用庞特里亚金最优化原理求解一维守恒律
Solving 1D Conservation Laws Using Pontryagin's Minimum Principle
AI总结：
本文探讨了标量凸守恒律与庞特里亚金最优化原理之间的联系，提出了一种求解守恒律的最小值解方法，并证明该解等价于熵解，扩展了Lax-Oleinik公式至非均匀凸通量函数。
摘要：
本文讨论了标量凸守恒律与庞特里亚金最优化原理之间的联系。对于可以找到关联最优控制问题的通量函数，提出了一种求解守恒律的最小值解。对于标量空间无关凸守恒律，此类控制问题存在，且守恒律的最小值解等同于熵解。这可视为将Lax-Oleinik公式扩展到凸（不一定是均匀凸）通量函数。利用庞特里亚金最优化原理，给出了求解标量凸守恒律最小值解的算法。提供了近似求解空间无关守恒律的数值示例，以展示所提算法的准确性和适用性。此外，还提供了一个使用Chebfun的MATLAB程序（附带演示代码以展示其使用方法）来近似求解具有空间无关通量函数的标量凸守恒律。
英文摘要：
This paper discusses a connection between scalar convex conservation laws and Pontryagin's minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar space-independent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the Lax--Oleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin's minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both space-dependent and space-independent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with space-independent flux functions.
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