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.


93C05 Linear systems in control theory
93B17 Transformations
93B25 Algebraic methods
Full Text: DOI


[1] A. G. Chentsov and S. I. Morina, Extensions and Relaxations (Kluwer, Dordrecht, 2002). · Zbl 1031.93001
[2] A. G. Chentsov, ”Nonsequential Approximate Solutions in Abstract Control Problems,” in Proceedings of International Semininar ”Control Theory and Theory of Generalized Solutions of Hamilton-Jacobi Equations” (Ural University, Ekaterinburg, 2006), Vol. 1, pp. 48–60.
[3] A. G. Chentsov, ”Nonsequential Approximate Solutions in Abstract Problems of Attainability,” Russian Academy of Sciences, Ural Branch, Institute of Mathematics and Mechanics 12, 216–241 (2006). · Zbl 1126.49008
[4] A. G. Chentsov, Finitely Additive Measures and Relaxations of Extremal Problems (Nauka, Ekaterinburg, 1993) [in Russian].
[5] A. G. Chentsov, Asymptotic Attainability (Kluwer, Dordrecht, 1997). · Zbl 0859.93002
[6] V. P. Serov and A.G. Chentsov, ”An Extension Construction for Control Problems with Integral Constraints,” Differents. Uravneniya 26(4), 607–618 (1990). · Zbl 0707.49002
[7] A. G. Chentsov and T. Yu. Kashirtseva, ”Generalized Trajectories of Linear Control Systems with Discontinuous Control Coefficients,” Vestn. Chelyabinsk. Univ., Ser. 3. Matem., Mekhan., Informatika, No. 2, 137–146 (1999).
[8] S. I. Morina and A. G. Chentsov, ”On the Extension of One Control Problem with Constrained Energetic Resource and Phase Constraints in Several Coordinates,” Izv. Ross. Akad. Nauk. Teoriya i Sistemy Upravleniya, No. 1, 39–48 (2004).
[9] J. L. Kelley, General Topology (Van Nostrand, New York, 1955; Nauka, Moscow, 1968).
[10] B. M. Miller and E. A. Rubinovich, Optimization of Dynamic Pulse Control Systems (Nauka, Moscow, 2005) [in Russian].
[11] J. Neveu, Bases Mathématiques du Calcul des Probabilités (Mason, Paris, 1964; Mir, Moscow, 1969).
[12] N. Dunford and J. T. Schwartz, Linear Operators. Part 1. General Theory (Interscience Publishers, New York, 1958; Inostrannaya Literatura, Moscow, 1962).
[13] A. A. Melentsov, V. A. Baidosov, and G. M. Zmeyev, Elements of Theory of Measure and Integral: Manual (Ural’sk. Gos. Univ., Sverdlovsk, 1980) [in Russian].
[14] J. Varga, Optimal Control of Differential and Functional Equations (Acad. Press, New York, 1966; Nauka, Moscow, 1977).
[15] S. I. Morina, ”On an Extension of a Linear Control Problem with Phase Constraints,” Differents. Uravneniya 41(4), 490–499 (2005). · Zbl 1124.93031
[16] K. P. S. Bhaskara Rao and M. Bhaskara Rao, Theory of Charges. A Study of Finitely Additive Measures (Acad. Press, New York, 1983). · Zbl 0516.28001
[17] R. Engelking, General Topology (Polish Scientific Publishers, Warsaw, 1977; Mir, Moscow, 1986).
[18] A. N. Sesekin, ”On Connectivity of the Set of Discontinuous Solutions of a Dynamic System with Impulsive Control,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 85–93 (1996) [Russian Mathematics (Iz. VUZ) 40 (11), 82–90 (1996)]. · Zbl 0916.93013
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