Solution for a problem of optimal allocation of resources

The article considers a generalized version of the bilinear problem of optimal investment planning. The economic interpretation of this statement is a simplified version of the general problem of allocation of resources in a two-sector economic model of a special form. Based on the classical principle of maximum, we have obtained conditions on the task options, which preserve the simplest structure of extremal control without special turnpike-type segments. In fact, these conditions provide "correctness" of the only switching point of extremal control. Here, this point is the only root of the nonlinear exponential equation, which has a convenient structure for iterative solution search. Final time conditions that characterize long-term and short-term planning strategies are obtained. Since the considered problem is nonconvex, we have carried out an additional analysis to determine the optimality of extremal controls. An optimality property of the constructed controls is verified by means of sufficient conditions, which are derived from shrewd formulas for the increment of a functional based on phase concavity of Pontryagin’s function and strong extremity of control.