Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces
Steady-State and Periodic Exponential Turnpike Property for Optimal Control Problems in Hilbert Spaces
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DOI：
10.1137/16m1097638
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发表时间：
2016-10
期刊：
影响因子：
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通讯作者：
E. Trélat;Can Zhang;E. Zuazua
中科院分区：
文献类型：
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作者：
E. Trélat;Can Zhang;E. Zuazua
本文研究了Hilbert空间中最优控制问题的稳态（或周期）指数收费公路性质。收费公路的性质，这基本上是由于双曲线特征的哈密顿系统所产生的庞特里亚金最大值原理，反映了这样一个事实，即在大的控制时间范围内，最佳状态和控制和伴随状态保持大部分时间接近最佳稳态。当用最优周期轨迹代替最优稳态时，类似的陈述也成立。为了建立结果，我们设计了一个适当的二分法变换，代数Riccati和李雅普诺夫方程的解决方案的基础上。我们说明我们的结果与例子，包括线性热和波方程的周期跟踪项。
In this work, we study the steady-state (or periodic) exponential turnpike property of optimal control problems in Hilbert spaces. The turnpike property, which is essentially due to the hyperbolic feature of the Hamiltonian system resulting from the Pontryagin maximum principle, reflects the fact that, in large control time horizons, the optimal state and control and adjoint state remain most of the time close to an optimal steady-state. A similar statement holds true as well when replacing an optimal steady-state by an optimal periodic trajectory. To establish the result, we design an appropriate dichotomy transformation, based on solutions of the algebraic Riccati and Lyapunov equations. We illustrate our results with examples including linear heat and wave equations with periodic tracking terms.