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On viscosity solutions of path-dependent Hamilton--Jacobi--Bellman--Isaacs equations for fractional-order systems

Mikhail I. Gomoyunov

Abstract

This paper deals with a two-person zero-sum differential game for a dynamical system described by a Caputo fractional differential equation of order $α\in (0, 1)$ and a Bolza cost functional. The differential game is associated to the Cauchy problem for the path-dependent Hamilton--Jacobi--Bellman--Isaacs equation with so-called fractional coinvariant derivatives of the order $α$ and the corresponding right-end boundary condition. A notion of a viscosity solution of the Cauchy problem is introduced, and the value functional of the differential game is characterized as a unique viscosity solution of this problem.

On viscosity solutions of path-dependent Hamilton--Jacobi--Bellman--Isaacs equations for fractional-order systems

Abstract

This paper deals with a two-person zero-sum differential game for a dynamical system described by a Caputo fractional differential equation of order and a Bolza cost functional. The differential game is associated to the Cauchy problem for the path-dependent Hamilton--Jacobi--Bellman--Isaacs equation with so-called fractional coinvariant derivatives of the order and the corresponding right-end boundary condition. A notion of a viscosity solution of the Cauchy problem is introduced, and the value functional of the differential game is characterized as a unique viscosity solution of this problem.

Paper Structure

This paper contains 11 sections, 10 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential game and HJBI equation
  4. Differential game
  5. Path-dependent HJBI equation
  6. Properties of value functional
  7. Viscosity solutions
  8. Uniqueness of viscosity solutions
  9. Auxiliary functional
  10. Proof of \ref{['theorem_uniqueness']}
  11. Proofs of \ref{['subsection_Lemmas']}

Key Result

Proposition \oldthetheorem

\newlabelproposition_the_value_is_the_minimax0 Let assumptions (i)--(iv) hold and let the functional $\sigma$ be continuous. Then, the value functional $\rho$ of differential game differential_game is a unique minimax solution of Cauchy problem Cauchy_problem where the Hamiltonian $H$ is given by

Theorems & Definitions (26)

  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Proof 1
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • ...and 16 more