员，该操作员通过人工扩散方法具有独特的解决方案。基础半群的紧凑性进一步用于在适当降低的状态空间上定义无限 - 马的最佳控制问题。在对初始数据的小假设下，讨论了最佳控制问题解决方案的可行性和存在。基于基于移位的Riccati方程的局部近似值的数值结果说明了理论发现。
The long time behavior and detailed convergence analysis of Langevin equations has received increased attention over the last years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator which is a consequence of the partially deterministic nature of a second order stochastic differential equation. In this manuscript, the effect of controlling the confinement potential without altering the original invariant measure is investigated. This leads to an abstract bilinear control system with an unbounded but infinite-time admissible control operator which, by means of an artificial diffusion approach, is shown to possess a unique solution. The compactness of the underlying semigroup is further used to define an infinite-horizon optimal control problem on an appropriately reduced state space. Under smallness assumptions on the initial data, feasibility of and existence of a solution to the optimal control problem are discussed. Numerical results based on a local approximation based on a shifted Riccati equation illustrate the theoretical findings.
