We consider a class of optimal control problems in which the cost to minimize comprises both a final cost and an integral term, and the data can be discontinuous with respect to the time variable in the following sense: they are continuous w.r.t. t on a set of full measure and have everywhere left and right limits. For this class of Bolza problems, employing techniques coming from viability theory, we give characterizations of the value function as the unique generalized solution to the corresponding Hamilton-Jacobi equation in the class of lower semicontinuous functions: if the final cost term is extended valued, the generalized solution to the Hamilton-Jacobi equation involves the concepts of lower Dini derivative and the proximal normal vectors; if the final cost term is a locally bounded lower semicontinuous function, then we can show that this has an equivalent characterization in a viscosity sense.
Accepté le :
Première publication :
Publié le :
DOI : 10.1051/cocv/2019041
Mots-clés : Hamilton-Jacobi equation, discontinuous data, value function
@article{COCV_2020__26_1_A66_0,
author = {Bernis, Julien and Bettiol, Piernicola},
title = {Solutions to the {Hamilton-Jacobi} equation for {Bolza} problems with discontinuous time dependent data},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019041},
mrnumber = {4152107},
zbl = {1448.49018},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019041/}
}
TY - JOUR AU - Bernis, Julien AU - Bettiol, Piernicola TI - Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019041/ DO - 10.1051/cocv/2019041 LA - en ID - COCV_2020__26_1_A66_0 ER -
%0 Journal Article %A Bernis, Julien %A Bettiol, Piernicola %T Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019041/ %R 10.1051/cocv/2019041 %G en %F COCV_2020__26_1_A66_0
Bernis, Julien; Bettiol, Piernicola. Solutions to the Hamilton-Jacobi equation for Bolza problems with discontinuous time dependent data. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 66. doi : 10.1051/cocv/2019041. http://www.numdam.org/articles/10.1051/cocv/2019041/
[1] and , Set-valued analysis. Springer Science & Business Media (2009). | DOI | MR | Zbl
[2] and , Optimal Control and Viscosity Solutions of the Hamilton-Jacobi Equations. Birkhäuser, Boston (1997). | DOI | MR | Zbl
[3] , Chimie Organique et Supramoléculaire. Ph.D. thesis, Université de Bretagne Occidentale (2007).
[4] and , The Hamilton Jacobi Equation For Optimal Control Problems with Discontinuous Time Dependence. SIAM J. Control Optim. 55 (2017) 1199–1225. | DOI | MR | Zbl
[5] and , Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Springer Science & Business Media (2004). | DOI | MR | Zbl
[6] , Optimization-theory and applications: problems with ordinary differential equations, vol. 17. Springer Science & Business Media (2012). | MR | Zbl
[7] , Functional analysis, calculus of variations and optimal control. Vol. 264 of Graduate Texts in Mathematics. Springer Science & Business Media (2013). | MR | Zbl
[8] , , and , Qualitative properties of trajectories of control systems: a survey. J. Dynam. Control Syst. 1 (1995) 1–48. | DOI | MR | Zbl
[9] , , and , Nonsmooth Analysis and Control Theory. Vol. 178 of Graduate Texts in Mathematics. Springer Verlag, New York (1998). | MR | Zbl
[10] and , Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities. ESAIM: COCV 5 (2000) 369–393. | Numdam | MR | Zbl
[11] , , et al., Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, Du Bois-Reymond necessary conditions, and Hamilton-Jacobi equations. Appl. Math. Optim. 48 (2003) 39–66. | DOI | MR | Zbl
[12] , Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257–272. | DOI | MR | Zbl
[13] and , Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints. Calc. Var. Partial Differ. Equ. 46 (2013) 725–747. | DOI | MR | Zbl
[14] , and , Measurable viability theorems and the Hamilton-Jacobi-Bellman equation. J. Differ.Equ. 116 (1995) 265–305. | DOI | MR | Zbl
[15] and , Existence of neighbouring feasible trajectories: applications to dynamic programming for state constrained optimal control problems. J. Optim. Theory Appl. 104 (2000) 21–40. | DOI | MR | Zbl
[16] , Calculus of Dini subdifferentials of functions and contingent coderivatives of set valued maps. Nonlinear Anal. Theory Methods Appl. 8 (1984) 517–539. | DOI | MR | Zbl
[17] , Weak lower semicontinuity of integral functionals. J. Optim. Theory Appl. 19 (1976) 3–16. | DOI | MR | Zbl
[18] , , et al., On representation formulas for Hamilton Jacobi’s equations related to calculus of variations problems. Topol. Methods Nonlinear Anal. 20 (2002) 85–118. | DOI | MR | Zbl
[19] , Convex analysis. Princeton University Press (2015). | Zbl
[20] , Optimal control. Springer Science & Business Media (2010). | DOI
[21] and , Hamilton-Jacobi theory for optimal control problems with data measurable in time. SIAM J. Control Optim. 28 (1990) 1404–1419. | DOI | MR | Zbl
Cité par Sources :
