Stability analysis of variational inequalities for bang-singular-bang controls

• Autorzy: Felgenhauer U.
• Języki publikacji: EN

Abstrakty

EN

The paper is related to parameter dependent optimal control problems for control-affine systems. The case of scalar reference control with bang-singular-bang structure is considered. The analysis starts from a variational inequality (VI) formulation of Pontryagin’s Maximum Principle. In a first step, under appropriate higher-order sufficient optimality conditions, the existence of solutions for the linearized problem (LVI) is proven. In a second step, for a certain class of right-hand side perturbation, it is show that the controls from LVI have bang-singular-bang structure and, in L₁ topology, depend Lipschitz continuously on the data. Applying finally a common fixed-point approach to VI, the results are brought together to obtain existence and structural stability results for extremals of the original control problem under parameter perturbation.

Słowa kluczowe

EN

parametric optimal control problems bang-singular control structure; approximation of extremals

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2013
• Tom: Vol. 42, no. 3
• Strony: 557--592
• Opis fizyczny: Bibliogr. 38 poz.

Twórcy

autor

Felgenhauer U.

• Brandenburgische Technische Universität Cottbus, Institut für Angewandte Mathematik und Wissenschaftliches Rechnen, Germany

Bibliografia

• 1. Agrachev, A.A. and Sachkov, Yu.L. (2004) Control Theory from the Geometric Viewpoint. Encyclopaedia of Mathematical Sciences 87. Control Theory and Optimization, II. Springer, Berlin.
• 2. Aronna, S.M., Bonnans, J.F., Dmitruk, A.V. and Lotito, P.A. (2012a) Quadratic order conditions for bang-singular extremals. Numer. Algebra, Contr. and Optim. 2 (3), 511–546.
• 3. Aronna, S.M., Bonnans, J.F. and Martinon, P. (2012b) A shooting algorithm for optimal control problems with singular arcs. J. Optim. Theory Appl., doi 10/10077s109577-012-0254-8 (electronically), 41 pp.
• 4. Dmitruk, A.V. (1977) Quadratic conditions for a weak minimum for singular regimes in optimal control problems. Soviet Math. Doklady 18(2).
• 5. Dmitruk, A.V. (2008) Jacobi type conditions for singular extremals. Control & Cybernetics, 37 (2), 285-306.
• 6. Dontchev, A.L. (1995) Characterization of Lipschitz stability in optimization, In: R. Luccheti and J. Revalski, eds., Recent Developments in Wellposed Variational Problems, Kluwer, Dordrecht, 95-116.
• 7. Dontchev, A.L. and Lewis, A. (2005) Perturbations and metric regularity. Set-Valued Anal. 13, 417-438.
• 8. Dontchev, A.L. and Malanowski, K. (2000) A characterization of Lipschitzian stability in optimal control. In: A. Ioffe, S. Reich and I. Shafri, eds. Calculus of variations and optimal control, Haifa 1998. Chapman & Hall/CRC, Boca Raton, FL., 62-76.
• 9. Dontchev, A.L., Quincampoix, M. and Zlateva, N. (2006) Aubin criterion for metric regularity. J. Convex Anal. 13, 281-297.
• 10. Dontchev, A.L. and Rockafellar, R.T. (1996) Characterization of strong regularity for variational inequalities over polyhedral sets. SIAM J. Optimization 6, 1087-1105.
• 11. Dontchev, A.L. and Veliov, V.M. (2009) Metric regularity under approximations. Control and Cybernetics 38, 1283-1303.
• 12. Felgenhauer, U. (2001) Structural properties and approximation of optima controls. Nonlinear Analysis 47(3), 1869-1880.
• 13. Felgenhauer U. (2003) On stability of bang–bang type controls. SIAM J. Control Optim. 41(6), 1843-1867.
• 14. Felgenhauer U. (2010) Directional sensitivity differentials for parametric bang-bang control problems. In: I. Lirkov et al., eds., Large-Scale Scientific
• 15. Computing, Lecture Notes Comp. Sci. 5910, Springer, Berlin, 264-271. Felgenhauer U. (2012) Structural stability investigation of bang-singularbang optimal controls. J. Optim. Theory Appl. 152(3), 605-631.
• 16. Felgenhauer U. (2013) Note on local quadratic growth estimates in bangbang optimal control problems. Optimization, doi 10.1080/02331934.2013. 773000 (electronically), 17 pp.
• 17. Felgenhauer U., Poggiolini, L. and Stefani, G. (2009) Optimality and stability result for bang-bang optimal controls with simple and double switch behavior. Control & Cybernetics 38(4B), 1305-1325.
• 18. Fiacco, A. V. (1983) Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York.
• 19. Fiacco, A.V. and McCormick, G.P. (1968) Nonlinear Programming: Sequential Unconstrained Minimization Techniques. John Wiley & Sons, New York.
• 20. Goh, B.S. (1966a) The second variation for the singular Bolza problem. SIAM J. Control 4 (2), 309–325.
• 21. Goh, B.S. (1966b) Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4 (4), 716-731.
• 22. Kinderlehrer, D. and Stampacchia, G. (1980) An Introduction to Variational Inequalities and their Applications. Academic Press, New York.
• 23. Knobloch, H.W. (1981) Higher Order Necessary Conditions in Optimal Control Theory. Lecture Notes Contr. Inf. Sci. 34, Springer, Berlin.
• 24. Krener, A.J. (1977) The high order maximum principle and its application to singular extremals. SIAM J. Control Optim. 15(2), 256-292.
• 25. Ledzewicz, U. and Schättler, H. (2007) Antiangiogenic therapy in cancer treatment as an optimal control problem. SIAM J. Control Optim. 46(3), 1052-1079.
• 26. Malanowski, K. (1995) Stability and sensitivity analysis of solutions to nonlinear optimal control problems. Appl. Math. Optim. 32, 111-141.
• 27. Malanowski, K. (2001) Stability and Sensitivity Analysis for Optimal Control Problems with Control-State Constraints. Dissertationes Mathematicae, Polska Akademia Nauk, Instytut Matematyczny, Warszawa.
• 28. Maurer, H. (1976) Numerical solution of singular control problems using multiple shooting techniques. J. Optim. Theory Appl. 18(2), 235-257.
• 29. Oberle, H.J. (1979) Numerical computation of singular control problems with application to optimal heating and cooling by solar energy. Appl. Math. Optim. 5, 297-314.
• 30. Poggiolini L. and Stefani G. (2005) Minimum time optimality for a bangsingular arc: second-order sufficient conditions. In: Decision and Control, 2005, and 2005 European Control Conference, CDC-ECC’05, 44th IEEE Conference on, Sevilla (Spain). IEEE, 1433-1438.
• 31. Poggiolini L. and Stefani G. (2008) Sufficient optimality conditions for a bang-singular extremal in the minimum time problem. Control & Cybernetics 37 (2), 469-490.
• 32. Poggiolini L. and Stefani G. (2011) Bang-singular-bang extremals: Sufficient optimality conditions. J. Dyn. Contr. Systems 17(4), 469-514.
• 33. Robinson, S.M. (1980) Strongly regular generalized equations. Math. Oper. Res. 5, 43–62
• 34. Rudin, W. (1987) Real and Complex Analysis. McGraw-Hill, New York.
• 35. Stefani, G. (2008) Strong optimality of singular trajectories. In: F. Ancona et al., eds., Geometric Control and Nonsmooth Analysis, Series Adv. Math. Appl. Sci., 76, World Sci. Publ., Hackensack, New Jersey, 300-327.
• 36. Vossen, G. (2005) Numerische Lösungsmethoden, hinreichende Optimalit¨atsbedingungen und Sensitivitätsanalyse für optimale bang-bang und singuläre Steuerungen. PhD Thesis, Faculty of Mathematics and Science, University Münster (Germany).
• 37. Vossen, G. (2010) Switching time optimization for bang-bang and singular controls. J. Optim. Theory Appl. 144, 409-429.
• 38. Zelikin, V.F. and Borisov, V.F. (1994) Theory of Chattering Control. Birkhäuser, Boston.