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Equivalence of second order optimality conditions for bang-bang control problems. Part 1: Main results

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EN
Abstrakty
EN
Second order optimality conditions have been derived in the literature in two different forms. Osmolovskii (1988a, 1995, 2000, 2004) obtained second order necessary and sufficient conditions requiring that, a certain quadratic form be positive (semi)-definite on a critical cone. Agrachev, Stefani, Zezza (2002) first, reduced the bang-bang control problem to finite-dimensional optimization and then show that well-known sufficient optimality conditions for this optimization problem supplemented by the strict bang-bang property furnish sufficient conditions for the bang-bang control problem. In this paper, we establish the equivalence of both forms of sufficient conditions and give explicit relations between corresponding Lagrange multipliers and elements of critical cones. Part 1 summarizes the main results while detailed proofs will be given in Part 2.
Rocznik
Strony
927--950
Opis fizyczny
Bibliogr. 24 poz.
Twórcy
  • Systems Research Institute, Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland, Institute of Mathematics and Physics, The Academy of Podlasie 3 Maja 54, 08-110 Siedlce, Poland
autor
  • Westfalische Wilhelms–Universitat Munster Institut fur Numerische und Angewandte Mathematik Einsteinstr. 62, D–48149 Munster, Germany
Bibliografia
  • Agrachev, A.A., Stefani, G. and Zezza, P.L. (2002) Strong optimality for a bang–bang trajectory. SIAM J. Control and Optimization 41, 991-1014.
  • Ben-Tal, A. and Zowe, J. (1982) A unified theory of first and second order conditions for extremum problems in topological vector spaces. Mathematical Programming Study 19, 39–76.
  • Ben-Tal, A. and Zowe, J. (1982) Necessary and sufficient conditions optimality conditions for a class of nonsmooth minimization problems. Mathematical Programming 24, 70-91.
  • Felgenhauer, U. (2003) On stability of bang–bang type controls. SIAM J. Control and Optimization 41, 1843–1867.
  • Felgenhauer, U. (2005) Optimality properties of controls with bang-bang components in problems with semilinear state equation. Control and Cybernetics 34, in this issue.
  • Hestens (1966) Calculus of Variations and Optimal Control. Kohn Wiley, New York.
  • Kim, J.-H.R., Lippi, G.L. and Maurer, H. (2004) Minimizing the transition time in lasers by optimal control methods. Single mode semiconductor lasers with homogeneous transverse profile. Physica-D – Nonlinear phenomena 191, 238–260.
  • Kim, J.-H.R. and Maurer, H. (2004) Sensitivity analysis of optimal control problems with bang-bang controls. Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Dec. 9-12, 2003, IEEE Control Society, 3281–3286.
  • Levitin, E.S., Milyutin, A.A. and Osmolovskii, N.P. (1974) Conditions for a local minimum in a problem with constraints. In: Mathematical economics and functional analysis. Nauka, Moscow, 130-202.
  • Levitin, E.S., Milyutin, A.A. and Osmolovskii, N.P. (1978) Conditions of high order for a local minimum in problems with constraints. Russian Math. Surveys 33 (6), 97–168.
  • Maurer, H., Buskens, C., Kim, J.-H.R. and Kaya, Y. (2004) Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Submitted to Optimal Control Applications and Methods.
  • Maurer, H. and Osmolovskii, N.P. (2004) Second order sufficient conditions for time optimal bang–bang control problems. SIAM J. Control and Optimization 42, 2239–2263.
  • Maurer, H. and Osmolovskii, N.P. (2003) Second order optimality conditions for bang–bang control problems. Control and Cybernetics 32 (3), 555-584.
  • Milyutin, A.A. and Osmolovskii, N.P. (1998) Calculus of Variations and Optimal Control. Translations of Mathematical Monographs, 180, American Mathematical Society, Providence.
  • Osmolovskii, N.P. (1988a) High-order necessary and sufficient conditions for Pontryagin and bounded-strong minima in the optimal control problems. Dokl. Akad. Nauk SSSR, Ser. Cybernetics and Control Theory 303, 1052–1056, English transl., Sov. Phys. Dokl. 33 (12), 1988, 883–885.
  • Osmolovskii, N.P. (1988b) Theory of higher order conditions in optimal control. Doctor of Science Thesis (in Russian), Moscow.
  • Osmolovskii, N.P. (1995) Quadratic conditions for nonsingular extremals in optimal control (A theoretical treatment). Russian J. of Mathematical Physics 2, 487–516.
  • Osmolovskii, N.P. (2000) Second order conditions for broken extremal. In: A. Ioffe, S. Reich and I. Shafir, eds., Calculus of variations and optimal control (Technion 1998), Chapman and Hall/CRC, Boca Raton, Florida, 198-216.
  • Osmolovskii, N.P. (2002) Second-order sufficient conditions for an extremum in optimal control. Control and Cybernetics 31 (3), 803-831.
  • Osmolovskii, N.P. (2004) Quadratic optimality conditions for broken extremals in the general problem of calculus of variations. Journal of Math. Science 123 (3), 3987-4122.
  • Osmolovskii, N.P. and Lempio, F. (2000) Jacobi–type conditions and Riccati equation for broken extremal. Journal of Math. Science 100 (5), 2572–2592.
  • Osmolovskii, N.P. and Lempio, F. (2002) Transformation of quadratic forms to perfect squares for broken extremals. Journal of Set Valued Analysis 10, 209-232.
  • Poggioloni, L. and Stefani, G. (2003) State-local optimality of a bangbang trajectory: a Hamiltonian approach. Report.
  • Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V. and Mishchenko, E.F. (1961) The Mathematical Theory of Optimal Processes. Fizmatgiz, Moscow; English translation: Pergamon Press, New York, 1964.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-BAT5-0008-0020
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