  We consider a Cauchy problem for a Hamilton--Jacobi equation with coinvariant
derivatives of an order $\alpha \in (0, 1)$. 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 $\alpha$. 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.
