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

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January 2012, 32(1): 1-26. doi: 10.3934/dcds.2012.32.1

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

Piernicola Bettiol ^1, and Hélène Frankowska ^2,

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

Received July 2010 Revised March 2011 Published September 2011

Abstract
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Consider a closed subset $K \subset \mathbb{R}^n$ and $f:[0,T]\times \mathbb{R}^n\times U \to \mathbb{R}^n$, where $U$ is a complete separable metric space. We associate to these data the control system under a state constraint \begin{equation*}\label{dm400} \left \{ \begin{array}{lll} x'(t) &=&f(t,x(t),u(t)), \; \; u(t)\in U \quad\; \mbox{ a.e. in }\; [0,T] \\ x(t) & \in & K \quad\; \mbox{ for all }\; t \in [0,T]\\ x(0)& = &x_0 . \end{array} \right. \end{equation*} When the boundary of $K$ is smooth, then an inward pointing condition guarantees that under standard assumptions on $f$ (measurable in $t$, Lipschitz in $x$, continuous in $u$) the sets of solutions to the above system depend on the initial state $x_0$ in a Lipschitz way. This follows from the so-called Neighboring Feasible Trajectories (NFT) theorems. Some recent counterexamples imply that NFT theorems are not valid when $f$ is discontinuous in time and $K$ is a finite intersection of sets with smooth boundaries, that is in the presence of multiple state constraints.
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 $K$.

Keywords: inverse mapping theorem., state constraints, metric regularity, control system, Lipschitz dependence on initial conditions, Differential inclusion.

Mathematics Subject Classification: Primary: 34A60, 34H05, 54C60, 93C15; Secondary: 49K15, 49K2.

Citation: Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1

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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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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