多维一阶双曲系统最优边界控制中的指数转盘性质
Exponential Turnpike in Optimal Boundary Control of Multi-Dimensional First-Order Hyperbolic Systems
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对多维一阶双曲系统的最优边界控制，在内部和边界耗散条件下，通过加权能量估计和对偶性证明了指数转盘性质，并应用于二维圣维南方程的闸门控制。
AI中文摘要：
我们针对多维一阶线性双曲系统的最优边界控制建立了指数转盘性质，其中控制算子和观测算子均为无界，且代价函数可以是非二次的。主要假设是内部耗散和边界耗散条件。这些条件确保相关的输入-状态和输入-输出映射有界，且算子范数与时间范围T无关。这种一致有界性对证明至关重要，证明结合了正问题和伴随问题的加权能量估计、输入-输出对偶性以及强凸性。最后，我们将该结果应用于带有闸门控制的线性化二维圣维南方程，并数值展示了转盘行为。
英文摘要：
We establish an exponential turnpike property for optimal boundary control of multi-dimensional first-order linear hyperbolic systems in which both the control and observation operators are unbounded and the cost functions may be nonquadratic. The main assumptions are internal and boundary dissipation conditions. These conditions ensure that the associated input-to-state and input-to-output maps are bounded, with operator norms independent of the time horizon T. This uniform boundedness is essential to the proof, which combines weighted energy estimates for the forward and adjoint equations, input-output duality, and strong convexity. Finally, we apply the result to the linearized two-dimensional Saint-Venant equations with sluice-gate control and illustrate the turnpike behavior numerically.
