Optimal control problems for the bilinear system of special structure

We consider three optimal control problems (linear, bilinear, and quadratic functionals) with respect to the special bilinear system with a matrix of rank $1$. For the first problem, we obtain two variants of conditions with respect to initial data of the system and functional such that the maximum principle becomes a sufficient optimality condition. In this case, the problem becomes very simple: the optimal control is determined in the integration process of the phase or conjugate system (one Cauchy problem). Then the optimization problem for a bilinear functional is considered. Sufficient optimality conditions for the boundary controls without switching points are obtained. These conditions are represented as inequalities for functions of one variable (time). The optimal control problem with the quadratic functional reduces to bilinear case on the basis of special increment formula.