Optimality and stability result for bang-bang optimal controls with simple and double switch behaviour

Control and Cybernetics

Volumn 38, Issue 4 B, 2009, Pages 1305-1325

  Felgenhauer, Ursula ^a   Poggiolini, Laura ^b   Stefani, Gianna ^b  

^a Institut für Angewandtc Mathematik und Wissenschaftliches Rechnen   (Germany)

^b UNIVERSITY OF FLORENCE   (Italy)

Author keywords

Bang bang control; Hamiltonian flows; Parametric control problems

Indexed keywords

EID: 77953740768     PISSN: 03248569     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (30)

1. AGRACHEV, A. and SACHKOV, Y. (2004) Control theory from the geometric viewpoint. Encyclopaedia of Mathematical Sciences, 87. Control Theory and Optimization, II. Springer, Berlin.
2. AGRACHEV, A., STEFANI, G. and ZEZZA, P. L. (2002) Strong optimality for a bang-bang trajectory. SIAM J. Control Optim. 41 (4), 991-1014.
3. ARUTYUNOV, A.V., AVAKOV, E.R. and IZMAILOV, A.F. (2007) Directional regularity and metric regularity. SIAM J. Optim. 18 (3), 810-833.
4. CLARKE, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley Inc., New York.
5. FELGENHAUER, U. (2003) On stability of bang-bang type controls. SIAM J. Control Optim. 41 (6), 1843-1867.
6. FELGENHAUER, U. (2004) Optimality and sensitivity for semilinear bang-bang type optimal control problems. Internat. J. Appl. Math. Computer Sc. 14 (4), 447-454.
7. FELGENHAUER, U. (2005) Optimality properties of controls with bang-bang components in problems with scmilincar state equation. Control & Cyber-netics 34 (3), 763-785.
8. FELGENHAUER, U. (2008a) Lipschitz stability of broken extremals in bang-bang control problems. In: I. Lirkov et al., eds., Proc. th Internat. Conf. Large-Scale Scientific Computing, Sozopol 2007, Lecture Notes Comp. Sci. 4818, Springer, 306-314.
9. FELGENHAUER, U. (2008b) The shooting approach in analyzing bang-bang extremals with simultaneous control switches. Control & Cybernetics 37 (2), 307-327.
10. GOODMAN, R.W. (1976) Nilpotcnt Lie groups: structure and applications to analysis, Lecture Notes in Mathematics 562, Springer
11. KIM, J. R. and MAURER, H. (2003) Sensitivity analysis of optimal control problems with bang-bang controls. In: Proc. 42nd IEEE Conference on Decision and Control, Hawaii 2003, 4, 3281-3286.
12. KLATTE, D. and KUMMER, B. (2002) Nonsmooth Equations in Optimization. Kluwer Acad. Publ, Dordrecht.
13. LEDZEWICZ, U., NOWAKOWSKI, A. and SCHÄTTLER, H. (2004) Stratifiable families of extremals and sufficient conditions for optimality in optimal control problems. J. Optim. Theory Appl. 122 (2), 345-370.
14. MAURER, H. and OSMOLOVSKII, N. P. (2004) Second order sufficient condi-tions for time-optimal bang-bang control. SIAM J. Control Optim. 42 (6), 2239-2263.
15. MAURER, H. and OSMOLOVSKII, N.P. (2005) Equivalence of second-order op-timality conditions for bang-bang control problems. Control & Cybernetics 34 (3), 927-950.
16. MAURER, H. and OSMOLOVSKII, N.P. (2007) Equivalence of second order op-timality conditions for bang-bang control problems. II. Proofs, variational derivatives and representations. Control & Cybernetics 36 (1), 5-45.
17. MlLYUTlN, A.A. and OSMOLOVSKII, N.P. (1998) Calculus of Variations and Optimal Control, Amcr. Mathcm. Soc., Providence, Rhode Island.
18. MORDUKHOVICH, B.S. (2006) Variational Analysis and Generalized Differen-tiation I,II. Springer, Berlin.
19. NOBLE, J. and SCHÄTTLER, H. (2002) Sufficient conditions for relative min-ima of broken extremals in optimal control theory. J. Math. Anal. Appl. 269, 98-128.
20. OBERLE H.J. (1987) Numerical computation of singular control functions for a two-link robot arm. In: R. Bur lisch et al., cds., Optimal Control. LNCIS 95, Springer, 244-253.
21. OSMOLOVSKII, N.P. (1995) Quadratic conditions for nonsingular extremals in optimal control (A theoretical treatment). Russian J. of Mathem. Physics 2, 487-512.
22. OSMOLOVSKII, N.P. (2004) Quadratic cxtrcmality conditions for broken ex-tremals in the general problem of the calculus of variations. Optimal con-trol and dynamical systems. J. of Mathem, Sciences 123 (3), 3987-4122.
23. OSMOLOVSKII, N.P. and LEMPIO, F. (2002) Transformation of quadratic forms to perfect squares for broken extremals. Set-Valued Analysis 10, 209-232.
24. POGGIOLINI, L. and SPADINI, M. (2008) Sufficient optimality conditions for a bang-bang trajectory in a Bolza Problem. In: A. Sarychev et al., eds., Mathematical Control Theory and Finance, Springer, 337-357.
25. POGGIOLINI, L. and SPADINI, M. (2009) Strong local optimality for a bang-bang trajectory in a Mayer problem. Submitted.
26. POGGIOLINI, L. and STEFANI, G. (2004) State-local optimality of a bang-bang trajectory: a Hamiltonian approach. Systems Control Lett. 53, 269-279.
27. POGGIOLINI, L. and STEFANI, G. (2006) Sufficient optimality conditions for a bang-bang trajectory. In: Proc. 45th IEEE Conference on Decision and Control, San Diego (USA).
28. ROCKAFELLAR, R.T. and WETS, R.J. (1998) Variational Analysis. Springer, Berlin.
29. SCHÄTTLER, H. (2006) Local fields of extremals for optimal control problems with state constraints of relative degree 1. J. Dyn. Control Syst. 12 (4), 563-599.
30. SARYCHEV, A.V. (1997) First- and second-order sufficient optimality condi-tions for bang-bang controls. SIAM J. Control Optim. 35 (1), 315-340.