Preview
Unable to display preview. Download preview PDF.
References
Agrachev, A.A. and R.V. Gamkrelidze, “A second-order optimality principle for a time-optimal problem,” Math. Sbornik 100, 142 (1976).
Agrachev, A.A. and R.V. Gamkrelidze, “The exponential representation of flows and the chronological calculus,” Math. Sbornik 109, 149 (1978).
Agrachev, A.A. and R.V. Gamkrelidze, “The Morse index and the Maslov index for smooth control systems,” Doklady Akad. Nauk USSR 287 (1986).
Ball, J.M. and V.J. Mizel, “One-dimensional variational problems whose minimizers do not satsify the Euler-Lagrange equation,” Arch. Rational Mech. Anal. 63 (1985), pp. 273–294.
Bell, D.J. and D.H. Jacobson, Singular Optimal Control Problems, Academic Press (1973).
Berkovitz, L.D. Optimal Control Theory, Springer-Verlag (1974).
Bianchini, R.M., and G. Stefani, “Sufficient conditions for local controllability,” in Proceedings 25th IEEE Conference on Decision and Control (1986).
Bianchini, R.M., and G. Stefani, “Local controllability about a reference trajectory,” in Analysis and Optimization of Systems, A. Bensoussan and J.L. Lions eds., Springer-Verlag Lect. Notes Contr. Inf. Sci. 83 (1986).
Boltyansky, V.G., “Sufficient conditions for optimality and the justification of the Dynamic Programming Principle,” SIAM J. Control 4 (1966), pp. 326–361.
Bressan, A., “A high-order test for optimality of bang-bang controls,” SIAM J. Control Opt. 23 (1985), pp. 38–48.
Bressan, A., “The generic optimal stabilizing controls in dimension 3,” SIAM J. Control Opt. 24 (1986), pp. 177–190.
Brockett, R.W., “Lie theory, functional expansions and necessary conditions in optimal control,” in Mathematical Control Theory, W.A. Coppel ed., Springer-Verlag (1978), pp. 68–76.
Brunovsky, P., “Every normal linear system has a regular synthesis,” Mathematica Slovaca, 28 (1978), pp. 81–100.
Brunovsky, P., “Existence of regular synthesis for general problems,” J. Diff. Equations, 38 (1980), pp. 317–343.
Coddington, E.A. and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill (1955).
Crandall, M.G. and P.L. Lions, “Viscosity solutions of Hamilton-Jacobi equations,” Trans. Amer. Math. Soc. 277 (1983), pp. 1–42.
Crouch, P.E. and F. Lamnabhi-Lagarrigue, “Algebraic and multiple integral identities,” to appear in Acta Applicandae Mathematicae.
Fleming, W.H. and R.W. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag (1975).
Frankowska, H., “An open mapping principle for set-valued maps,” Centre de Recherches Mathématiques, Univ. de Montréal, rapport CRM-1364 (1986).
Gabasov, V., and F.M. Kirillova, “High-order necessary conditions for optimality,” SIAM J. Control 10 (1972), pp. 127–168.
Goh, B.S., “The second variation for the Bolza problem,” SIAM J. Control 4 (1966), pp. 309–325.
Hardt, R.M., “Stratifications of real analytic maps and images,” Inventiones Math. 28 (1975), pp. 193–208.
Hermes, H., “On local and global controllability,” SIAM J. Control 12 (1974), pp. 252–261.
Hermes, H., “Lie algebras of vector fields and local approximation of attainable sets,” SIAM J. Control Opt. 16 (1978), pp. 715–727.
Hermes, H., “Control systems which generate decomposable Lie algebras,” J. Diff. Equations 44 (1982), pp. 166–187.
Hironaka, H., Subanalytic Sets, Lect. Notes Istituto Matematico “Leonida Tonelli,” Pisa, Italy (1973).
Jacobson, D.H. and J.L. Speyer, “Necessary and sufficient conditions for optimality for singular control problems: a limit approach,” J. Math. Anal. Appl. 34 (1971), pp. 239–266.
Kawski, M., “A new necessary condition for local controllability,” to appear in Proceedings Conference on Differential Geometry, San Antonio, Texas, 1988, Amer. Math. Society Contemporary Mathematics Series.
Kawski, M., “High order small-time local controllability,” to appear in Nonlinear Optimal Control and Controllability, H.J. Sussmann ed., M. Dekker, Inc.
Kelley, H.J., R.E. Kopp and H.G. Moyer, “Singular extremals,” in Topics in Optimization, G. Leitman ed., Academic Press (1967).
Knobloch, H.W., High Order Necessary Conditions in Optimal Control, Springer-Verlag (1975).
Krener, A.J., “The higher order maximum principle and its application to singular extremals,” SIAM J. Control Opt. 15 (1977), pp. 256–293.
Kupka, I., “The ubiquity of Fuller's phenomenon,” to appear in Nonlinear Optimal Control and Controllability, H.J. Sussmann ed., M. Dekker, Inc.
Lamnabhi-Lagarrigue, F., “Singular optimal control problems: on the order of a singular arc,” Systems and Control Letters 9 (1987), pp. 173–182.
Lamnabhi-Lagarrigue, F., and G. Stefani, “Singular optimal control problems: on the necessary conditions for optimality,” to appear in SIAM J. Control Opt.
Lee, E.B. and L. Markus, Foundations of Optimal Control Theory, J. Wiley, New York (1967).
Lions, P.L. and T. Souganidis, “Differential games, optimal control and directional derivatives of viscosity solutions of Bellman's and Isaacs' equations,” SIAM J. Control Opt. 23 (1985), pp. 566–583.
Marchal, C., “Chattering arcs and chattering controls,” J. Optim Theory Appl. 11 (1973), pp. 441–468.
Pontryagin, L.S., V.G. Boltyansky, R.V. Gamkrelidze and E.F. Mischenko, The Mathematical Theory of Optimal Processes, J. Wiley (1962).
Schättler, H., “On the local structure of time-optimal bang-bang trajectories in ℝ3,” SIAM J. Control Opt. 26 (1988), pp. 186–204.
Schättler, H., “The local structure of time-optimal trajectories in ℝ3 under generic conditions,” SIAM J. Control Opt. 26 (1988), pp. 899–918.
Schättler, H., and H.J. Sussmann, “On the regularity of optimal controls,” J. Appl. Math. Physics (ZAMP) 38 (1987), pp. 292–301.
Serre, J.P., Lie algebras and Lie groups, W.A. Benjamin, New York (1975).
Stefani, G., “Local controllability of nonlinear systems: an example,” Systems and Control Letters 6 (1985), pp. 123–125.
Stefani, G., “On the local controllability of a scalar-input control system,” in Theory and Applications of Nonlinear Control systems, C.l. Byrnes and A. Lindquist eds., Elsevier (1986).
Stefani, G., “A sufficient condition for extremality,” in Analysis and Optimization of Systems, A. Bensoussan and J.L. lions eds., Springer-Verlag Lect. Notes Contr. Inf. Sci. 111 (1988), pp. 270–281.
Sussmann, H.J. and V. Jurdjevic, “Controllability of nonlinear systems,” J. Diff. Equations 12 (1972), pp. 95–116.
Sussmann, H.J., “A bang-bang theorem with bounds on the number of switchings,” SIAM J. Control Opt. 17 (1979), pp. 629–651.
Sussmann, H.J., “Lie brackets and local controllability: a sufficient condition for scalar input systems,” SIAM J. Control Opt. 21 (1983), pp. 686–713.
Sussmann, H.J., “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), pp. 1–115.
Sussmann, H.J., “A Lie-Volterra expansion for nonlinear systems,” in Mathematical Theory of Networks and Systems, Proceedings of the MTNS-83 International Symposium, Beer-Sheva, Israel, P.A. Fuhrmann Ed., Springer-Verlag (1984), pp. 822–828.
Sussmann, H.J., “Lie Brackets and real analyticity in control theory,” in Mathematical Control Theory, C. Olech ed., Banach Center Publications, Volume 14, PWN-Polish Scientific Publishers, Warsaw, Poland, 1985, pp. 515–542.
Sussmann, H.J., “Resolution of singularities and linear time-optimal control,” in Proceedings 23rd IEEE Conference on Decision and Control, Las Vegas, Nevada (Dec. 1984), pp. 1043–1046.
Sussmann, H.J., “A general theorem on local controllability,” SIAM J. Control Opt. 25 (1987), pp. 158–194.
Sussmann, H.J., “The structure of time-optimal trajectories for single-input systems in the plane: the C ∞ nonsingular case,” SIAM J. Control Opt. 25 (1987), pp. 433–465.
Sussmann, H.J., “The structure of time-optimal trajectories for single-input systems in the plane: the general real-analytic case,” SIAM J. Control Opt. 25 (1987) pp. 868–904.
Sussmann, H.J., “Regular synthesis for time-optimal control of single-input realanalytic systems in the plane,” SIAM J. Control Opt. 25 (1987), pp. 1145–1162.
Sussmann, H.J., “A product expansion for the Chen series,” in Theory and Applications of Nonlinear Control Systems, C. Byrnes and A. Lindquist Eds., North-Holland (1986), pp. 323–335.
Sussmann, H.J., “Envelopes, conjugate points and optimal bang-bang extremals,” in Algebraic and Geometric Methods in Nonlinear Control Theory, M. Fliess and M. Hazewinkel Eds., D. Reidel Publishing Co., Dordrecht, The Netherlands (1986), pp.325–346.
Sussmann, H.J., “A weak regularity theorem for real analytic optimal control problems,” Revista Matemálica Iberoamericana 2 (1986), pp. 307–317.
Sussmann, H.J., “Recent developments in the regularity theory of optimal trajectories,” in Linear and Nonlinear Mathematical Control Theory, Rendiconti del Seminario Matematico, Università e Politecnico di Torino, Fascicolo Speciale 1987, pp. 149–182.
Sussmann, H.J., “Real analytic designularization and subanalytic sets: an elementary approach,” to appear in Transactions Amer. Math. Soc.
Viennot, G., Algèbres de Lie et Monoïdes Libres, Springer-Verlag (1978).
Wagner, K., “Über den Steuerbarkeitsbegriff bei nichtlinearen Kontrollsystemen,” Arch. Math. 47 (1986), pp. 29–40.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1989 Springer-Verlag
About this chapter
Cite this chapter
Sussmann, H.J. (1989). Optimal Control. In: Nijmeijer, H., Schumacher, J.M. (eds) Three Decades of Mathematical System Theory. Lecture Notes in Control and Information Sciences, vol 135. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0008471
Download citation
DOI: https://doi.org/10.1007/BFb0008471
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-51605-7
Online ISBN: 978-3-540-46709-0
eBook Packages: Springer Book Archive