On the Minimization of a Degenerate Quadratic FunctionalстатьяЭлектронная публикация
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Дата последнего поиска статьи во внешних источниках: 11 ноября 2019 г.
Аннотация:On the linear control system, we consider an integral quadratic functional with a degenerate coefficient at the square of control. The problem is to find its minimum under a given initial value of the state variable and a free terminal value which comes into the terminal part of the functional. Using a change of variable, the so-called Goh transformation, a passage to a new control and thus to an extended space of admissible controls is performed. Rewritten in the new variables, the functional may be then nondegenerate, i.e., may satisfy the strengthened Legendre condition with respect to the new control. Assuming the positive definiteness of the transformed functional for zero initial state value, we prove the existence of its minimum for any initial values of the state variable and then show that this minimum is a quadratic form of the initial value, the matrix of which satisfies the corresponding Riccati equation. It is also proved that the minimum of the transformed functional is equal to the infinum of the initial one, and a minimization sequence for the later one is constructed.