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2004 | 14 | 4 | 447-454

Tytuł artykułu

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

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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.








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  • Institut für Mathematik, Brandenburgische Technische Universität Cottbus, PF 101344, 03013 Cottbus, Germany


  • 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.
  • 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.
  • Felgenhauer U. (2003a): On stability of bang-bang type controls. - SIAM J. Contr. Optim., Vol. 41, No. 6, pp. 1843-1867.
  • 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.
  • Felgenhauer U. (2003c): On sensitivity results for bang-bang type controls of linear systems.- Preprint M-01/2003, Technical University Cottbus, available at
  • 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).
  • Fiacco A.V. (1983): Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. - New York: Academic Press.
  • 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.
  • Malanowski K. (2001): Stability and Sensitivity Analysis for Optimal Control Problems with Control-State Constraints. - Warsaw: Polish Academy of Sciences.
  • 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.
  • 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.
  • Milyutin A.A. and Osmolovskii N.P. (1998): Calculus of Variations and Optimal Control. - Providence: AMS.
  • 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.
  • 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.
  • 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.
  • 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.

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