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We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariantderivatives of an order $\alpha \in (0, 1)$. Such problems arise naturally inoptimal control problems for dynamical systems which evolution is described byordinary differential equations with the Caputo fractional derivatives of theorder $\alpha$. We propose a notion of a generalized in the minimax sensesolution of the considered problem. We prove that a minimax solution exists, isunique, and is consistent with a classical solution of this problem. Inparticular, 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