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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

Piernicola Bettiol 1, and  Richard Vinter 2,

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.
Keywords: Control systems, optimal control., state constraints.
Mathematics Subject Classification: Primary: 93C10, 93C15; Secondary: 34H05, 49K1.
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.

show all references

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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