The value function representing Hamilton–Jacobi equation with hamiltonian depending on value of solution
ESAIM: Control, Optimisation and Calculus of Variations (2014)
- Volume: 20, Issue: 3, page 771-802
- ISSN: 1292-8119
Access Full Article
topAbstract
topHow to cite
topMisztela, A.. "The value function representing Hamilton–Jacobi equation with hamiltonian depending on value of solution." ESAIM: Control, Optimisation and Calculus of Variations 20.3 (2014): 771-802. <http://eudml.org/doc/272786>.
@article{Misztela2014,
abstract = {In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of variation. In the paper we prove that absolutely continuous functions that are solutions of Mayer problem satisfy the Lipschitz condition. Using this fact we show that the value function is a bilateral solution of Hamilton–Jacobi equation. Moreover, we prove that continuity or the local Lipschitz condition of the function of final cost g is inherited by the value function. Our results allow to state the theorem about existence and uniqueness of bilateral solutions in the class of functions that are bounded from below and satisfy the local Lipschitz condition. In proving the main results we use recently derived necessary optimality conditions of Loewen–Rockafellar [P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 32 (1994) 442–470; P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 35 (1997) 2050–2069].},
author = {Misztela, A.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamilton–Jacobi equation; optimal control; nonsmooth analysis; viability theory; viscosity solution; Hamilton-Jacobi equation},
language = {eng},
number = {3},
pages = {771-802},
publisher = {EDP-Sciences},
title = {The value function representing Hamilton–Jacobi equation with hamiltonian depending on value of solution},
url = {http://eudml.org/doc/272786},
volume = {20},
year = {2014},
}
TY - JOUR
AU - Misztela, A.
TI - The value function representing Hamilton–Jacobi equation with hamiltonian depending on value of solution
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 3
SP - 771
EP - 802
AB - In the paper we investigate the regularity of the value function representing Hamilton–Jacobi equation: − Ut + H(t, x, U, − Ux) = 0 with a final condition: U(T,x) = g(x). Hamilton–Jacobi equation, in which the Hamiltonian H depends on the value of solution U, is represented by the value function with more complicated structure than the value function in Bolza problem. This function is described with the use of some class of Mayer problems related to the optimal control theory and the calculus of variation. In the paper we prove that absolutely continuous functions that are solutions of Mayer problem satisfy the Lipschitz condition. Using this fact we show that the value function is a bilateral solution of Hamilton–Jacobi equation. Moreover, we prove that continuity or the local Lipschitz condition of the function of final cost g is inherited by the value function. Our results allow to state the theorem about existence and uniqueness of bilateral solutions in the class of functions that are bounded from below and satisfy the local Lipschitz condition. In proving the main results we use recently derived necessary optimality conditions of Loewen–Rockafellar [P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 32 (1994) 442–470; P.D. Loewen and R.T. Rockafellar, SIAM J. Control Optim. 35 (1997) 2050–2069].
LA - eng
KW - Hamilton–Jacobi equation; optimal control; nonsmooth analysis; viability theory; viscosity solution; Hamilton-Jacobi equation
UR - http://eudml.org/doc/272786
ER -
References
top- [1] L. Ambrosio, O. Ascenzi and G. Buttazzo, Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl.142 (1989) 301–316. Zbl0689.49025MR1014576
- [2] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). Zbl0890.49011MR1484411
- [3] G. Barles, Solutions de viscosité des équations de Hamilton–Jacobi. Springer-Verlag, Berlin Heidelberg (1994). Zbl0819.35002MR1613876
- [4] E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Eqs.15 (1990) 1713–1742. Zbl0732.35014MR1080619
- [5] L. Cesari, Optymization – theory and applications, problems with ordinary differential equations. Springer, New York (1983). Zbl0506.49001MR688142
- [6] F.H. Clarke, Optimization and nonsmooth analysis. Wiley, New York (1983). Zbl0582.49001MR709590
- [7] F.H. Clarke and P.D. Loewen, Variational problems with Lipschitzian minimizers. Ann. Inst. Henri Poincare, Anal. Nonlinaire 6 (1989) 185–209. Zbl0677.49006MR1019114
- [8] F.H. Clarke and R.B. Vinter, Regularity properties of solutions to the basic problem in the calculus of variations. Trans. Amer. Math. Soc.289 (1985) 73–98. Zbl0563.49009MR779053
- [9] M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.277 (1983) 1–42. Zbl0599.35024MR690039
- [10] M.G. Crandall, L.C. Evans and P.-L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc.282 (1984) 487–502. Zbl0543.35011MR732102
- [11] G. Dal Maso, H. Frankowska, Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities. ESAIM: COCV 5 (2000) 369–393. Zbl0952.49024MR1765430
- [12] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim.31 (1993) 257–272. Zbl0796.49024MR1200233
- [13] H. Frankowska, S. Plaskacz and T. Rzeʆuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation. J. Differ. Eqs. 116 (1995) 265–305. Zbl0836.34016MR1318576
- [14] G.H. Galbraith, Extended Hamilton – Jacobi characterization of value functions in optimal control. SIAM J. Control Optim.39 (2000) 281–305. Zbl0971.49017MR1780920
- [15] G.H. Galbraith, Cosmically Lipschitz Set-Valued Mappings. Set-Valued Analysis10 (2002) 331–360. Zbl1025.26020MR1934750
- [16] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings. Springer (1999). Zbl1107.55001
- [17] P.D. Loewen and R.T. Rockafellar, Optimal control of unbounded differential inclusions. SIAM J. Control Optim.32 (1994) 442–470. Zbl0823.49016MR1261148
- [18] P.D. Loewen, R.T. Rockafellar, New necessary conditions for the generalized problem of Bolza. SIAM J. Control Optim.34 (1996) 1496–1511. Zbl0871.49023MR1404843
- [19] P.D. Loewen and R.T. Rockafellar, Bolza problems with general time constraints. SIAM J. Control Optim.35 (1997) 2050–2069. Zbl0904.49014MR1478652
- [20] S. Plaskacz and M. Quincampoix, On representation formulas for Hamilton Jacobi’s equations related to calculus of variations problems. Topol. Methods Nonlinear Anal.20 (2002) 85–118. Zbl1021.49024MR1940532
- [21] M. Quincampoix, N. Zlateva, On lipschitz regularity of minimizers of a calculus of variations problem with non locally bounded Lagrangians CR Math. 343 (2006) 69–74. Zbl1093.49023MR2241962
- [22] R.T. Rockafellar, Equivalent subgradient versions of Hamiltonian and Euler – Lagrange equations in variational analysis. SIAM J. Control Optim.34 (1996) 1300–1314. Zbl0878.49012MR1395835
- [23] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag, Berlin (1998). Zbl0888.49001MR1491362
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.