Lipschitz regularity of solution map of control systems with multiple state constraints

Discrete and Continuous Dynamical Systems

Volumn 32, Issue 1, 2012, Pages 1-26

  Bettiol, Piernicola ^a   Frankowska, Hélène ^b  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b INSTITUT DE MATHÉMATIQUES DE JUSSIEU   (France)

Author keywords

Control system; Differential inclusion; Inverse mapping theorem; Lipschitz dependence on initial conditions; Metric regularity; State constraints

Indexed keywords

EID: 84859525543     PISSN: 10780947     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcds.2012.32.1     Document Type: Article

References (30)

1. J.-P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Mathematics of Oper. Res., 9 (1984), 87-111.
2. J.-P. Aubin and A. Cellina, "Differential Inclusions," Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1984.
3. J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Birkhäuser Boston, Inc., Boston, MA, 1990.
4. P. Bettiol, P. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, International Journal of Game Theory, 34 (2006), 495-527. (Pubitemid 44728947)
5. P. Bettiol, A. Bressan and R. B. Vinter, On trajectories satisfying a state constraint: W1;1 estimates and counter-examples, SIAM J. Control and Optimization, 48 (2010), 4664-4679.
6. P. Bettiol, A. Bressan and R. B. Vinter, Estimates for trajectories conńed to a cone in Rn, SIAM J. Control Optimization, 49 (2011), 21-42.
7. P. Bettiol and H. Frankowska, Regularity of solution maps of differential inclusions under state constraints, Set-Valued Analysis, 15 (2007), 21-45.
8. P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems, J. Differential Equations, 243 (2007), 256-269.
9. P. Bettiol and R. B. Vinter, Sensitivity interpretations of the costate variable for optimal control problems with state constraints, SIAM J. Control and Optimization, 48 (2010), 3297-3317.
10. A. Cernea and H. Frankowska, A connection between the maximum principle and dynamic programming for constrained control problems, SIAM J. Control and Optimization, 44 (2005), 673-703.
11. M. C. Delfour and J. P. Zolesio, Oriented distance function and its evolution equation for initial sets with thin boundary, SIAM J. Control and Optimization, 42 (2004), 2286-2304.
12. A. L. Dontchev and R. T. Rockafellar, "Implicit Functions and Solution Mappings," Springer Mathematics Monographs, Springer, Dodrecht, 2009.
13. H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491.
14. A. F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control and Optimization, 5 (1967), 609-621.
15. F. Forcellini and F. Rampazzo, On non-convex differential inclusions whose state is con- strained in the closure of an open set. Applications to dynamic programming, J. Differential Integral Equations, 12 (1999), 471-497.
16. H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations, 84 (1990), 100-128.
17. H. Frankowska, Some inverse mapping theorems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 183-234.
18. H. Frankowska, Regularity of minimizers and of adjoint states in optimal control under state constraints, J. of Convex Analysis, 13 (2006), 299-328. (Pubitemid 44272480)
19. H. Frankowska, Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints, Control and Cybernetics, 38 (2009), 1327-1340.
20. H. Frankowska and E. Marchini, Lipschitzianity of optimal trajectories for the Bolza optimal control problem, Calculus of Variations and PDE's, 27 (2006), 467-492. (Pubitemid 44483359)
21. H. Frankowska and S. Plaskacz, A measurable upper semicontinuous viability theorem for tubes, J. of Nonlinear Analysis, TMA, 26 (1996), 565-582. (Pubitemid 126396295)
22. H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Equations, 161 (2000), 449-478.
23. H. Frankowska and R. B. Vinter, Existence of neighbouring feasible trajectories: Applica- tions to dynamic programming for state-constrained optimal control problems, Journal of Optimization Theory and Applications, 104 (2000), 21-40.
24. A. D. Ioé, Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk, 55 (2000), 103-162.
25. English translation Math. Surveys, 55 (2000), 501-558.
26. R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.
27. F. Rampazzo and R. B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA J. Math. Control Inform, 16 (1999), 335-351. (Pubitemid 32081372)
28. F. Rampazzo and R. B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control and Optimization, 39 (2000), 989-1007. (Pubitemid 32813683)
29. H. M. Soner, Optimal control problems with state-space constraints, SIAM J. Control and Optimization, 24 (1986), 552-561.
30. R. B. Vinter, "Optimal Control," Birkhäuser Boston Inc., Boston, MA, 2000.