14. On the problem of optimum control

Annotation: The use of Langrangian multipliers at solution of optimal control problems in linear statement with qua dratic quality criterion leads to the necessity of solving boundary value problem with conditions for multipliers at the right end of control interval. Solution of the obtained equations for the purpose of regulation synthesis in forward time in this case does not produce stabilizing effect, as a rule. For regulation synthesis, the met hod is widely used of analytical construction of optimal regulator based on stabilizing matrix, which is obtained by solution of algebraic Riccati equation. However, in this case, there are some difficulties ‒ the necessity of calculating the stabilizing matrix, impossibility of calculating this matrix in non-stationary problem. The article proposes the regulation synthesis method by way of solving boundary value problem on regulation cycle i nterval. For this purpose, the differential equations for state parameters and Langrangian multipliers are expressed in the form of finite-difference linear relations. Taking into account that the state parameters and Langrangian multipliers are equal to zero at the end of cycle, the Langrangian multipliers at the beginning of cycle are determined by known values of state parameters for the same moment through solving the above linear system. The obtained values form the regulation law. In consequence of small duration of regulation cycle, an amplifying coefficient is introduced in the regulation law. Its value is determined based on results of preliminary modeling. Efficiency of the proposed method was verified by the example of adopted dynamic system, including non-stationary. The amplifying coefficient is fairly simply selected by the type of stabilization process. The proposed method may be used in the control systems of rockets of various purpose for motion parameters regulation.