Publication Cover
Optimization
A Journal of Mathematical Programming and Operations Research
Volume 64, 2015 - Issue 3
111
Views
2
CrossRef citations to date
0
Altmetric
Articles

Note on local quadratic growth estimates in bang–bang optimal control problems

Pages 521-537 | Received 05 Oct 2012, Accepted 25 Jan 2013, Published online: 15 Mar 2013

References

  • Bonnans JF, Shapiro A. Perturbation analysis of optimization problems. New York, NY: Springer; 2000.
  • Fiacco AV. Introduction to sensitivity and stability analysis in nonlinear programming. New York, NY: Academic Press; 1983.
  • Agrachev A, Stefani G, Zezza PL. Strong optimality for a bang-bang trajectory. SIAM J. Control Optim. 2002;41:991–1014.
  • Milyutin AA, Osmolovskii NP. Calculus of variations and optimal control. Providence, RI: Amer. Mathem. Soc; 1998.
  • Noble J, Schättler H. Sufficient conditions for relative minima of broken extremals in optimal control theory. J. Math. Anal. Appl. 2002;269:98–128.
  • Osmolovskii NP. Second-order conditions for broken extremals. In: Ioffe A, Reich S, Shafrir J, editors. Calculus of variations and optimal control. Boca Raton, FL: Chapman & Hall/CRC. Res. Notes Math. 2000;411:198–216.
  • Osmolovskii NP. Quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. Optimal control and dynamical systems. J. of Mathem. Sci. 2004;123:3987–4122.
  • Osmolovskii NP, Lempio F. Transformation of quadratic forms to perfect squares for broken extremals. Set-Valued Anal. 2002;10:209–232.
  • Poggiolini L, Stefani G. State-local optimality of a bang-bang trajectory: a Hamiltonian approach. Systems Control Lett. 2004;53:269–279.
  • Schättler H, Ledzewicz U. Geometric optimal control. Theory, methods and examples. New York, NY: Springer; 2012.
  • Felgenhauer U, Poggiolini L, Stefani G. Optimality and stability result for bang-bang optimal controls with simple and double switch behavior. Control & Cybern. 2009;38:1305–1325.
  • Poggiolini L, Spadini M. Sufficient optimality conditions for a bang-bang trajectory. SIAM J. Control Optim. 2011;49:140–161.
  • Maurer H, Osmolovskii NP. Equivalence of second-order optimality conditions for bang-bang control problems. Control & Cybernetics. 2005;34:927–950.
  • Maurer H, Osmolovskii NP. Equivalence of second order optimality conditions for bang-bang control problems. II. Proofs, variational derivatives and representations. Control & Cybernetics. 2007;36:5–45.
  • Osmolovskii NP. Quadratic conditions for nonsingular extremals in optimal control (A theoretical treatment). Russian J. of Mathem. Physics. 1995;2:487–512.
  • Felgenhauer U. On stability of bang-bang type controls. SIAM J. Control Optim. 2003;41:1843–1867.
  • Kim JR, Maurer H. Sensitivity analysis of optimal ontrol problems with bang-bang controls. 42th IEEE Conference on Decision Control. Maui, Hawaii: Decision. Control. 2003;4:3281–3286.
  • Felgenhauer U. Directional sensitivity differentials for parametric bang-bang control problems. In: Large-Scale Scientific Computing, Lirkov I, Margenov S, Wasniewski J, editors. Lecture notes comp. sci. Vol. 5910. Berlin: Springer. 2010; p. 264–271.
  • Alt W, Baier R, Lempio F, Gerdts M. Approximations of linear control problems with bang-bang solutions. Optimization. 2013;62:9–32.
  • Alt W, Seydenschwanz M. Regularization and discretization of linear-quadratic control problems. Control & Cybernetics. 2012;40:903–920.
  • Klötzler R. On a general conception of duality in optimal control. In: Equadiff IV, Fabera J, editors. Lect. notes math. Vol. 703. New York, NY: Springer; 1979. p. 189–196.
  • Schättler H, Ledzewicz U, Dekhordi SM, Reisi M. A geometric analysis of bang-bang extremals in optimal control problems for combination cancer chemotherapy, 51st IEEE Conf. Maui, Hawaii: Decision Control; 2012.
  • Maurer H, Osmolovskii NP. Second-order sufficient optimality conditions for time-optimal bang-bang control. SIAM J. Control Optim. 2004;42:2239–2263.
  • Maurer H, Pickenhain S. Second order sufficient conditions for optimal control problems with mixed control-state constraints. J. Optim. Theor. Appl. 1995;86:649–667.
  • Felgenhauer U. The shooting approach in analyzing bang-bang extremals with simultaneous control switches. Control & Cybernetics. 2008;37:307–327.
  • Maurer H, Büskens C, Kim J-HR, Kaya CY. Optimization methods for the verification of second-order sufficient conditions for bang-bang controls. Optim. Contr. Appl. Meth. 2005;26:129–156.
  • Felgenhauer U. Optimality properties of controls with bang-bang components in problems with semilinear state equation. Control & Cybernetics. 2005;34:763–785.
  • Adams RA. Sobolev spaces. London: Academic Press; 1978.
  • Agrachev AA, Sachkov YuL. Control theory from the geometric viewpoint, encyclopaedia of mathematical sciences, 87, control theory and optimization II. Berlin: Springer; 2004.
  • Ledzewicz U, Nowakowski A, Schättler H. Stratifiable families of extremals and sufficient conditions for optimality in optimal control problems. J. Optim. Theory Appl. 2004;122:345–370.
  • Schättler H. Local fields of extremals for optimal control problems with state constraints of relative degree 1. J. Dyn. Control Syst. 2006;12:563–599.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.