We construct an algorithm for solving a constrained optimal control problem for a first-order evolutionary system governed by a positive self-adjoint operator. The problem consists in identifying distributed control that minimizes a given cost functional, which comprises a cost of the control and a trajectory regulation term, while steering the final state close to a given target. The approach explores the dual problem and it generalizes the Hilbert Uniqueness Method (HUM). The practical implementation of the algorithm is based on a spectral decomposition of the operator determining the dynamics of the system. Once this decomposition is available – which can be done offline and saved for future use – the optimal control problem is solved almost instantaneously. It is practically reduced to a scalar nonlinear equation for the optimal Lagrange multiplier. The efficiency of the algorithm is demonstrated through numerical examples corresponding to different types of control operators and penalization terms.

中文翻译：

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线性抛物线方程的谱分解最优分布控制

我们构建了一种算法，用于解决由正自伴随算子控制的一阶进化系统的约束最优控制问题。问题在于识别最小化给定成本函数的分布式控制，其中包括控制成本和轨迹调节项，同时使最终状态接近给定目标。该方法探索了对偶问题，并推广了希尔伯特唯一性方法 (HUM)。该算法的实际实现基于确定系统动态的算子的频谱分解。一旦这种分解可用——可以离线完成并保存以备将来使用——优化控制问题几乎立即得到解决。它实际上被简化为最优拉格朗日乘子的标量非线性方程。通过不同类型的控制算子和惩罚项对应的数值算例，证明了算法的有效性。