Optimality conditions for extremal controls in bilinear and quadratic problems

We consider the optimization problem for a bilinear functional with respect to a linear phase system with a modularly constrained control. On the base of exact formulas for the functional increment we establish sufficient optimality conditions for extremal controls. These conditions are stated as inequalities for one-dimensional functions on a time interval. They supplement the maximum principle, keeping the implementation complexity at the same level. The optimization problem for a quadratic functional is reduced to the bilinear case with the help of the matrix conjugate function.