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Sparsity preserving optimal control of discretized PDE systems

机译：离散PDE系统的稀疏保留最优控制

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We focus on the problem of optimal control of large-scale systems whose models are obtained by discretization of partial differential equations using the Finite Element (FE) or Finite Difference (FD) methods. The motivation for studying this pressing problem originates from the fact that the classical numerical tools used to solve low-dimensional optimal control problems are computationally infeasible for large-scale systems. Furthermore, although the matrices of large-scale FE or FD models are usually sparse banded or highly structured, the optimal control solution computed using the classical methods is dense and unstructured. Consequently, it is not suitable for efficient centralized and distributed real-time implementations. We show that the a priori (sparsity) patterns of the exact solutions of the generalized Lyapunov equations for FE and FD models are banded matrices. The a priori pattern predicts the dominant non-zero entries of the exact solution. We furthermore show that for well-conditioned problems, the a priori patterns are not only banded but also sparse matrices. On the basis of these results, we develop two computationally efficient methods for computing sparse approximate solutions of generalized Lyapunov equations. Using these two methods and the inexact Newton method, we show that the solution of the generalized Riccati equation can be approximated by a banded matrix. This enables us to develop a novel computationally efficient optimal control approach that is able to preserve the sparsity of the control law. We perform extensive numerical experiments that demonstrate the effectiveness of our approach. (C) 2018 Elsevier B.V. All rights reserved.
机译：我们关注于大型系统的最优控制问题，该系统的模型是通过使用有限元（FE）或有限差分（FD）方法离散化偏微分方程获得的。研究这一紧迫问题的动机源自这样一个事实，即用于解决低维最优控制问题的经典数值工具对于大型系统在计算上是不可行的。此外，尽管大型FE或FD模型的矩阵通常是稀疏带状的或高度结构化的，但使用经典方法计算的最优控制解决方案却是密集且无结构的。因此，它不适用于高效的集中式和分布式实时实施。我们表明，FE和FD模型的广义Lyapunov方程的精确解的先验（稀疏）模式是带状矩阵。先验模式预测精确解的主要非零条目。我们进一步表明，对于条件良好的问题，先验模式不仅具有带状而且具有稀疏矩阵。在这些结果的基础上，我们开发了两种计算有效的方法来计算广义Lyapunov方程的稀疏近似解。使用这两种方法和不精确的牛顿法，我们表明广义Riccati方程的解可以通过带状矩阵来近似。这使我们能够开发一种新颖的计算有效的最优控制方法，该方法能够保留控制律的稀疏性。我们进行了广泛的数值实验，证明了我们方法的有效性。 （C）2018 Elsevier B.V.保留所有权利。

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