Abstract <p> This article is devoted to the study of control problems for dynamical systems whoseevolution is described using differential equations with fractional derivatives. The focus is onconstructing a theory of Hamilton–Jacobi equations (and their generalized solutions) adequate forsuch problems and developing methods for constructing optimal feedback control strategies. Mostof the presented results are obtained for zero-sum differential minimax–maximin games with agiven cost functional, which can be considered a natural formalization of control problems forfractional-order systems under conflict and/or uncertainty with an optimal outcome guarantee.Certain propositions are formulated for the special case of optimal control problems. The study isbased on approaches and methods developed at the Ural scientific school of mathematical controltheory.</p>