论文标题
通过动态编程方程控制分数扩散问题
Control of fractional diffusion problems via dynamic programming equations
论文作者
论文摘要
我们探讨了包含分数laplacian项的全差异方程的反馈控制的近似。 为了获得该非局部方程的状态变量的反馈控制，我们使用汉密尔顿 - 雅各布 - 贝尔曼方程。 众所周知，这种方法遭受了维度的诅咒，为了减轻这个问题，我们将半拉格朗日方案伴随着使用Shepard近似的动态编程原理离散化。这种耦合可以使高维问题的近似值近似。与线性和非线性示例一起提供了针对连续问题解决方案的数值收敛。通过与开环控制方法的比较来说明该方法在系统干扰方面的鲁棒性。
We explore the approximation of feedback control of integro-differential equations containing a fractional Laplacian term. To obtain feedback control for the state variable of this nonlocal equation we use the Hamilton--Jacobi--Bellman equation. It is well-known that this approach suffers from the curse of dimensionality, and to mitigate this problem we couple semi-Lagrangian schemes for the discretization of the dynamic programming principle with the use of Shepard approximation. This coupling enables approximation of high dimensional problems. Numerical convergence toward the solution of the continuous problem is provided together with linear and nonlinear examples. The robustness of the method with respect to disturbances of the system is illustrated by comparisons with an open-loop control approach.
