# American Institute of Mathematical Sciences

September  2013, 3(3): 245-267. doi: 10.3934/mcrf.2013.3.245

## Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data

 1 Laboratoire de Mathematiques, Université de Bretagne Occidentale, 6 Avenue Victor Le Gorgeu, 29200 Brest, France 2 Department of Electrical and Electronic Engineering, Imperial College London, SW7 2BT

Received  November 2012 Revised  February 2013 Published  September 2013

Estimates on the distance of a given process from the set of processes that satisfy a specified state constraint in terms of the state constraint violation are important analytical tools in state constrained optimal control theory; they have been employed to ensure the validity of the Maximum Principle in normal form, to establish regularity properties of the value function, to justify interpreting the value function as a unique solution of the Hamilton-Jacobi equation, and for other purposes. A range of estimates are required, which differ according the metrics used to measure the `distance' and the modulus $\theta(h)$ of state constraint violation $h$ in terms of which the estimates are expressed. Recent research has shown that simple linear estimates are valid when the state constraint set $A$ has smooth boundary, but do not generalize to a setting in which the boundary of $A$ has corners. Indeed, for a velocity set $F$ which does not depend on $(t,x)$ and for state constraints taking the form of the intersection of two closed spaces (the simplest case of a boundary with corners), the best distance estimates we can hope for, involving the $W^{1,1,}$ metric on state trajectories, is a super-linear estimate expressed in terms of the $h|\log(h)|$ modulus. But, distance estimates involving the $h|\log (h)|$ modulus are not in general valid when the velocity set $F(.,x)$ is required merely to be continuous, while not even distance estimates involving the weaker, Hölder modulus $h^{\alpha}$ (with $\alpha$ arbitrarily small) are in general valid, when $F(.,x)$ is allowed to be discontinuous. This paper concerns the validity of distance estimates when the velocity set $F(t,x)$ is $(t,x)$-dependent and satisfy standard hypotheses on the velocity set (linear growth, Lipschitz $x$-dependence and an inward pointing condition). Hypotheses are identified for the validity of distance estimates, involving both the $h|\log(h)|$ and linear moduli, within the framework of control systems described by a controlled differential equation and state constraint sets having a functional inequality representation.
Citation: Piernicola Bettiol, Richard Vinter. Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data. Mathematical Control and Related Fields, 2013, 3 (3) : 245-267. doi: 10.3934/mcrf.2013.3.245
##### References:
 [1] J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [2] P. Bettiol, A. Bressan and R. B. Vinter, On trajectories satisfying a state constraint: $W^{1,1}$ estimates and counter-examples, SIAM J. Control Optim., 48 (2010), 4664-4679. doi: 10.1137/090769788. [3] P. Bettiol, A. Bressan and R. B. Vinter, Estimates for trajectories confined to a cone in $\mathbb{R}^{N}$, SIAM J. Control Optim., 49 (2011), 21-42. doi: 10.1137/09077240X. [4] P. Bettiol, P. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Int. J. Game Theory, 34 (2006), 495-527. doi: 10.1007/s00182-006-0030-9. [5] P. Bettiol and H. Frankowska, Regularity of solution maps of differential inclusions for systems under state constraints, Set-Valued Anal., 15 (2007), 21-45. doi: 10.1007/s11228-006-0018-4. [6] P. Bettiol and H. Frankowska, Lipschitz regularity of solution map to control systems with multiple state constraints, Discrete Contin. Dyn. Syst., 32 (2012), 1-26. [7] P. Bettiol, H. Frankowska and R. B. Vinter, $L^{\infty}$ estimates on trajectories confined to a closed subset, J. Differential Eq., 252 (2012), 1912-1933. doi: 10.1016/j.jde.2011.09.007. [8] P. Bettiol and R. B. Vinter, Existence of feasible approximating trajectories for differential inclusions with obstacles as state constraints, Proc. of the 48th IEEE CDC 2009. doi: 10.1109/CDC.2009.5400266. [9] P. Bettiol and R. B. Vinter, Sensitivity interpretations of the co-state variable for optimal control problems with state constraints, SIAM J. Control Optim., 48 (2010), 3297-3317. doi: 10.1137/080732614. [10] P. Bettiol and R. B. Vinter, Trajectories satisfying a state constraint: Improved estimates and new non-degeneracy conditions, IEEE Trans. Automat. Control, 56 (2011), 1090-1096. doi: 10.1109/TAC.2010.2088670. [11] A. Bressan and G. Facchi, Trajectories of differential inclusions with state constraints, J. Differential Eq., 250 (2011), 2267-2281. doi: 10.1016/j.jde.2010.12.021. [12] F. H. Clarke, The maximum principle under minimal hypotheses, SIAM J. Control Optim., 14 (1976), 1078-1091. doi: 10.1137/0314067. [13] F. H. Clarke, L. Rifford and R. J. Stern, Feedback in state constrained optimal control, ESAIM Control Optim. Calc. Var., 7 (2002), 97-133. doi: 10.1051/cocv:2002005. [14] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [15] F. Forcellini and F. Rampazzo, On nonconvex differential inclusions whose state is constrained in the closure of an open set. Applications to dynamic programming, Differential Integral Equations, 12 (1999), 471-497. [16] H. Frankowska and M. Mazzola, Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints, Calculus Var. Partial Differ. Equ., 46 (2013), 725-747. doi: 10.1007/s00526-012-0501-8. [17] H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, NoDEA Nonlinear Differ. Equ. Appl., 20 (2013), 361-383. doi: 10.1007/s00030-012-0183-0. [18] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Eq., 161 (2000), 449-478. doi: 10.1006/jdeq.2000.3711. [19] H. Frankowska and R. B. Vinter, Existence of neighbouring feasible trajectories: Applications to dynamic programming for state constrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 21-40. doi: 10.1023/A:1004668504089. [20] 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. [21] F. Rampazzo and R. B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim., 39 (2000), 989-1007. doi: 10.1137/S0363012998340223. [22] H. M. Soner, Optimal control problems with state-space constraints,II, SIAM J. Control Optim., 24 (1986), 552-562 and 1110-1122. doi: 10.1137/0324067. [23] R. B. Vinter, "Optimal Control," Systems & Control: Foundations & Applications. Birkhaüser Boston, Inc., Boston, MA, 2000.

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##### References:
 [1] J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. [2] P. Bettiol, A. Bressan and R. B. Vinter, On trajectories satisfying a state constraint: $W^{1,1}$ estimates and counter-examples, SIAM J. Control Optim., 48 (2010), 4664-4679. doi: 10.1137/090769788. [3] P. Bettiol, A. Bressan and R. B. Vinter, Estimates for trajectories confined to a cone in $\mathbb{R}^{N}$, SIAM J. Control Optim., 49 (2011), 21-42. doi: 10.1137/09077240X. [4] P. Bettiol, P. Cardaliaguet and M. Quincampoix, Zero-sum state constrained differential games: Existence of value for Bolza problem, Int. J. Game Theory, 34 (2006), 495-527. doi: 10.1007/s00182-006-0030-9. [5] P. Bettiol and H. Frankowska, Regularity of solution maps of differential inclusions for systems under state constraints, Set-Valued Anal., 15 (2007), 21-45. doi: 10.1007/s11228-006-0018-4. [6] P. Bettiol and H. Frankowska, Lipschitz regularity of solution map to control systems with multiple state constraints, Discrete Contin. Dyn. Syst., 32 (2012), 1-26. [7] P. Bettiol, H. Frankowska and R. B. Vinter, $L^{\infty}$ estimates on trajectories confined to a closed subset, J. Differential Eq., 252 (2012), 1912-1933. doi: 10.1016/j.jde.2011.09.007. [8] P. Bettiol and R. B. Vinter, Existence of feasible approximating trajectories for differential inclusions with obstacles as state constraints, Proc. of the 48th IEEE CDC 2009. doi: 10.1109/CDC.2009.5400266. [9] P. Bettiol and R. B. Vinter, Sensitivity interpretations of the co-state variable for optimal control problems with state constraints, SIAM J. Control Optim., 48 (2010), 3297-3317. doi: 10.1137/080732614. [10] P. Bettiol and R. B. Vinter, Trajectories satisfying a state constraint: Improved estimates and new non-degeneracy conditions, IEEE Trans. Automat. Control, 56 (2011), 1090-1096. doi: 10.1109/TAC.2010.2088670. [11] A. Bressan and G. Facchi, Trajectories of differential inclusions with state constraints, J. Differential Eq., 250 (2011), 2267-2281. doi: 10.1016/j.jde.2010.12.021. [12] F. H. Clarke, The maximum principle under minimal hypotheses, SIAM J. Control Optim., 14 (1976), 1078-1091. doi: 10.1137/0314067. [13] F. H. Clarke, L. Rifford and R. J. Stern, Feedback in state constrained optimal control, ESAIM Control Optim. Calc. Var., 7 (2002), 97-133. doi: 10.1051/cocv:2002005. [14] I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353. doi: 10.1016/0022-247X(74)90025-0. [15] F. Forcellini and F. Rampazzo, On nonconvex differential inclusions whose state is constrained in the closure of an open set. Applications to dynamic programming, Differential Integral Equations, 12 (1999), 471-497. [16] H. Frankowska and M. Mazzola, Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints, Calculus Var. Partial Differ. Equ., 46 (2013), 725-747. doi: 10.1007/s00526-012-0501-8. [17] H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, NoDEA Nonlinear Differ. Equ. Appl., 20 (2013), 361-383. doi: 10.1007/s00030-012-0183-0. [18] H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Eq., 161 (2000), 449-478. doi: 10.1006/jdeq.2000.3711. [19] H. Frankowska and R. B. Vinter, Existence of neighbouring feasible trajectories: Applications to dynamic programming for state constrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 21-40. doi: 10.1023/A:1004668504089. [20] 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. [21] F. Rampazzo and R. B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim., 39 (2000), 989-1007. doi: 10.1137/S0363012998340223. [22] H. M. Soner, Optimal control problems with state-space constraints,II, SIAM J. Control Optim., 24 (1986), 552-562 and 1110-1122. doi: 10.1137/0324067. [23] R. B. Vinter, "Optimal Control," Systems & Control: Foundations & Applications. Birkhaüser Boston, Inc., Boston, MA, 2000.
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