Solving Two-Point Boundary Value Problems for a Wave Equation via the Principle of Stationary Action and Optimal Control
Solving Two-Point Boundary Value Problems for a Wave Equation via the Principle of Stationary Action and Optimal Control
复制标题
利用平稳作用和最优控制原理求解波动方程的两点边值问题
DOI：
10.1137/151003167
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
W. McEneaney
中科院分区：
文献类型：
--
作者：
P. Dower;W. McEneaney
提出了一种求解波动方程两点边值问题的新方法。这种新方法利用平稳作用原理，在最优控制的框架内重新表述和解决这类问题。具体地说，提出了无限维最优控制问题，使得波动方程动力学和时间边界数据分别通过相关的哈密顿量和终端收益的选择来捕捉。为了解决任何这种终端收益的最优控制问题，从而解决与该终端收益所封装的边界数据相对应的任何两点边值问题，构造了最优控制问题的基本解。具体地说，对应于任何给定终端收益的最优控制问题可以通过该基本解与指定终端收益的最大加卷积来求解。重要的是，基本解被证明是关于一组算子微分方程组的唯一解定义的二次泛函，并且可以用谱方法计算。给出了一个算例，计算了该基本解并将其应用于求解感兴趣的波动方程的两点边值问题。
A new approach to solving two-point boundary value problems for a wave equation is developed. This new approach exploits the principle of stationary action to reformulate and solve such problems in the framework of optimal control. In particular, an infinite dimensional optimal control problem is posed so that the wave equation dynamics and temporal boundary data of interest are captured via the characteristics of the associated Hamiltonian and choice of terminal payoff respectively. In order to solve this optimal control problem for any such terminal payoff, and hence solve any two-point boundary value problem corresponding to the boundary data encapsulated by that terminal payoff, a fundamental solution to the optimal control problem is constructed. Specifically, the optimal control problem corresponding to any given terminal payoff can be solved via a max-plus convolution of this fundamental solution with the specified terminal payoff. Crucially, the fundamental solution is shown to be a quadratic functional that is defined with respect to the unique solution of a set of operator differential equations, and computable using spectral methods. An example is presented in which this fundamental solution is computed and applied to solve a two-point boundary value problem for the wave equation of interest.