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Degenerate problems of optimal control. I. (English. Russian original) Zbl 1230.49018
Autom. Remote Control 72, No. 3, 497-511 (2011); translation from Avtom. Telemekh. 2011, No. 3, 36-50 (2011).
Summary: A practically important and theoretically challenging class of optimal control problems which can be integrated within the common notion of “degenerate problems” is considered. The general definition of such problems is given, which arises from the connection between the degeneracy and the presence of hidden passive differential constraints or discrete chains in the problem. This definition is analyzed with the focus on its relation with the classical notion of degeneracy in the variational calculus and the notion of singular and sliding modes well known in the control theory. This paper is the first one in the series of three, which are aimed at presenting a survey of the main facts and applications of the special theory of such problems, which is essentially based on finding and eliminating passive constraints. New results and generalizations are also reported.

49K15 Optimality conditions for problems involving ordinary differential equations
93B17 Transformations
Full Text: DOI
[1] Krotov, V.F., Solution of Variational Problems Using Sufficient Conditions of the Absolute Minimum. I, Autom. Remote Control, 1962, vol. 23, no. 12, pp. 1571–1583.
[2] Krotov, V.F. and Gurman, V.I, Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973. · Zbl 0271.49003
[3] Gurman, V.I., Vyrozhdennye zadachi optimal’nogo upravleniya (Degenerate Problems of Optimal Control), Moscow: Nauka, 1977. · Zbl 0463.49002
[4] Gurman, V.I., Printsip rasshireniya v zadachakh upravleniya (Extension Principle in Control Problems), Moscow: Nauka, 1985. · Zbl 0905.49001
[5] Alekseev, V.M., Tikhomirov, V.M., and Fomin, S.V., Optimal’noe upravlenie (Optimal Control), Moscow: Nauka, 1979. · Zbl 0516.49002
[6] Moskalenko, A.I., Sufficient Conditions of Joint Optimality of Systems, Dokl. Akad. Nauk SSSR, 1977, vol. 232, no. 3, pp. 524–527.
[7] Moskalenko, A.I., Metody nelineinykh otobrazhenii v optimal’nom upravlenii (Nonlinear Mapping Methods in Optimal Control), Novosibirsk: Nauka, 1983.
[8] Agrachev, A.A. and Sachkov, Yu.L., Geometricheskaya teoriya upravleniya (Geometric Control Theory), Moscow: Fizmatlit, 2005. · Zbl 1078.93001
[9] Gurman, V.I., Optimality Controlled Processes with Unbounded Derivatives, Autom. Remote Control, 1972, no. 12, pp. 1924–1930. · Zbl 0274.49024
[10] Filippov, A.F., Differential Equations with Discontinuous Right-Hand Side, Mat. Sb., 1960, vol. 51(93), no. 1, pp. 99–28. · Zbl 0138.32204
[11] Warga, J., Relaxed Variational Problems, J. Math. Anal. Appl., 1962, vol. 4, no. 1, pp. 111–128. · Zbl 0102.31801 · doi:10.1016/0022-247X(62)90033-1
[12] Webster, R., Convexity, Oxford: Oxford Univ. Press, 1994.
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