论文标题
汉密尔顿 - 雅各比贝尔曼方程的稳定流形的深神网络近似
Deep neural network approximations for the stable manifolds of the Hamilton-Jacobi-Bellman equations
论文作者
论文摘要
对于无限 - 摩尼斯控制问题，最佳控制可以由半球形域中的汉密尔顿 - 雅各比 - 贝尔曼（HJB）方程的特征性汉密尔顿系统的稳定歧管表示。在本文中，我们首先从理论上证明，如果一个近似值足够接近某种意义上的HJB方程的确切稳定歧管，则该近似值从该近似范围得出的控制稳定了系统，并且几乎是最佳的。然后，基于理论结果，我们提出了一种深度学习算法，以近似稳定的歧管和计算最佳反馈控制。该算法通过在稳定歧管中随机查找轨迹来依赖自适应数据生成。这种算法基本上是无网格的，因此它可能适用于广泛的高维非线性系统。我们通过两个例子来证明我们的方法的有效性：稳定反应轮摆和控制抛物线抗抛物线的Allen-CAHN方程。
For an infinite-horizon control problem, the optimal control can be represented by the stable manifold of the characteristic Hamiltonian system of Hamilton-Jacobi-Bellman (HJB) equation in a semiglobal domain. In this paper, we first theoretically prove that if an approximation is sufficiently close to the exact stable manifold of the HJB equation in a certain sense, then the control derived from this approximation stabilizes the system and is nearly optimal. Then, based on the theoretical result, we propose a deep learning algorithm to approximate the stable manifold and compute optimal feedback control numerically. The algorithm relies on adaptive data generation through finding trajectories randomly within the stable manifold. Such kind of algorithm is grid-free basically, making it potentially applicable to a wide range of high-dimensional nonlinear systems. We demonstrate the effectiveness of our method through two examples: stabilizing the Reaction Wheel Pendulums and controlling the parabolic Allen-Cahn equation.
