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Lipschitz regularity of solution map of control systems with multiple state constraints
1. | Department of Electrical and Electronic Engineering, Imperial College London, Exhibition Road, London SW7 2BT, United Kingdom |
2. | CNRS and Institut de Mathématiques de Jussieu, UMR 7586, Université Pierre et Marie Curie (Paris 6), 4 place Jussieu, 75252 Paris cedex 05, France |
In this paper we prove that for multiple state constraints the inward pointing condition yields local Lipschitz dependence of solution sets on the initial states from the interior of $K$. Furthermore we relax the usual inward pointing condition. The novelty of our approach lies in an application of a generalized inverse mapping theorem to investigate feasible solutions of control systems. Our results also imply a viability theorem without convexity of right-hand sides for initial states taken in the interior of $K$.
References:
[1] |
J.-P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Mathematics of Oper. Res., 9 (1984), 87-111.
doi: 10.1287/moor.9.1.87. |
[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.
doi: 10.1007/s00182-006-0030-9. |
[5] |
P. Bettiol, A. Bressan and R. B. Vinter, On trajectories satisfying a state constraint: $W^{1,1}$ estimates and counter-examples, SIAM J. Control and Optimization, 48 (2010), 4664-4679.
doi: 10.1137/090769788. |
[6] |
P. Bettiol, A. Bressan and R. B. Vinter, Estimates for trajectories confined to a cone in $\mathbbR^n$, SIAM J. Control Optimization, 49 (2011), 21-42.
doi: 10.1137/09077240X. |
[7] |
P. Bettiol and H. Frankowska, Regularity of solution maps of differential inclusions under state constraints, Set-Valued Analysis, 15 (2007), 21-45.
doi: 10.1007/s11228-006-0018-4. |
[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.
doi: 10.1137/080732614. |
[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.
doi: 10.1137/S0363012902411945. |
[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.
doi: 10.1090/S0002-9947-1959-0110078-1. |
[14] |
A. F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control and Optimization, 5 (1967), 609-621.
doi: 10.1137/0305040. |
[15] |
F. Forcellini and F. Rampazzo, On non-convex differential inclusions whose state is constrained 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. |
[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. |
[21] |
H. Frankowska and S. Plaskacz, A measurable upper semicontinuous viability theorem for tubes, J. of Nonlinear Analysis, TMA, 26 (1996), 565-582.
doi: 10.1016/0362-546X(94)00299-W. |
[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: Applications to dynamic programming for state-constrained optimal control problems, Journal of Optimization Theory and Applications, 104 (2000), 21-40.
doi: 10.1023/A:1004668504089. |
[24] |
A. D. Ioffe, Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk, 55 (2000), 103-162; English translation Math. Surveys, 55 (2000), 501-558. |
[25] |
R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.
doi: 10.1090/S0002-9947-00-02550-2. |
[26] |
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.
doi: 10.1093/imamci/16.4.335. |
[27] |
F. Rampazzo and R. B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control and Optimization, 39 (2000), 989-1007.
doi: 10.1137/S0363012998340223. |
[28] |
H. M. Soner, Optimal control problems with state-space constraints, SIAM J. Control and Optimization, 24 (1986), 552-561.
doi: 10.1137/0324032. |
[29] |
R. B. Vinter, "Optimal Control,'' Birkhäuser Boston Inc., Boston, MA, 2000. |
show all references
References:
[1] |
J.-P. Aubin, Lipschitz behavior of solutions to convex minimization problems, Mathematics of Oper. Res., 9 (1984), 87-111.
doi: 10.1287/moor.9.1.87. |
[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.
doi: 10.1007/s00182-006-0030-9. |
[5] |
P. Bettiol, A. Bressan and R. B. Vinter, On trajectories satisfying a state constraint: $W^{1,1}$ estimates and counter-examples, SIAM J. Control and Optimization, 48 (2010), 4664-4679.
doi: 10.1137/090769788. |
[6] |
P. Bettiol, A. Bressan and R. B. Vinter, Estimates for trajectories confined to a cone in $\mathbbR^n$, SIAM J. Control Optimization, 49 (2011), 21-42.
doi: 10.1137/09077240X. |
[7] |
P. Bettiol and H. Frankowska, Regularity of solution maps of differential inclusions under state constraints, Set-Valued Analysis, 15 (2007), 21-45.
doi: 10.1007/s11228-006-0018-4. |
[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.
doi: 10.1137/080732614. |
[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.
doi: 10.1137/S0363012902411945. |
[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.
doi: 10.1090/S0002-9947-1959-0110078-1. |
[14] |
A. F. Filippov, Classical solutions of differential equations with multi-valued right-hand side, SIAM J. Control and Optimization, 5 (1967), 609-621.
doi: 10.1137/0305040. |
[15] |
F. Forcellini and F. Rampazzo, On non-convex differential inclusions whose state is constrained 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. |
[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. |
[21] |
H. Frankowska and S. Plaskacz, A measurable upper semicontinuous viability theorem for tubes, J. of Nonlinear Analysis, TMA, 26 (1996), 565-582.
doi: 10.1016/0362-546X(94)00299-W. |
[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: Applications to dynamic programming for state-constrained optimal control problems, Journal of Optimization Theory and Applications, 104 (2000), 21-40.
doi: 10.1023/A:1004668504089. |
[24] |
A. D. Ioffe, Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk, 55 (2000), 103-162; English translation Math. Surveys, 55 (2000), 501-558. |
[25] |
R. A. Poliquin, R. T. Rockafellar and L. Thibault, Local differentiability of distance functions, Trans. Amer. Math. Soc., 352 (2000), 5231-5249.
doi: 10.1090/S0002-9947-00-02550-2. |
[26] |
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.
doi: 10.1093/imamci/16.4.335. |
[27] |
F. Rampazzo and R. B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control and Optimization, 39 (2000), 989-1007.
doi: 10.1137/S0363012998340223. |
[28] |
H. M. Soner, Optimal control problems with state-space constraints, SIAM J. Control and Optimization, 24 (1986), 552-561.
doi: 10.1137/0324032. |
[29] |
R. B. Vinter, "Optimal Control,'' Birkhäuser Boston Inc., Boston, MA, 2000. |
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