Refined Isaacs conditions for fractional differential games with memory

Refine the Isaacs condition for zero-sum differential games whose state dynamics are governed by a Caputo fractional differential equation of order 0<alpha<1, and determine conditions under which the lower and upper Isaacs Hamiltonians coincide so that a value exists for the associated time-fractional Hamilton–Jacobi–Bellman–Isaacs equation.

Background

Earlier in the paper, the authors formulate fractional dynamic programming and the time-fractional HJBI equation for zero-sum games with Caputo dynamics, noting that memory introduces path dependence and right-sided fractional time operators.

They emphasize that establishing Isaacs-type conditions ensuring equality of the upper and lower Hamiltonians is crucial for guaranteeing the existence of a value and well-posedness of viscosity solutions in the fractional setting.

References

Open questions. Refinements are still needed on: (a) Isaacs–type conditions under memory; (b) existence/uniqueness for general–sum FO games; (c) verifiable sufficient conditions for Stackelberg equilibria in FDGs; and (d) scalable algorithms with a priori/a posteriori certificates that couple fractional–approximation error with game–theoretic suboptimality.

— Fractional Calculus in Optimal Control and Game Theory: Theory, Numerics, and Applications -- A Survey (2512.12111 - Mojahed et al., 13 Dec 2025) in Section 4, Subsection “Fractional Differential Games (FDGs)”, Open questions paragraph