## On an extension of a linear control problem with phase constraints.(English. Russian original)Zbl 1124.93031

Differ. Equ. 41, No. 4, 518-528 (2005); translation from Differ. Uravn. 41, No. 4, 490-499 (2005).
In the paper nonautonomous linear systems in the form $$\dot x(t) = A(t) x(t) + B(t) u(t)$$ with control functions measurable by some finitely additive measures and satisfying to integral constraints of the form $$\int_{[0,\theta]} {\|u(t)\|_r \lambda(dt)} \leq c$$ are under consideration. For such systems generalized control problem is stated. In generalized problem the system trajectory is considered as the limit of ordinary problem trajectories and can be discontinuous. Corresponding limit is considered in the topology of pointwise convergence. It is shown that ordinary solutions are embedded in the space of generalized solutions. In the paper the description of the attraction sets in terms of generalized control problem is obtained and the asymptotic equivalence of two different approaches for the perturbation of phase constraints (with respect to all coordinates of the phase vector and with respect to certain part of coordinates) is proved. The author gives an example in which the limit trajectory is realized by a finitely additive measure and shows that corresponding motion cannot be realized in the class of countably additive measures.

### MSC:

 93C15 Control/observation systems governed by ordinary differential equations 93-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory
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