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The study considers the problem of optimal control for linear discrete systems with a free right end of the trajectory and constraints on control. A new approach to constructing a discrete system is proposed and control is determined at discrete instants of time. Necessary and sufficient conditions for optimality are obtained and a method is proposed for the exact solution of the boundary value problem, which is reduced to solving a finite number of systems of algebraic equations. The proposed method for solving the optimal control problem for a discrete system allows to represent the desired optimal control in the form of synthesising control. For this, a positively definite symmetric matrix is defined that satisfies a difference equation of Riccati type. An algorithm for constructing control for discrete systems is developed, based on the feedback principle, taking into account constraints on the values of controls. The problem is solved using the Lagrange multipliers of a special form, which depend on the phase coordinates at discrete instants of time. The proposed method for solving the problem of optimal control with constraints on control values is implemented on a computer with an application package and tested for the task of planning production and storage of products. Numerical calculations are carried out on a computer using the described problem-solving algorithm in which it is possible to take into account restrictions on the values of controls. Optimal values are determined and appropriate schedules of the production plan, storage of products and limited management at discrete instants of time are constructed.

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For the mathematical model of a three-sector economic cluster, the problem of optimal control with fixed ends of trajectories is considered. An algorithm for solving the optimal control problem for a system with a quadratic functional is proposed. Control is defined on the basis of the principle of feedback. The problem is solved using the Lagrange multipliers of a special form, which makes it possible to find a synthesizing control. The problem of optimal stabilization for a class of nonlinear systems with coefficients that depend on the state of the control object is considered. The results obtained for nonlinear systems are used in the construction of control parameters for a three-sector economic cluster on an infinite time interval.

The problem of optimal control over a finite time interval for a mathematical model of a three-sector economic cluster is posed. The economic system is reduced by means of transformations to the optimal control problem for one class of nonlinear systems with coefficients depending on the state of the control object. Two optimal control problems for one class of nonlinear systems with and without control constraints are considered. The nonlinear objective functional in these problems depends on the control and state of the object. Then, using the results of solving optimal control problems on a finite interval, an algorithm for solving the problem for a nonlinear system of a three-sector economic cluster is developed. A nonlinear control based on the feedback principle using Lagrange multipliers of a special kind is found. The results obtained for nonlinear systems are used to construct the control parameters of a mathematical model of a three-sector economic cluster at a finite time interval with a given functional and various initial conditions. The results of the system state calculation are shown in the figures, the optimal controls satisfy the given constraints. The optimal distribution of labor and investment resources for a three-sector economic cluster is determined. They ensure that the system is brought into an equilibrium state and satisfy balance ratios. These results are useful for practice and are important because there are a number of optimal control problems when it is necessary to transfer a system from an initial state to a desired final state for a given time interval. Such problems often arise for an economic system when a certain level of development is required.