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.
Mots clés : state constraints, optimal control problems, nonlinear controlled systems, Hamilton-Jacobi equations, viscosity solutions
@article{COCV_2013__19_2_337_0, author = {Altarovici, Albert and Bokanowski, Olivier and Zidani, Hasnaa}, title = {A general {Hamilton-Jacobi} framework for non-linear state-constrained control problems}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {337--357}, publisher = {EDP-Sciences}, volume = {19}, number = {2}, year = {2013}, doi = {10.1051/cocv/2012011}, mrnumber = {3049714}, zbl = {1273.35089}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012011/} }
TY - JOUR AU - Altarovici, Albert AU - Bokanowski, Olivier AU - Zidani, Hasnaa TI - A general Hamilton-Jacobi framework for non-linear state-constrained control problems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 337 EP - 357 VL - 19 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012011/ DO - 10.1051/cocv/2012011 LA - en ID - COCV_2013__19_2_337_0 ER -
%0 Journal Article %A Altarovici, Albert %A Bokanowski, Olivier %A Zidani, Hasnaa %T A general Hamilton-Jacobi framework for non-linear state-constrained control problems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 337-357 %V 19 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012011/ %R 10.1051/cocv/2012011 %G en %F COCV_2013__19_2_337_0
Altarovici, Albert; Bokanowski, Olivier; Zidani, Hasnaa. A general Hamilton-Jacobi framework for non-linear state-constrained control problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 2, pp. 337-357. doi : 10.1051/cocv/2012011. http://www.numdam.org/articles/10.1051/cocv/2012011/
[1] Viability theory. Birkäuser, Boston (1991). | MR | Zbl
,[2] Viability solutions to structured Hamilton-Jacobi equations under constraints. SIAM J. Control Optim. 49 (2011) 1881-1915. | MR | Zbl
,[3] Differential inclusions, Comprehensive Studies in Mathematics. Springer, Berlin, Heidelberg, New York, Tokyo 264 (1984). | MR | Zbl
and ,[4] Set-valued analysis, Birkhäuser Boston Inc., Boston, MA. Systems and Control : Foundations and Applications 2 (1990). | MR | Zbl
and ,[5] The viability kernel algorithm for computing value functions of infinite horizon optimal control problems. J. Math. Anal. Appl. 201 (1996) 555-576. | MR | Zbl
and ,[6] Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, Systems and Control : Foundations and Applications. Birkhäuser, Boston (1997). | MR | Zbl
and ,[7] Pursuit-evasion games with state constraints : dynamic programming and discrete-time approximations. Discrete Contin. Dyn. Syst. 6 (2000) 361-380. | MR | Zbl
, and ,[8] Solutions de viscosité des équations de Hamilton-Jacobi, Springer, Paris. Math. Appl. 17 (1994). | MR | Zbl
,[9] The minimal time function on stratified domains. Submitted (2011).
and ,[10] Viscosity solutions and analysis in L∞, in Proc. of the NATO Advanced Study Institute (1999) 1-60. | MR | Zbl
,[11] The bellman equation for minimizing the maximum cost. Nonlinear Anal. 13 (1989) 1067-1090. | Zbl
and ,[12] Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15 (1990) 1713-1742. | MR | Zbl
and ,[13] Relaxation of constrained control problems. SIAM J. Control Optim. 34 (1996) 2077-2091. | MR | Zbl
and ,[14] An efficient data structure and accurate scheme to solve front propagation problems. J. Sci. Comput. 42 (2010) 251-273. | MR | Zbl
, and ,[15] Reachability and minimal times for state constrained nonlinear problems without any controllability assumption. SIAM J. Control Optim. 48 (2010) 4292-4316. | MR | Zbl
, and ,[16] Deterministic state constrained optimal control problems without controllability assumptions. ESAIM : COCV 17 (2011) 995-1015. | Numdam | MR | Zbl
, and ,[17] Binope-HJ : a d-dimensional C++ parallel HJ solver. http://www.ensta-paristech.fr/~zidani/BiNoPe-HJ/ (2011).
, and ,[18] Hamilton-Jacobi equations with state constraints. Trans. Amer. Math. Soc. 318 (1990) 643-683. | MR | Zbl
and ,[19] Optimal times for constrained nonlinear control problems without local controllability. Appl. Math. Optim. 36 (1997) 21-42. | MR | Zbl
, and ,[20] Numerical schemes for discontinuous value function of optimal control. Set-Valued Analysis 8 (2000) 111-126. | MR | Zbl
, and ,[21] Pursuit differential games with state constraints. SIAM J. Control Optim. 39 (2000) 1615-1632 (electronic). | MR | Zbl
, and ,[22] Nonsmooth analysis and control theory. Springer (1998). | MR | Zbl
, , and ,[23] Viscosity solutions of Hamilton Jacobi equations. Bull. Amer. Math. Soc. 277 (1983) 1-42. | MR | Zbl
and ,[24] Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487-502. | MR | Zbl
, and ,[25] The existence of value in differential games, American Mathematical Society, Providence, RI. Memoirs of the American Mathematical Society 126 (1972). | MR | Zbl
and ,[26] Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. | MR | Zbl
,[27] Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251 (2000) 818-838. | MR | Zbl
and ,[28] Relaxation of control systems under state constraints. SIAM J. Control Optim. 37 (1999) 1291-1309. | MR | Zbl
and ,[29] Existence of neighboring feasible trajectories : applications to dynamic programming for state-constrained optimal control problems. J. Optim. Theory Appl. 104 (2000) 21-40. | MR | Zbl
and ,[30] Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations. Indiana Univ. Math. J. 33 (1984) 721-748. | MR | Zbl
,[31] A new formulation of state constraint problems for first-order PDEs. SIAM J. Control Optim. 34 (1996) 554-571. | MR | Zbl
and ,[32] Some properties of constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 25 (1987) 1244-1252. | MR | Zbl
,[33] Approximation and regularity results on constrained viscosity solutions of Hamilton-Jacobi-Bellman equations. J. Math. Systems, Estimation Control 4 (1994) 467-483. | MR | Zbl
and ,[34] Hamilton-Jacobi formulation for reach-avoid differential games. IEEE Trans. Automat. Control 56 (2011) 1849-1861. | MR
and ,[35] On nonlinear optimal control problems with state constraints. SIAM J. Control Optim. 33 (1995) 1411-1424. | MR | Zbl
,[36] Multivalued dynamics on a closed domain with absorbing boundary. applications to optimal control problems with integral constraints. Nonlinear Anal. 41 (2000) 631-647. | MR | Zbl
and ,[37] A PDE-based fast local level set method. J. Comput. Phys. 155 (1999) 410-438. | MR | Zbl
, , , and ,[38] Approximation of viability kernel. Appl. Math. Optim. 29 (1994) 187-209. | MR | Zbl
,[39] Optimal control with state-space constraint I. SIAM J. Control Optim. 24 (1986) 552-561. | MR | Zbl
,[40] Optimal control with state-space constraint II. SIAM J. Control Optim. 24 (1986) 1110-1122. | MR | Zbl
,[41] Pursuit-evasion problems and viscosity solutions of Isaacs equations. SIAM J. Control Optim. 31 (1993) 604-623. | MR | Zbl
,[42] On the existence of solutions to a differential game. SIAM J. Control 5 (1967) 153-162. | MR | Zbl
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