Optimality and sensitivity for semilinear bang-bang type optimal control problems

Ursula Felgenhauer

International Journal of Applied Mathematics and Computer Science (2004)

  • Volume: 14, Issue: 4, page 447-454
  • ISSN: 1641-876X

Abstract

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In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In particular, we will exclude the so-called singular control arcs, see Assumptions 1 and 2, Section 2. It is known from the results by Agrachev et al. (2002) stating that regularity assumptions, together with a certain strict second-order condition for the optimization problem formulated in switching points, are sufficient for strong local optimality of a state-control solution pair. This finite-dimensional problem is analyzed in Section 3 and optimality conditions are formulated (Lemma 2). Using well-known results concerning solution sensitivity for mathematical programs in R^n (Fiacco, 1983) one may further conclude that, under parameter changes in the problem data, the switching points will change Lipschitz continuously. The last section completes these qualitative statements by calculating sensitivity differentials (Theorem 2, Lemma 6). The method requires a simultaneous solution of certain linearized multipoint boundary value problems.

How to cite

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Felgenhauer, Ursula. "Optimality and sensitivity for semilinear bang-bang type optimal control problems." International Journal of Applied Mathematics and Computer Science 14.4 (2004): 447-454. <http://eudml.org/doc/207709>.

@article{Felgenhauer2004,
abstract = {In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In particular, we will exclude the so-called singular control arcs, see Assumptions 1 and 2, Section 2. It is known from the results by Agrachev et al. (2002) stating that regularity assumptions, together with a certain strict second-order condition for the optimization problem formulated in switching points, are sufficient for strong local optimality of a state-control solution pair. This finite-dimensional problem is analyzed in Section 3 and optimality conditions are formulated (Lemma 2). Using well-known results concerning solution sensitivity for mathematical programs in R^n (Fiacco, 1983) one may further conclude that, under parameter changes in the problem data, the switching points will change Lipschitz continuously. The last section completes these qualitative statements by calculating sensitivity differentials (Theorem 2, Lemma 6). The method requires a simultaneous solution of certain linearized multipoint boundary value problems.},
author = {Felgenhauer, Ursula},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {strong local optima; optimality conditions; sensitivity differentials; bang-bang control; stability in optimal control; solution structure},
language = {eng},
number = {4},
pages = {447-454},
title = {Optimality and sensitivity for semilinear bang-bang type optimal control problems},
url = {http://eudml.org/doc/207709},
volume = {14},
year = {2004},
}

TY - JOUR
AU - Felgenhauer, Ursula
TI - Optimality and sensitivity for semilinear bang-bang type optimal control problems
JO - International Journal of Applied Mathematics and Computer Science
PY - 2004
VL - 14
IS - 4
SP - 447
EP - 454
AB - In optimal control problems with quadratic terminal cost functionals and systems dynamics linear with respect to control, the solution often has a bang-bang character. Our aim is to investigate structural solution stability when the problem data are subject to perturbations. Throughout the paper, we assume that the problem has a (possibly local) optimum such that the control is piecewise constant and almost everywhere takes extremal values. The points of discontinuity are the switching points. In particular, we will exclude the so-called singular control arcs, see Assumptions 1 and 2, Section 2. It is known from the results by Agrachev et al. (2002) stating that regularity assumptions, together with a certain strict second-order condition for the optimization problem formulated in switching points, are sufficient for strong local optimality of a state-control solution pair. This finite-dimensional problem is analyzed in Section 3 and optimality conditions are formulated (Lemma 2). Using well-known results concerning solution sensitivity for mathematical programs in R^n (Fiacco, 1983) one may further conclude that, under parameter changes in the problem data, the switching points will change Lipschitz continuously. The last section completes these qualitative statements by calculating sensitivity differentials (Theorem 2, Lemma 6). The method requires a simultaneous solution of certain linearized multipoint boundary value problems.
LA - eng
KW - strong local optima; optimality conditions; sensitivity differentials; bang-bang control; stability in optimal control; solution structure
UR - http://eudml.org/doc/207709
ER -

References

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  1. Agrachev A., Stefani G. and Zezza P.L. (2002): Strong optimality for a bang-bang trajectory. - SIAM J. Contr. Optim., Vol. 41, No. 4, pp. 991-1014. Zbl1020.49021
  2. Dontchev A. and Malanowski K. (2000): A characterization of Lipschitzian stability in optimal control, In: Calculus of Variations and Optimal Control (A. Ioffe et al., Eds.). - Boca Raton, FL: Chapman and Hall CRC, Vol. 411, pp. 62-76. Zbl0970.49021
  3. Felgenhauer U. (2003a): On stability of bang-bang type controls. - SIAM J. Contr. Optim., Vol. 41, No. 6, pp. 1843-1867. Zbl1031.49026
  4. Felgenhauer U. (2003b): Stability and local growth near bounded-strong local optimal controls, In: System Modelling and Optimization XX (E. Sachs and R. Tichatschke, Eds.). - Dordrecht, The Netherlands: Kluwer, pp. 213-227. Zbl1050.49021
  5. Felgenhauer U. (2003c): On sensitivity results for bang-bang type controls of linear systems.- Preprint M-01/2003, Technical University Cottbus, available at http://www.math.tu-cottbus.de/INSTITUT/lsopt/publication/preprints.html. Zbl1031.49026
  6. Felgenhauer U. (2003d): Optimality and sensitivity properties of bang-bang controls for linear systems. - Proc. 21-st IFIP Conf. System Modeling and Optimization, Sophia Antipolis, France, Dordrecht, The Netherlands: Kluwer, (submitted). Zbl1031.49026
  7. Fiacco A.V. (1983): Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. - New York: Academic Press. Zbl0543.90075
  8. Kim J.-H.R. and Maurer H. (2003): Sensitivity analysis of optimal control problems with bang-bang controls. -Proc. 42nd IEEE Conf. Decision and Control, CDC'2003, Maui, Hawaii, USA, Vol. 4, pp. 3281-3286. 
  9. Malanowski K. (2001): Stability and Sensitivity Analysis for Optimal Control Problems with Control-State Constraints. - Warsaw: Polish Academy of Sciences. 
  10. Maurer H. and Osmolovskii N.P. (2004): Second order sufficient conditions for time-optimal bang-bang control problems. - SIAM J. Contr. Optim., Vol. 42, No. 6, pp. 2239-2263. Zbl1068.49015
  11. Maurer H. and Pickenhain S. (1995): Second order sufficient conditions for optimal control problems with mixed control-state constraints. - J. Optim. Theor. Appl., Vol. 86, No. 3, pp. 649-667. Zbl0874.49020
  12. Milyutin A.A. and Osmolovskii N.P. (1998): Calculus of Variations and Optimal Control. - Providence: AMS. Zbl1331.49007
  13. Noble J. and Schaettler H. (2002): Sufficient conditions for relative minima of broken extremals in optimal control theory. - J. Math. Anal. Appl., Vol. 269, No. 1, pp. 98-128. Zbl1012.49023
  14. Osmolovskii N.P. (2000): Second-order conditions for broken extremals, In: Calculus of Variations and Optimal Control (A. Ioffe et al., Eds.). - Boca Raton, FL: Chapman and HallCRC, Vol. 411, pp. 198-216. Zbl0971.49010
  15. Osmolovskii N.P. and Lempio F. (2002): Transformation of quadratic forms to perfect squares for broken extremals. - Set-Valued Anal., Vol. 10, No. 2-3, pp. 209-232. Zbl1050.49016
  16. Sarychev A.V. (1997): First- and second-order sufficient optimality conditions for bang-bang controls. - SIAM J. Contr. Optim., Vol. 35, No. 1, pp. 315-340. Zbl0868.49019

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