We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].
Accepté le :
DOI : 10.1051/cocv/2018003
Mots-clés : Optimal control, networks, Hamilton-Jacobi equation, viscosity solutions, uniqueness, switching cost
@article{COCV_2019__25__A15_0, author = {Dao, Manh Khang}, title = {Hamilton-Jacobi equations for optimal control on networks with entry or exit costs}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, publisher = {EDP-Sciences}, volume = {25}, year = {2019}, doi = {10.1051/cocv/2018003}, zbl = {1437.49041}, mrnumber = {3963664}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2018003/} }
TY - JOUR AU - Dao, Manh Khang TI - Hamilton-Jacobi equations for optimal control on networks with entry or exit costs JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2018003/ DO - 10.1051/cocv/2018003 LA - en ID - COCV_2019__25__A15_0 ER -
%0 Journal Article %A Dao, Manh Khang %T Hamilton-Jacobi equations for optimal control on networks with entry or exit costs %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2018003/ %R 10.1051/cocv/2018003 %G en %F COCV_2019__25__A15_0
Dao, Manh Khang. Hamilton-Jacobi equations for optimal control on networks with entry or exit costs. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 15. doi : 10.1051/cocv/2018003. http://archive.numdam.org/articles/10.1051/cocv/2018003/
[1] Hamilton-Jacobi equations on networks. IFAC Proc. 44 (2011) 2577–2582.
, , and ,[2] Hamilton-Jacobi equations constrained on networks. NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445. | DOI | MR | Zbl
, , and ,[3] Hamilton-Jacobi equations for optimal control on junctions and networks. ESAIM: COCV 21 (2015) 876–899. | Numdam | MR | Zbl
, and ,[4] Discontinuous viscosity solutions of first-order Hamilton-Jacobi equations: a guided visit. Nonlinear Anal. 20 (1993) 1123–1134. | DOI | MR | Zbl
,[5] An introduction to the theory of viscosity solutions for first-order Hamilton-Jacobi equations and applications, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Vol. 2074 of Lecture Notes in Mathematics. Springer, Heidelberg (2013) 49–109. | DOI | MR | Zbl
,[6] A Bellman approach for two-domains optimal control problems in ℝN. ESAIM: COCV 19 (2013) 710–739. | Numdam | MR | Zbl
, and ,[7] A Bellman approach for regional optimal control problems in ℝN. SIAM J. Control Optim. 52 (2014) 1712–1744. | DOI | MR | Zbl
, and ,[8] Deterministic exit time control problems with discontinuous exit costs. SIAM J. Control Optim. 35 (1997) 399–434. | DOI | MR | Zbl
,[9] Comparison principle for the Cauchy problem for Hamilton-Jacobi equations with discontinuous data. Nonlinear Anal. 45 (2001) 1015–1037. | DOI | MR | Zbl
,[10] A comparison among various notions of viscosity solution for Hamilton-Jacobi equations on networks. J. Math. Anal. Appl. 407 (2013) 112–118. | DOI | MR | Zbl
and ,[11] Hamilton-Jacobi equations with state constraints. Trans. Am. Math. Soc. 318 (1990) 643–683. | DOI | MR | Zbl
and ,[12] Vertex control of flows in networks. Netw. Heterog. Media 3 (2008) 709–722. | DOI | MR | Zbl
, , and ,[13] Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints. Calc. Var. Partial Differ. Equ. 46 (2013) 725–747. | DOI | MR | Zbl
and ,[14] Conservation laws models, in Traffic Flow on Networks. Vol. 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). | MR | Zbl
and ,[15] Discontinuous solutions of Hamilton-Jacobi equations on networks. J. Differ. Equ. 263 (2017) 8418–8466. | DOI | MR | Zbl
, and ,[16] Flux-limited solutions for quasi-convex Hamilton-Jacobi equations on networks. Ann. Sci. Éc. Norm. Supér. 50 (2017) 357–448. | DOI | MR | Zbl
and ,[17] A Hamilton-Jacobi approach to junction problems and application to traffic flows. ESAIM: COCV 19 (2013) 129–166. | Numdam | MR | Zbl
, and ,[18] A short introduction to viscosity solutions and the large time behavior of solutions of Hamilton-Jacobi equations, in Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Vol. 2074 of Lecture Notes in Mathematics Springer, Heidelberg (2013) 111–249. | DOI | MR | Zbl
,[19] Viscosity solutions for junctions: well posedness and stability. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545. | MR | Zbl
and ,[20] Well Posedness for Multi-dimensional Junction Problems With Kirchoff-Type Conditions. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017) 807–816. | MR | Zbl
and ,[21] Hamilton-Jacobi Equations for Optimal Control on Multidimensional Junctions. Preprint (2014). | arXiv
,[22] Viscosity solutions of Eikonal equations on topological networks. Calc. Var. Partial Differ. Equ. 46 (2013) 671–686. | DOI | MR | Zbl
and ,[23] Optimal control with state-space constraint. I. SIAM J. Control Optim. 24 (1986) 552–561. | DOI | MR | Zbl
,[24] Optimal control with state-space constraint. II. SIAM J. Control Optim. 24 (1986) 1110–1122. | DOI | MR | Zbl
,Cité par Sources :