Мирослав Михайлович Копець, Сергій Францович Сабол


The article investigates the linear-quadratic problem of optimal control for the process of the vibrating string. The urgency of this problem is not in doubt. In contrast, the most common methods of investigation of this problem (the Pontryagin maximum principle, dynamic programming Bellman method), in the article the method of Lagrange is implemented. As a result, necessary optimality conditions received. The conditions identified to ensure the uniqueness of the optimal control. A system of integral-differential Riccati equations and additional conditions for it obtained. The solution of this system gives the opportunity to provide optimal control as explicit form. The concrete examples and graphic illustration of the main results observed. In the future, it is promising to study the resulting functions of systems (14) and (16). Also the analysis of a similar mathematical model with stochastic parameters represents an interest for investigation.


quadratic functional; string vibrations; Bellman method of dynamic programming; Lagrange multipliers method; optimal control; maximum principle of Pontryagin; system of integro-differential Riccati equations


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