Differential Games in Fractional-Order Systems: Inequalities for Directional Derivatives of the Value Functional

For a dynamical system described by differential equations with Caputo fractional derivatives of order $\alpha \in (0,1)$, we consider a minimax–maximin differential game with a performance index that estimates the motion of the system on a fixed finite time interval. We obtain differential inequalities that characterize the value functional of the game in terms of appropriate directional derivatives.