最优控制与Hamilton-Jacobi方程中的状态依赖时滞
State-Dependent Delays in Optimal Control and Hamilton--Jacobi Equations
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- School of Mathematical Sciences and LPMC Nankai University（南开大学数学科学学院）
- Institut de Mathématique de Bourgogne - UMR 5584 CNRS, Université Bourgogne Europe（勃艮第大学欧洲大学）
- School of Mathematical Sciences Nankai University（南开大学数学科学学院）
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中文总结 AI 辅助
本文为状态依赖时滞方程控制问题建立Hamilton-Jacobi理论，通过粘性解刻画值函数，并推导Pontryagin原理与半凹性估计。
中文摘要 AI 辅助
我们为受状态依赖时滞方程支配的有限时域最优控制问题发展了Hamilton-Jacobi理论。在此设定中，时滞时间映射依赖于受控轨迹本身，因此状态和被评估过去状态的点同时变化。自然状态变量因此是整个历史，值泛函定义在Lipschitz历史空间上。在适当的增长、Lipschitz和单调性假设下，我们建立了动态规划原理，并将值泛函刻画为具有协不变导数的相关Hamilton-Jacobi方程的唯一边界粘性解。为适应Lipschitz历史，我们引入了基于多边形扩展生成的有限维投影的粘性解概念，并在所得泛函类中证明了比较原理。在额外正则性假设下，我们推导了Pontryagin最小值原理，其伴随方程包含由时滞状态依赖性引起的前项。我们还获得了通过值泛函的时滞超微分表述的广义横截性关系。最后，我们建立了历史变量中的半凹性以及适当解流形上的联合半凹性估计。
英文摘要
We develop a Hamilton--Jacobi theory for finite-horizon optimal control problems governed by state-dependent delay equations. In this setting, the delayed-time map depends on the controlled trajectory itself, so that both the state and the point at which the past state is evaluated vary simultaneously. The natural state variable is therefore the entire history, and the value functional is defined on a space of Lipschitz histories. Under suitable growth, Lipschitz, and monotonicity assumptions, we establish the dynamic programming principle and characterize the value functional as the unique viscosity solution of the associated Hamilton--Jacobi equation with co-invariant derivatives. To accommodate Lipschitz histories, we introduce a notion of viscosity solution based on finite-dimensional projections generated by polygonal extensions, and prove a comparison principle in the resulting class of functionals. Under additional regularity assumptions, we derive a Pontryagin minimum principle whose adjoint equation contains advanced terms induced by the state dependence of the delay. We also obtain a generalized transversality relation formulated through the delay superdifferential of the value functional. Finally, we establish semiconcavity in the history variable and a joint semiconcavity estimate on an appropriate solution manifold.
