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Geometrical and topological methods in optimal control theory. (English) Zbl 0846.93016

This is yet another survey paper on geometric and topological methods in control theory, adding to at least 36 other surveys on the same topic published in the last 20-25 years. The present paper contains 165 pages from which the largest part (91 pages) is occupied by a long bibliographical list of 1680 items.
Starting with a short but clear overview of the so-called “chronological calculus”, the author presents in the first part of the paper the main concepts and results obtained in the last 30 years on the orbits of families of vector fields, global and local controllability of smooth nonlinear systems, state-space and feedback equivalence, minimal realization and necessary optimality conditions.
In the second part of the paper, the author presents a survey, mostly of his own results, concerning bang-bang theorems for smooth systems of constant rank and some applications of the classical critical point theory of Morse, Lusternik-Schnirelman and minimax principles of the Mountain-Pass-Theorem type to optimal control problems.
Accurately and clearly written, covering most of the main aspects of the subject and providing an almost complete and up-to-date bibliographical list, the present survey may prove to be a useful working-tool for the specialists and even for the biginners in the field.

MSC:

93B29 Differential-geometric methods in systems theory (MSC2000)
93-02 Research exposition (monographs, survey articles) pertaining to systems and control theory
49K15 Optimality conditions for problems involving ordinary differential equations
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References:

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[1449] H. J. Sussmann, ”Minimal realizations and canonical forms for bilinear systems,”J. Franklin Inst.,301, No. 6, 593–604 (1976). · Zbl 0335.93021 · doi:10.1016/0016-0032(76)90080-6
[1450] H. J. Sussmann, ”Semigroup representations, bilinear approximations of input-output maps and generalized inputs,”Lect. Notes Econ. Math. Sci.,131, 172–191 (1976). · Zbl 0353.93025
[1451] H. J. Sussmann, ”Existence and uniqueness of minimal realizations of nonlinear systems,”Math. Syst. Theory,10, No. 2, 263–284 (1977). · Zbl 0354.93017 · doi:10.1007/BF01683278
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[1453] H. J. Sussmann, ”On the gap between deterministic and ordinary differential equations,”Appl. Probability,6, 19–41 (1978). · Zbl 0391.60056
[1454] H. J. Sussmann, ”Subanalytic sets and feedback control,”J. Differ. Equat.,31, No. 1, 31–52 (1979). · Zbl 0407.93010 · doi:10.1016/0022-0396(79)90151-7
[1455] H. J. Sussmann, ”Generic single-input observability of continuous time polynomial systems,” In:Proc. IEEE Conf. Decis. and Contr., San Diego, Calif., 1979, New York (1979), pp. 566–571. · Zbl 0428.93030
[1456] H. J. Sussmann, ”Single-input observability of continuous-time systems,”Math. Syst. Theory,12, No. 4, 371–393 (1979). · Zbl 0422.93019 · doi:10.1007/BF01776584
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[1460] H. J. Sussmann, ”Les semi-groups sous-analytiques et le regularite des commandes en boucle fermee,”Asterisque, No. 75–76, 219–226 (1980).
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[1463] H. J. Sussmann, ”Time-optimal control in the plane,”Lect. Notes Contr. Inf. Sci.,39, 244–260 (1982). · Zbl 0516.93025 · doi:10.1007/BFb0006833
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[1466] H. J. Sussmann, ”Lie brackets and local controllability: a sufficient condition for scalar-input systems,”SIAM J. Contr. Optimiz.,21, No. 5, 686–713 (1983). · Zbl 0523.49026 · doi:10.1137/0321042
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[1469] H. J. Sussmann, ”Lie brackets and real analycity in control theory,” In:Banach Center Publ. Vol. 14, PWN, Warsaw (1985), pp. 415–422.
[1470] H. J. Sussmann, ”A general theorem on local controllability,”New Brunswick, N. J., Math. Dep. Rutgers Univ. (1985). · Zbl 0629.93012
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[1476] H. J. Sussmann, ”Small time local controllability and continuity of the optimal time functions for linear systems,”J. Optimiz. Theory Appl.,43, No. 2, 281–296 (1987). · Zbl 0596.93012 · doi:10.1007/BF00939220
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