    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

 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

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 $K$.
Citation: Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete & Continuous Dynamical Systems, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1
##### References:
  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.  Google Scholar  J.-P. Aubin and A. Cellina, "Differential Inclusions,'' Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1984. Google Scholar  J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Birkhäuser Boston, Inc., Boston, MA, 1990. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems, J. Differential Equations, 243 (2007), 256-269. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  A. L. Dontchev and R. T. Rockafellar, "Implicit Functions and Solution Mappings," Springer Mathematics Monographs, Springer, Dodrecht, 2009. Google Scholar  H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491. doi: 10.1090/S0002-9947-1959-0110078-1.  Google Scholar  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.  Google Scholar  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. Google Scholar  H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations, 84 (1990), 100-128. Google Scholar  H. Frankowska, Some inverse mapping theorems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 183-234. Google Scholar  H. Frankowska, Regularity of minimizers and of adjoint states in optimal control under state constraints, J. of Convex Analysis, 13 (2006), 299-328. Google Scholar  H. Frankowska, Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints, Control and Cybernetics, 38 (2009), 1327-1340. Google Scholar  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. Google Scholar  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.  Google Scholar  H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Equations, 161 (2000), 449-478. Google Scholar  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.  Google Scholar  A. D. Ioffe, Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk, 55 (2000), 103-162; English translation Math. Surveys, 55 (2000), 501-558. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  H. M. Soner, Optimal control problems with state-space constraints, SIAM J. Control and Optimization, 24 (1986), 552-561. doi: 10.1137/0324032.  Google Scholar  R. B. Vinter, "Optimal Control,'' Birkhäuser Boston Inc., Boston, MA, 2000. Google Scholar

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##### References:
  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.  Google Scholar  J.-P. Aubin and A. Cellina, "Differential Inclusions,'' Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 1984. Google Scholar  J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Birkhäuser Boston, Inc., Boston, MA, 1990. Google Scholar  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.  Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  P. Bettiol and H. Frankowska, Normality of the maximum principle for nonconvex constrained Bolza problems, J. Differential Equations, 243 (2007), 256-269. Google Scholar  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.  Google Scholar  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. Google Scholar  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.  Google Scholar  A. L. Dontchev and R. T. Rockafellar, "Implicit Functions and Solution Mappings," Springer Mathematics Monographs, Springer, Dodrecht, 2009. Google Scholar  H. Federer, Curvature measures, Trans. Amer. Math. Soc., 93 (1959), 418-491. doi: 10.1090/S0002-9947-1959-0110078-1.  Google Scholar  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.  Google Scholar  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. Google Scholar  H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations, 84 (1990), 100-128. Google Scholar  H. Frankowska, Some inverse mapping theorems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 183-234. Google Scholar  H. Frankowska, Regularity of minimizers and of adjoint states in optimal control under state constraints, J. of Convex Analysis, 13 (2006), 299-328. Google Scholar  H. Frankowska, Normality of the maximum principle for absolutely continuous solutions to Bolza problems under state constraints, Control and Cybernetics, 38 (2009), 1327-1340. Google Scholar  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. Google Scholar  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.  Google Scholar  H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Equations, 161 (2000), 449-478. Google Scholar  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.  Google Scholar  A. D. Ioffe, Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk, 55 (2000), 103-162; English translation Math. Surveys, 55 (2000), 501-558. Google Scholar  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.  Google Scholar  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.  Google Scholar  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.  Google Scholar  H. M. Soner, Optimal control problems with state-space constraints, SIAM J. Control and Optimization, 24 (1986), 552-561. doi: 10.1137/0324032.  Google Scholar  R. B. Vinter, "Optimal Control,'' Birkhäuser Boston Inc., Boston, MA, 2000. Google Scholar
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