Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T16:31:31.942Z Has data issue: false hasContentIssue false
  • English
  • Français

A general Hamilton-Jacobi framework for non-linear state-constrained control problems∗

Published online by Cambridge University Press:  21 June 2012

Albert Altarovici ,
Olivier Bokanowski  and
Hasnaa Zidani
Albert Altarovici
Affiliation:
Projet Commands, INRIA Saclay & ENSTA ParisTech, 32 Bd. Victor, 75739 Paris Cedex 15, France. Albert.Altarovici@inria.fr; Hasnaa.Zidani@ensta.fr
Olivier Bokanowski
Affiliation:
Projet Commands, INRIA Saclay & ENSTA ParisTech, 32 Bd. Victor, 75739 Paris Cedex 15, France. Albert.Altarovici@inria.fr; Hasnaa.Zidani@ensta.fr Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Jacques-Louis Lions, UMR 7598, UPMC, CNRS, 75205 Paris, France; boka@math.jussieu.fr
Hasnaa Zidani
Affiliation:
Projet Commands, INRIA Saclay & ENSTA ParisTech, 32 Bd. Victor, 75739 Paris Cedex 15, France. Albert.Altarovici@inria.fr; Hasnaa.Zidani@ensta.fr
Get access

Abstract

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

Keywords

State constraintsoptimal control problemsnonlinear controlled systemsHamilton-Jacobi equationsviscosity solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 337 - 357
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.-P. Aubin, Viability theory. Birkäuser, Boston (1991).
Aubin, J.-P., Viability solutions to structured Hamilton-Jacobi equations under constraints. SIAM J. Control Optim. 49 (2011) 1881–1915. Google Scholar
J.-P. Aubin and A. Cellina, Differential inclusions, Comprehensive Studies in Mathematics. Springer, Berlin, Heidelberg, New York, Tokyo 264 (1984).
J.-P. Aubin and H. Frankowska, Set-valued analysis, Birkhäuser Boston Inc., Boston, MA. Systems and Control : Foundations and Applications 2 (1990).
Aubin, J.-P. and Frankowska, H., The viability kernel algorithm for computing value functions of infinite horizon optimal control problems. J. Math. Anal. Appl. 201 (1996) 555–576. Google Scholar
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).
Bardi, M., Koike, S. and Soravia, P., Pursuit-evasion games with state constraints : dynamic programming and discrete-time approximations. Discrete Contin. Dyn. Syst. 6 (2000) 361–380. Google Scholar
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi, Springer, Paris. Math. Appl. 17 (1994).
R.C. Barnard and P.R. Wolenski, The minimal time function on stratified domains. Submitted (2011).
E.N. Barron, Viscosity solutions and analysis in L ∞, in Proc. of the NATO Advanced Study Institute (1999) 1–60.
Barron, E.N. and Ishii, H., The bellman equation for minimizing the maximum cost. Nonlinear Anal. 13 (1989) 1067–1090. Google Scholar
Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15 (1990) 1713–1742. Google Scholar
Barron, E.N. and Jensen, R., Relaxation of constrained control problems. SIAM J. Control Optim. 34 (1996) 2077–2091. Google Scholar
Bokanowski, O., Cristiani, E. and Zidani, H., An efficient data structure and accurate scheme to solve front propagation problems. J. Sci. Comput. 42 (2010) 251–273. Google Scholar
Bokanowski, O., Forcadel, N. and Zidani, H., Reachability and minimal times for state constrained nonlinear problems without any controllability assumption. SIAM J. Control Optim. 48 (2010) 4292–4316. Google Scholar
Bokanowski, O., Forcadel, N. and Zidani, H., Deterministic state constrained optimal control problems without controllability assumptions. ESAIM : COCV 17 (2011) 995–1015. Google Scholar
O. Bokanowski, J. Zhao and H. Zidani, Binope-HJ : a d-dimensional C++ parallel HJ solver. http://www.ensta-paristech.fr/~zidani/BiNoPe-HJ/ (2011).
Capuzzo-Dolcetta, I. and Lions, P.-L., Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643–683. Google Scholar
Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P., Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim. 36 (1997) 21–42. Google Scholar
Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P., Numerical schemes for discontinuous value function of optimal control. Set-Valued Analysis 8 (2000) 111–126. Google Scholar
Cardaliaguet, P., Quincampoix, M. and Saint-Pierre, P., Pursuit differential games with state constraints. SIAM J. Control Optim. 39 (2000) 1615–1632 (electronic). Google Scholar
F. Clarke, Y.S. Ledyaev, R. Stern and P. Wolenski, Nonsmooth analysis and control theory. Springer (1998).
Crandall, M. and Lions, P.-L., Viscosity solutions of Hamilton Jacobi equations. Bull. Amer. Math. Soc. 277 (1983) 1–42. Google Scholar
Crandall, M., Evans, L. and Lions, P.-L., Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487–502. Google Scholar
R.J. Elliott and N.J. Kalton, The existence of value in differential games, American Mathematical Society, Providence, RI. Memoirs of the American Mathematical Society 126 (1972).
Frankowska, H., Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257–272. Google Scholar
Frankowska, H. and Plaskacz, S., Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251 (2000) 818–838. Google Scholar
Frankowska, H. and Rampazzo, F., Relaxation of control systems under state constraints. SIAM J. Control Optim. 37 (1999) 1291–1309. Google Scholar
Frankowska, H. and Vinter, R.B., Existence of neighboring feasible trajectories : applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl. 104 (2000) 21–40. Google Scholar
Ishii, H., Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984) 721–748. Google Scholar
Ishii, H. and Koike, S., A new formulation of state constraint problems for first-order PDEs. SIAM J. Control Optim. 34 (1996) 554–571. Google Scholar
Loreti, P., Some properties of constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 25 (1987) 1244–1252. Google Scholar
Loreti, P. and Tessitore, E., Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. J. Math. Systems, Estimation Control 4 (1994) 467–483. Google Scholar
Margellos, K. and Lygeros, J., Hamilton-Jacobi formulation for reach-avoid differential games. IEEE Trans. Automat. Control 56 (2011) 1849–1861. Google Scholar
Motta, M., On nonlinear optimal control problems with state constraints. SIAM J. Control Optim. 33 (1995) 1411–1424. Google Scholar
Motta, M. and Rampazzo, F., Multivalued dynamics on a closed domain with absorbing boundary. applications to optimal control problems with integral constraints. Nonlinear Anal. 41 (2000) 631–647. Google Scholar
Peng, D.P., Merriman, B., Osher, S., Zhao, H.K. and Kang, M.J., A PDE-based fast local level set method. J. Comput. Phys. 155 (1999) 410–438. Google Scholar
Saint-Pierre, P., Approximation of viability kernel. Appl. Math. Optim. 29 (1994) 187–209. Google Scholar
Soner, H.M., Optimal control with state-space constraint I. SIAM J. Control Optim. 24 (1986) 552–561. Google Scholar
Soner, H.M., Optimal control with state-space constraint II. SIAM J. Control Optim. 24 (1986) 1110–1122. Google Scholar
Soravia, P., Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control Optim. 31 (1993) 604–623. Google Scholar
Varaiya, P.P., On the existence of solutions to a differential game. SIAM J. Control 5 (1967) 153–162. Google Scholar