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Minimax Solutions of Hamilton--Jacobi Equations with Fractional Coinvariant Derivatives

Mikhail Gomoyunov

Abstract

We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order $α\in (0, 1)$. Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by ordinary 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.

Minimax Solutions of Hamilton--Jacobi Equations with Fractional Coinvariant Derivatives

Abstract

We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant derivatives of an order . Such problems arise naturally in optimal control problems for dynamical systems which evolution is described by ordinary differential equations with the Caputo fractional derivatives of the order . 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.

Paper Structure

This paper contains 18 sections, 161 equations.

Table of Contents

  1. Preliminaries
  2. Differential Inclusions with Fractional Derivatives
  3. Ordinary Differential Inclusions
  4. Functional Differential Inclusions
  5. Hamilton--Jacobi Equation with Fractional Coinvariant Derivatives
  6. Fractional Coinvariant Derivatives
  7. Hamilton--Jacobi Equation
  8. Minimax Solution
  9. Consistency
  10. Comparison Principle
  11. Lyapunov--Krasovskii Functionals
  12. Functional $V_{\gamma, \mu}$
  13. Functional $V^\ast_{\beta, \mu}$
  14. Functional $V_\ast$
  15. Functional $\mathcal{V}_\varepsilon$
  16. ...and 3 more sections

Theorems & Definitions (9)

  • proof
  • proof
  • proof : Proof of Theorem \ref{['theorem_minimax_is_differentiable']}.
  • proof
  • proof
  • proof
  • proof : Proof of Lemma \ref{['lemma_V']}.
  • proof : Proof of Lemma \ref{['lemma_nu_varepsilon']}.
  • proof