Abstract
We consider a Cauchy problem for a Hamilton–Jacobi equation with coinvariant derivatives of an order α ∈ (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 α. 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.
Highlights
Nowadays, the theory of differential equations with fractional-order derivatives is an actively developing branch of mathematics, which attracts the interest of many researchers
Attention is paid to optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives
This technique was extended to Hamilton–Jacobi equations with first-order ci-derivatives, which arise in optimization problems for dynamical systems described by functional differential equations of a retarded type [37] and of a neutral type [38, 39, 42]
Summary
The theory of differential equations with fractional-order derivatives (see, e.g., [8, 22, 41, 43, 46]) is an actively developing branch of mathematics, which attracts the interest of many researchers. Attention is paid to optimal control problems for dynamical systems which evolution is described by differential equations with the Caputo fractional derivatives. By analogy with the case of optimal control problems for dynamical systems described by ordinary differential equations (i.e., when α = 1), the value functional usually does not possess the required smoothness properties, which leads to the need to introduce and study generalized solutions of the obtained Cauchy problem. Minimax solutions of Hamilton–Jacobi equations with first-order partial derivatives were proposed and comprehensively studied in [48] (see [49]) This technique was extended to Hamilton–Jacobi equations with first-order ci-derivatives, which arise in optimization problems for dynamical systems described by functional differential equations of a retarded type [37] (see [17, 31–34, 36], and [3] for an infinite dimensional case) and of a neutral type [38, 39, 42].
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