# Deterministic state-constrained optimal control problems without controllability assumptions

Olivier Bokanowski; Nicolas Forcadel; Hasnaa Zidani

ESAIM: Control, Optimisation and Calculus of Variations (2011)

- Volume: 17, Issue: 4, page 995-1015
- ISSN: 1292-8119

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topBokanowski, Olivier, Forcadel, Nicolas, and Zidani, Hasnaa. "Deterministic state-constrained optimal control problems without controllability assumptions." ESAIM: Control, Optimisation and Calculus of Variations 17.4 (2011): 995-1015. <http://eudml.org/doc/221896>.

@article{Bokanowski2011,

abstract = {
In the present paper, we consider nonlinear optimal control problems
with constraints on the state of the system. We are interested in
the characterization of the value function without any
controllability assumption. In the unconstrained case, it is possible to derive a
characterization of the value function by means of a
Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the
behavior of the value function along the trajectories arriving or
starting from any position x. In the constrained case, when no
controllability assumption is made, the HJB equation may have
several solutions. Our first result aims to give the precise
information that should be added to the HJB equation in order to
obtain a characterization of the value function. This result is very
general and holds even when the dynamics is not continuous and the
state constraints set is not smooth. On the other hand we study also some stability results for relaxed
or penalized control problems.
},

author = {Bokanowski, Olivier, Forcadel, Nicolas, Zidani, Hasnaa},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Optimal control problem; state constraints; Hamilton-Jacobi equation; optimal control problem; Hamilton-Jacobi-Bellman equation},

language = {eng},

month = {11},

number = {4},

pages = {995-1015},

publisher = {EDP Sciences},

title = {Deterministic state-constrained optimal control problems without controllability assumptions},

url = {http://eudml.org/doc/221896},

volume = {17},

year = {2011},

}

TY - JOUR

AU - Bokanowski, Olivier

AU - Forcadel, Nicolas

AU - Zidani, Hasnaa

TI - Deterministic state-constrained optimal control problems without controllability assumptions

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2011/11//

PB - EDP Sciences

VL - 17

IS - 4

SP - 995

EP - 1015

AB -
In the present paper, we consider nonlinear optimal control problems
with constraints on the state of the system. We are interested in
the characterization of the value function without any
controllability assumption. In the unconstrained case, it is possible to derive a
characterization of the value function by means of a
Hamilton-Jacobi-Bellman (HJB) equation. This equation expresses the
behavior of the value function along the trajectories arriving or
starting from any position x. In the constrained case, when no
controllability assumption is made, the HJB equation may have
several solutions. Our first result aims to give the precise
information that should be added to the HJB equation in order to
obtain a characterization of the value function. This result is very
general and holds even when the dynamics is not continuous and the
state constraints set is not smooth. On the other hand we study also some stability results for relaxed
or penalized control problems.

LA - eng

KW - Optimal control problem; state constraints; Hamilton-Jacobi equation; optimal control problem; Hamilton-Jacobi-Bellman equation

UR - http://eudml.org/doc/221896

ER -

## References

top- J.-P. Aubin and A. Cellina, Differential inclusions, Comprehensive studies in mathematics264. Springer, Berlin, Heidelberg, New York, Tokyo (1984).
- J.-P. Aubin and H. Frankowska, Set-valued analysis, Systems and Control: Foundations and Applications2. Birkhäuser Boston Inc., Boston (1990).
- M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems and Control: Foundations and Applications. Birkhäuser, Boston (1997).
- M. Bardi, P. Goatin and H. Ishii, A Dirichlet type problem for nonlinear degenerate elliptic equations arising in time-optimal stochastic control. Adv. Math. Sci. Appl.10 (2000) 329–352.
- G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Mathématiques et Applications17. Springer, Paris (1994).
- G. Barles and B. Perthame, Discontinuous solutions of deterministic optimal stopping time problems. RAIRO: Modél. Math. Anal. Numér.21 (1987) 557–579.
- G. Barles and B. Perthame, Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim.26 (1988) 1133–1148.
- G. Barles and B. Perthame, Comparaison principle for Dirichlet-type Hamilton-Jacobi equations and singular perturbations of degenerated elliptic equations. Appl. Math. Optim.21 (1990) 21–44.
- E.N. Barron, Viscosity solutions and analysis in L∞, in Proceedings of the NATO advanced Study Institute (1999) 1–60.
- E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Diff. Equ.15 (1990) 1713–1742.
- A. Blanc, Deterministic exit time problems with discontinuous exit cost. SIAM J. Control Optim.35 (1997) 399–434.
- O. Bokanowski, N. Forcadel and H. Zidani, Reachability and minimal times for state constrained nonlinear problems without any controllability assumption. SIAM J. Control Optim.48 (2010) 4292–4316.
- I. Capuzzo-Dolcetta and P.-L. Lions, Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc.318 (1990) 643–683.
- P. Cardaliaguet, M. Quincampoix and P. Saint-Pierre, Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim.36 (1997) 21–42.
- F. Clarke, Y.S. Ledyaev, R. Stern and P. Wolenski, Nonsmooth analysis and control theory. Springer (1998).
- H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim.31 (1993) 257–272.
- H. Frankowska and S. Plaskacz, Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints. J. Math. Anal. Appl.251 (2000) 818–838.
- H. Frankowska and R.B. Vinter, Existence of neighboring feasible trajectories: applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl.104 (2000) 21–40.
- H. Ishii and S. Koike, A new formulation of state constraint problems for first-order PDEs. SIAM J. Control Optim.34 (1996) 554–571.
- M. Motta, On nonlinear optimal control problems with state constraints. SIAM J. Control Optim.33 (1995) 1411–1424.
- H.M. Soner, Optimal control with state-space constraint, I. SIAM J. Control Optim.24 (1986) 552–561.
- H.M. Soner, Optimal control with state-space constraint, II. SIAM J. Control Optim.24 (1986) 1110–1122.
- P. Soravia, Optimality principles and representation formulas for viscosity solutions of Hamilton-Jacobi equations. II. Equations of control problems with state constraints. Diff. Int. Equ.12 (1999) 275–293.

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