## The closure of the sheaf of trajectories of a linear control system with integral constraints.(English. Russian original)Zbl 1180.93047

Russ. Math. 53, No. 12, 50-58 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 12, 59-68 (2009).
Summary: We consider a linear system with discontinuous coefficients controlled by a parameter under an integral constraint imposed on the control resource. It is well known that in such problems the closure of the sheaf of trajectories that correspond to ordinary controls (piecewise constant or measurable functions) coincides with the sheaf of trajectories in a generalized problem, where for generalized controls one uses finite additive measures of bounded variation. Therewith the closure is defined in the topology of pointwise convergence, because the limit elements (the generalized trajectories) may be discontinuous functions. In this paper we prove that any generalized trajectory can be approximated by a sequence of ordinary solutions to the initial system. We propose a concrete technique for constructing such sequences.

### MSC:

 93C05 Linear systems in control theory 93B17 Transformations 93B25 Algebraic methods
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### References:

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