Abstract
Consideration was given to various cases of controllability of the limit system describing behavior of a system with unlimited set of rates under (practically sufficiently high) pulse control actions. In compliance with these cases, the structure of the generalized solutions of the initial system was determined as regular solutions of a smaller-order system called the derivative system. Using the facts of the geometrical control theory, proposed was a multistage procedure for construction of the sequences of permissible solutions of the initial system approximating solutions of the derivative system. The presentation was illustrated by an example.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Krotov, V.F., Discontinuous Solutions of the Variational Problems, Izv. Vuzov, Mat., 1961, no. 1, pp. 86–98.
Gurman, V.I., On Optimal Processes of Special Control, Avtom. Telemekh., 1965, no. 5, pp. 782–791.
Gurman, V.I., On Optimal Processes with Unlimited Derivatives, Avtom. Telemekh., 1972, no. 12, pp. 14–21.
Gurman, V.I., Vyrozhdennye zadachi optimal’nogo upravleniya (Degenerate Problems of Optimal Control), Moscow: Nauka, 1977.
Gurman, V.I., Printsip rasshireniya v zadachakh upravleniya (Principle of Extension in the Problems of Control), Moscow: Nauka, 1997.
Kolokol’nikova, G.A., Study of the Generalized Solutions of the Problems of Optimal Control with Unlimited Linear Controls on the Basis of Multiple Transformations, Diff. Uravn., 1992, vol. 28, no. 11, pp. 1919–1932.
Dykhta, V.A. and Samsonyuk, O.N., Optimal’noe impul’snoe upravlenie s prilozheniyami (Optimal Pulse Control with Applications), Moscow: Fizmatlit, 2000.
Rashevskii, P.K., On Connectibility of Any Two Points of the Completely Nonholonomic Space by an Admissible Line, in Uch. Zap. Mosk. Gos. Ped. Inst., Ser. Fiz.-mat., 1938, vol. 3, no. 2, pp. 83–94.
Agrachev, A.A. and Sachkov, Yu.L., Geometricheskaya teoriya upravleniya (Geometrical Control Theory), Moscow: Fizmatlit, 2005.
Bacciotti, A. and Stefani, G., On the Relationship between Global and Local Controllability, Math. Syst. Theory, 1983, vol. 16, pp. 79–91.
Krotov, V.F. and Gurman, V.I., Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973.
Gaishun, I.V., Vpolne razreshimye mnogomernye differentsial’nye uravneniya (Completely Solvable Multidimensional Differential Equations), Minsk: Nauka and Tekhnika, 1983.
Author information
Authors and Affiliations
Additional information
Original Russian Text © V.I. Gurman, Yu.L. Sachkov, 2008, published in Avtomatika i Telemekhanika, 2008, No. 4, pp. 72–80.
This work was supported by the Russian Foundation for Basic Research, project no. 06-01-00330.
Rights and permissions
About this article
Cite this article
Gurman, V.I., Sachkov, Y.L. Representation and realization of the generalized solutions of the unlimited-locus controllable systems. Autom Remote Control 69, 609–617 (2008). https://doi.org/10.1134/S0005117908040073
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117908040073