×

Infinite time regular synthesis. (English) Zbl 0921.93016

This paper brings us back to the golden years of open-loop control optimization of Pontryagin, Boltjanskij, Gamkrelidze and Brunovsky. The control system is nonlinear in the state and control, the criterion to be minimized is again nonlinear in these variables with a bound control. The Hamiltonian formalism is introduced. The solution to two second-order systems is obtained.

MSC:

93B50 Synthesis problems
70H05 Hamilton’s equations
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] M. Bardi, I. Capuzzo Dolcetta: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equation, Birkäuser, Boston, 1998. Zbl0890.49011 MR1484411 · Zbl 0890.49011
[2] V.G. Boltianskii: Sufficient conditions for optimality and the justification of the dynamic programming principle, SIAM J. Contr. and Opt., 4, 1966, 326-361. Zbl0143.32004 MR197205 · Zbl 0143.32004 · doi:10.1137/0304027
[3] A. Bressan, B. Piccoli: Structural stability for time-optimal planar syntheses, Dynamics of Continuous, Discrete and Impulsive Systems 3, 1997, 335-371. Zbl0885.93032 MR1461687 · Zbl 0885.93032
[4] A. Bressan, B. Piccoli: A generic classification of time optimal planar stabilizing feedbacks’, SIAM J. Contr. and Opt. 36, 1998, 12-32. Zbl0910.93044 MR1616525 · Zbl 0910.93044 · doi:10.1137/S0363012995291117
[5] P. Brunovsky: Every normal linear system has a regular time-optimal synthesis, Math. Slovaca 28, 1978, 81-100. Zbl0369.49013 MR527776 · Zbl 0369.49013
[6] P. Brunovsky: Existence of regular syntheses for general problems, J. Diff. Eq., 38, 1980, 317-343. Zbl0417.49030 MR605053 · Zbl 0417.49030 · doi:10.1016/0022-0396(80)90011-X
[7] L. Cesari: Optimization - Theory and applications, Springer-Verlag, New York, 1983. Zbl0506.49001 MR688142 · Zbl 0506.49001
[8] W.H. Fleming, R. W. Rishel: Deterministic and stochastic optimal control, Springer-Verlag, New York, 1975. Zbl0323.49001 MR454768 · Zbl 0323.49001
[9] W.H. Fleming, M. Soner: Controlled Markov processes and viscosity solutions, Springer-Verlag, New York, 1993. Zbl0773.60070 MR1199811 · Zbl 0773.60070
[10] B. Piccoli: Regular time-optimal syntheses for smooth planar Systems, Rend. Sem. Mat. Univ. Padova 95, 1996, 59-79. Zbl0912.49018 MR1405355 · Zbl 0912.49018
[11] B. Piccoli: Classification of generic singularities for the planar time optimal syntheses, SIAM J. Contr. Opt. 34, 1996, 1914-1946. Zbl0865.49022 MR1416494 · Zbl 0865.49022 · doi:10.1137/S0363012993256149
[12] B. Piccoli, H.J. Sussmann: Regular synthesis and sufficiency conditions for optimality, to appear in SIAM J. Contr. Opt.. Zbl0961.93014 MR1788064 · Zbl 0961.93014 · doi:10.1137/S0363012999322031
[13] E.P. Ryan: Singular optimal controls for second-order saturating systems, Int. J. Control 30, No. 4 1979, 549-564. Zbl0422.49006 MR554963 · Zbl 0422.49006 · doi:10.1080/00207177908922792
[14] H.J. Sussmann: Subanalytic sets and feedback control, J. Diff. Eq. 31, No.l, 1979, 31-52. Zbl0407.93010 MR524816 · Zbl 0407.93010 · doi:10.1016/0022-0396(79)90151-7
[15] H.J. Sussmann: Lie brackets, real analyticity and geometric control theory, in Differential Geometric Control Theory, R. W. Brockett, R.S. Millman and H.J. Sussmann Eds., Birkhäuser Boston Inc., 1983, 1-115. Zbl0545.93002 MR708500 · Zbl 0545.93002
[16] H.J. Sussmann: Regular synthesis for time-optimal control of single-input real-analytic systems in the plane, SIAM J. Contr. Opt. 25, No. 5, 1987, 1145-1162. Zbl0696.93026 MR905037 · Zbl 0696.93026 · doi:10.1137/0325062
[17] H.J. Sussmann: Recent developments in the regularity theory of optimal trajectories, in Linear and nonlinear mathematical control theory, Rend. Sem. Mat. Univ. e Pol. Torino, Fascicolo speciale, 1987, 149-182. Zbl0649.49003 MR948974 · Zbl 0649.49003
[18] H.J. Sussmann: Synthesis, presynthesis, sufficient conditions for optimality and subanaltic sets, in Nonlinear controllability and optimal control, H.J. Sussmann ed., Marcel Dekker, New York, 1990, 1-19. Zbl0712.49015 MR1061381 · Zbl 0712.49015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.