On the set of necessary optimality conditions with positional controls generated by weakly decreasing solutions of the Hamilton-Jacobi inequality
Any weakly decreasing solution of the Hamilton-Jacobi inequality generates a so-called accessory problem of dynamic optimization over Krasovskii-Subbotin constructive motions (Euler curves) produced by extremal feedback control strategies. We derive conditions under which an optimal trajectory of the considered Mayer optimal control problem is a minimizer of the accessory problem for a fixed majorant - a certain solution of the Hamilton-Jacobi inequality. The result is formulated in terms of the compatibility of the latter solution with an optimal trajectory. In the general case of a nonsmooth majorant (and a nonsmooth problem), the optimality condition means that there is a component of the proximal subdifferential of the majorant along the optimal trajectory that coincides with a certain solution of an adjoint inclusion arising in the maximum principle of Kaskosz and Łojasiewicz. This implies the general feedback minimum principle - a global necessary optimality condition, which strengthens all known formulations of the maximum principle for problems without terminal constraints.