Sufficient Optimality Conditions for a Bang-bang Trajectory in a Bolza Problem

  • Laura Poggiolini
  • Marco Spadini


This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of the ones considered in [4, 11] and [12]. We require both the strict bang-bang Legendre condition, a non degeneracy condition at multiple switching times, and the second order conditions for the finite dimensional problems obtained by moving the switching times of the reference trajectory.


Optimal Control Problem Switching Time Hamiltonian Function Reference Trajectory Hamiltonian Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    . Agrachev AA, Gamkrelidze RV (1990) Symplectic geometry for optimal control. In: Sussmann HJ (ed) Nonlinear Controllability and Optimal Control, Pure and Applied Mathematics, vol 133. Marcel Dekker.Google Scholar
  2. 2.
    Agrachev AA, Gamkrelidze RV (1997) Symplectic methods for optimization and control. In: Jacubczyk B, Respondek W (eds) Geometry of Feedback and Optimal Control, Pure and Applied Mathematics, 1–58, Marcel Dekker, New York.Google Scholar
  3. 3.
    . Agrachev AA, Sachkov, YuL (2004) Control Theory from the Geometric View point Springer-Verlag.Google Scholar
  4. 4.
    Agrachev AA, Stefani G, Zezza P (2002) Strong optimality for a bang-bang trajectory. SIAM J. Control Optimization, 41(4):991–1014.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arnold VI (1980) Mathematical Methods in Classical Mechanics. Springer, New York.Google Scholar
  6. 6.
    Koslik B, Breitner MH (1997) In optimal control problem in economics with four linear control. J. Optim. Theory and App., 94(3):619–634.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Kuntz L, Scholtes S (1994) Structural analysis of nonsmooth mappings, inverse.functions and metric projections. Journal of Mathematical Analysis and Applications, 188:346–386.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Maurer H, Osmolovskii NP (2003) Second order optimality conditions for bang bang control problems. Control and Cybernetics, 3(32):555–584.Google Scholar
  9. 9.
    . Milnor J (1965) Topology from the Differentiable Viewpoint. The University.Press of Virginia.Google Scholar
  10. 10.
    Jong-Shi Pang J-S, Ralph D (1996) Piecewise smoothness, local invertibility, and parametric analysis of normal maps. Mathematics of Operations Research, 21(2):401–426.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    . Poggiolini L (2006) On local state optimality of bang-bang extremals in a free horizon bolza problem. Rendiconti del Seminario Matematico dell’Universitá e del Politecnico di Torino.Google Scholar
  12. 12.
    Poggiolini L, Stefani G (2004) State-local optimality of a bang-bang trajectory: a Hamiltonian approach. Systems & Control Letters, 53:269–279.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sarychev AV (1997) First and second order sufficient optimality conditions for bang-bang controls. SIAM J. Control Optimization, 1(35):315–340.CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Laura Poggiolini
    • 1
  • Marco Spadini
    • 1
  1. 1.Dipartimento di Matematica Applicata “G. Sansone”Università di FirenzeFirenzeItalia

Personalised recommendations