Minimax solutions of Hamilton--Jacobi equations with fractional coinvariant derivatives
Minimax solutions of Hamilton--Jacobi equations with fractional coinvariant derivatives
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具有分数共变导数的 Hamilton--Jacobi 方程的极小极大解
DOI：
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发表时间：
2020
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通讯作者：
M. Gomoyunov
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文献类型：
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作者：
M. Gomoyunov
我们考虑一个Hamilton—Jacobi方程的Cauchy问题，该方程在（0,1）$中具有一阶的协不变导数。这类问题自然出现在动态系统的最优控制问题中，这些系统的演化是由具有阶阶卡普托分数阶导数的微分方程描述的。我们提出了这个问题在极大极小意义上的广义解的概念。证明了该问题的极大极小解存在、唯一且与经典解一致。特别地，我们特别注意比较原理的证明，这需要构造一个合适的Lyapunov—Krasovskii泛函。
We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $alpha in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives of the order $alpha$. We propose a notion of a generalized in the minimax sense solution of the considered problem. We prove that a minimax solution exists, is unique, and is consistent with a classical solution of this problem. In particular, we give a special attention to the proof of a comparison principle, which requires construction of a suitable Lyapunov--Krasovskii functional.