This paper addresses the problem of solving a class of optimal control problems (OCPs) with infinite-dimensional linear state constraints involving Riesz-spectral operators. Each instance within this class has time/control-dependent polynomial Lagrangian cost and control constraints described by polynomials. We first perform a state-mode discretization of the Riesz-spectral operator. Then we approximate the resulting finite-dimensional OCPs by using a previously known hierarchy of semidefinite relaxations. Under certain compactness assumptions, we provide a converging hierarchy of semidefinite programming relaxations whose optimal values yield lower bounds for the initial OCP. We illustrate our method by two numerical examples, involving a diffusion partial differential equation and a wave equation. We also report on the related experiments.

中文翻译：

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使用占用量和SDP松弛来优化线性PDE的控制

本文解决的问题是解决一类具有Riesz谱算子的具有无限维线性状态约束的最优控制问题（OCP）。此类中的每个实例都具有与时间/控制相关的多项式拉格朗日成本和由多项式描述的控制约束。我们首先执行Riesz谱算子的状态模式离散化。然后，我们使用先前已知的半定性松弛层次来近似所得的有限维OCP。在某些紧凑性假设下，我们提供了半定规划松弛的收敛层次，其最优值产生了初始OCP的下限。我们通过两个数值示例来说明我们的方法，这些示例涉及扩散偏微分方程和波动方程。我们还将报告相关实验。