Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 79.

This paper deals with junction conditions for Hamilton–Jacobi–Bellman (HJB) equations for finite horizon control problems on multi-domains. We consider two different cases where the final cost is continuous or lower semi-continuous. In the continuous case, we extend the results in Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, in Control and Optimization with PDE Constraints, Vol. 164 of International Series of Numerical Mathematics. Birkhäuser, Basel (2013) 93–116. in a more general framework with switching running costs and weaker controllability assumptions. The comparison principle has been established to guarantee the uniqueness and the stability results for the HJB system on such multi-domains. In the lower semi-continuous case, we characterize the value function as the unique lower semi-continuous viscosity solution of the HJB system, under a local controllability assumption.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018072
Classification : 49L20, 49J15, 35F21
Mots-clés : Optimal control, Hamilton–Jacobi–Bellman equations, multi-domains, junction conditions
Ghilli, Daria 1 ; Rao, Zhiping 1 ; Zidani, Hasnaa 1

1 
@article{COCV_2019__25__A79_0,
     author = {Ghilli, Daria and Rao, Zhiping and Zidani, Hasnaa},
     title = {Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {25},
     year = {2019},
     doi = {10.1051/cocv/2018072},
     zbl = {1437.49040},
     mrnumber = {4040712},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2018072/}
}
TY  - JOUR
AU  - Ghilli, Daria
AU  - Rao, Zhiping
AU  - Zidani, Hasnaa
TI  - Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2019
VL  - 25
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv/2018072/
DO  - 10.1051/cocv/2018072
LA  - en
ID  - COCV_2019__25__A79_0
ER  - 
%0 Journal Article
%A Ghilli, Daria
%A Rao, Zhiping
%A Zidani, Hasnaa
%T Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2019
%V 25
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv/2018072/
%R 10.1051/cocv/2018072
%G en
%F COCV_2019__25__A79_0
Ghilli, Daria; Rao, Zhiping; Zidani, Hasnaa. Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 79. doi : 10.1051/cocv/2018072. http://www.numdam.org/articles/10.1051/cocv/2018072/

[1] Y. Achdou, F. Camilli, A. Cutri and N. Tchou, Hamilton-Jacobi equations constrained on networks. Nonlinear Differ. Equ. Appl. 20 (2013) 413–445. | DOI | MR | Zbl

[2] J.-P. Aubin and A. Cellina, Differential Inclusions. Vol. 264 of Comprehensive Studies in Mathematics. Springer, Berlin (1984). | Zbl

[3] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for two-domains optimal control problems in ℝN. ESAIM: COCV 19 (2013) 710–739. | Numdam | MR | Zbl

[4] G. Barles, A. Briani and E. Chasseigne, A Bellman approach for regional optimal control problems in ℝN. SIAM J. Control Optim. 52 (2014) 1712–1744. | DOI | MR | Zbl

[5] G. Barles and E. Chasseigne, (Almost) everything you always wanted to know about deterministic control problems in stratified domains. Netw. Heterog. Media 10 (2015) 809–836. | DOI | MR | Zbl

[6] 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). | MR | Zbl

[7] A. Briani and A. Davini, Monge solutions for discontinuous Hamiltonians. ESAIM: COCV 11 (2005) 229–251. | Numdam | MR | Zbl

[8] R.C. Barnard and P.R. Wolenski, Flow invariance on stratified domains. Set-Valued Variational Anal. 21 (2013) 377–403. | DOI | MR | Zbl

[9] A. Bressan and Y. Hong, Optimal control problems for control systems on stratified domains. Netw. Heterog. Media 2 (2007) 313–331. | DOI | MR | Zbl

[10] F. Camilli and C. Marchi, A comparison among various notions of viscosity solutions for Hamilton-Jacobi equations on networks. J. Math. Anal. Appl. 407 (2013) 112–118. | DOI | MR | Zbl

[11] F. Camilli and A. Siconolfi, Hamilton-Jacobi equations with measurable dependence on the state variable. Adv. Differ. Equ. 8 (2003) 733–768. | MR | Zbl

[12] F.H. Clarke, Optimization and Nonsmooth Analysis, Society for Industrial Mathematics (1990). | DOI | MR | Zbl

[13] F.H. Clarke, Functional analysis, calculus of variations and optimal control, Vol. 264 of Graduate Text in Mathematics. Springer, NY (2013). | MR | Zbl

[14] F.H. Clarke, Yu.S. Ledyaev, R.J. Stern and P.R. Wolenski, Nonsmooth Analysis and Control Theory. Vol. 178 of Graduate Texts in Mathematics. Springer-Verlag, New York (1997). | MR | Zbl

[15] G. Dal Maso and H. Frankowska, Value function for Bolza problem with discontinuous Lagrangian and Hamilton-Jacobi inequalities. ESAIM: COCV 5 (2000) 369–394. | Numdam | MR | Zbl

[16] A.F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic Publishers, MA, (1988). | DOI | MR | Zbl

[17] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257–272. | DOI | MR | Zbl

[18] H. Frankowska and S. Plaskacz, Semicontinuous solutions of Hamilton-Jacobi-Bellman equations with degenerate state constraints. J. Math. Anal. Appl. 251 (2000) 818–838. | DOI | MR | Zbl

[19] C. Hermosilla and H. Zidani, Infinite horizon problems on stratifiable state constraints sets. J. Differ. Equ. 258 (2015) 1430–1460. | DOI | MR | Zbl

[20] C. Hermosilla, H. Zidani and P. Wolenski, The mayer and minimum time problems with stratified state constraints. Set-Valued Var. Anal. 26 (2017) 643–662. | DOI | MR | Zbl

[21] C. Imbert and R. Monneau, Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks, Hamilton-Jacobi Equations on Networks (2011).

[22] C. Imbert, R. Monneau and H. Zidani, A Hamilton-Jacobi approach to junction problems and application to traffic flows. ESAIM: COCV 19 (2013) 129–166. | Numdam | MR | Zbl

[23] H. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Sc. Norm. Sup. Pisa (IV) 16 (1989) 105–135. | Numdam | MR | Zbl

[24] Z. Rao, A. Siconolfi and H. Zidani, Stationary Hamilton-Jacobi-Bellman equations on multi-domains. J. Differ. Equ. 257 (2014) 3978–4014. | DOI

[25] Z. Rao and H. Zidani, Hamilton-Jacobi-Bellman equations on multi-domains, in Control and Optimization with PDE Constraints, Vol. 164 of International Series of Numerical Mathematics. Birkhäuser, Basel (2013) 93–116. | DOI | MR | Zbl

[26] P. Soravia, Boundary value problems for Hamilton-Jacobi equations with discontinuous Lagrangian. Indiana Univ. Math. J. 51 (2002) 451–477. | DOI | MR | Zbl

[27] P. Wolenski and Y. Zhuang, Proximal analysis and the minimal time function. SIAM J. Control Optim. 36 (1998) 1048–1072. | DOI | MR | Zbl

Cité par Sources :