Optimality conditions for extremal controls in bilinear and quadratic problems

We consider the problem of optimization of bilinear functional with respect to linear phase system and modular control constraint. We obtain sufficient conditions of optimality for extremal controls. They complement the maximum principle and preserve the complexity of its implementation. These conditions are presented in the form of inequalities for functions of one variable on the time interval. The optimization problem for quadratic functional is reduced to the bilinear case with the help of the adjoint matrix function.