Second Order Optimality Conditions for Controls with Continuous and Bang-Bang Components

  • N. P. Osmolovskii
  • H. Maurer
Part of the IFIP International Federation for Information Processing book series (IFIPAICT, volume 202)


Second order necessary and sufficient optimality conditions for bang-bang control problems in a very general form have been obtained by the first author. These conditions require the positive (semi)-definiteness of a certain quadratic form on the finite-dimensional critical cone. In the present paper we formulate a generalization of these results to optimal control problems where the control variable has two components: a continuous unconstrained control appearing nonlinearly and a bang-bang control appearing linearly and belonging to a convex polyhedron. Many examples of control of this kind may be found in the literature.


bang-bang control Pontryagin minimum principle second order necessary and sufficient conditions critical cone quadratic form strengthened Legendre condition 


  1. [1]
    A.A. Agrachev, G. Stefani, R.L. Zezza Strong optimality for a bang-bang trajectory. SIAM J. Control and Optimization 41:991–1014, 2002.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    H. Maurer, N.R. Osmolovskii. Second order sufficient conditions for time optimal bang-bang control problems. SIAM J. Control and Optimization 42:2239–2263, 2004.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    H. Maurer, N.R. Osmolovskii. Second order optimality conditions for bang-bang control problems. Control & Cybernetics 32:555–584, 2003.Google Scholar
  4. [4]
    A.A. Milyutin, N.P. Osmolovskii. Calculus of Variations and Optimal Control. Translations of Mathematical Monographs, Vol. 180, American Mathematical Society, Providence, 1998.MATHGoogle Scholar
  5. [5]
    N.R Osmolovskii. 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, 1988; English transl., Sov. Phys. Dokl. 33, N. 12:883–885, 1988.MATHMathSciNetGoogle Scholar
  6. [6]
    N.R Osmolovskii. Quadratic conditions for nonsingular extremals in optimal control (A theoretical treatment). Russian J. of Mathematical Physics 2:487–516, 1995.MathSciNetGoogle Scholar
  7. [7]
    N.R Osmolovskii. Second order conditions for broken extremal. In: Calculus of variations and optimal control. (Technion 1998), A. Ioffe, S. Reich and I. Shafir, eds., Chapman and Hall/CRC, Boca Raton, Florida, 198–216, 2000.Google Scholar
  8. [8]
    N.P. Osmolovskii. Second-order sufficient conditions for an extremum in optimal control. Control and Cybernetics 31 803–831, 2002.MATHMathSciNetGoogle Scholar
  9. [9]
    N.P. Osmolovskii. Quadratic optimality conditions for broken extremals in the general problem of calculus of variations. Journal of Math. Science 123: 3987–4122, 2004.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© International Federation for Information Processing 2006

Authors and Affiliations

  • N. P. Osmolovskii
    • 1
    • 2
  • H. Maurer
    • 3
  1. 1.Systems Research InstitutePolish Academy of SciencesWarszawaPoland
  2. 2.University of Podlasie in SiedlceSiedlcePoland
  3. 3.Institut für Numerische und Angewandte MathematikWilhelms-Universität MünsterMünsterGermany

Personalised recommendations