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].
Mots-clés : Hamilton-Jacobi equation, optimal control, nonsmooth analysis, viability theory, viscosity solution
@article{COCV_2014__20_3_771_0,
author = {Misztela, A.},
title = {The value function representing {Hamilton-Jacobi} equation with hamiltonian depending on value of solution},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {771--802},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
doi = {10.1051/cocv/2013083},
mrnumber = {3264223},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2013083/}
}
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 SP - 771 EP - 802 VL - 20 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013083/ DO - 10.1051/cocv/2013083 LA - en ID - COCV_2014__20_3_771_0 ER -
%0 Journal Article %A Misztela, A. %T The value function representing Hamilton-Jacobi equation with hamiltonian depending on value of solution %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 771-802 %V 20 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013083/ %R 10.1051/cocv/2013083 %G en %F COCV_2014__20_3_771_0
Misztela, A. The value function representing Hamilton-Jacobi equation with hamiltonian depending on value of solution. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 771-802. doi : 10.1051/cocv/2013083. http://www.numdam.org/articles/10.1051/cocv/2013083/
[1] , and , Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands. J. Math. Anal. Appl. 142 (1989) 301-316. | MR | Zbl
[2] and , Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). | MR | Zbl
[3] , Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Berlin Heidelberg (1994). | MR | Zbl
[4] and , Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Eqs. 15 (1990) 1713-1742. | MR | Zbl
[5] , Optymization - theory and applications, problems with ordinary differential equations. Springer, New York (1983). | MR | Zbl
[6] , Optimization and nonsmooth analysis. Wiley, New York (1983). | MR | Zbl
[7] and , Variational problems with Lipschitzian minimizers. Ann. Inst. Henri Poincare, Anal. Nonlinaire 6 (1989) 185-209. | Numdam | MR | Zbl
[8] and , Regularity properties of solutions to the basic problem in the calculus of variations. Trans. Amer. Math. Soc. 289 (1985) 73-98. | MR | Zbl
[9] and , Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. | MR | Zbl
[10] , and , Some properties of viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 282 (1984) 487-502. | MR | Zbl
[11] , , Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities. ESAIM: COCV 5 (2000) 369-393. | Numdam | MR | Zbl
[12] , Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. | MR | Zbl
[13] H. Frankowska, S. Plaskacz and T. Rzeʆuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation. J. Differ. Eqs. 116 (1995) 265-305. | MR | Zbl
[14] , Extended Hamilton - Jacobi characterization of value functions in optimal control. SIAM J. Control Optim. 39 (2000) 281-305. | MR | Zbl
[15] , Cosmically Lipschitz Set-Valued Mappings. Set-Valued Analysis 10 (2002) 331-360. | MR | Zbl
[16] , Topological Fixed Point Theory of Multivalued Mappings. Springer (1999). | Zbl
[17] and , Optimal control of unbounded differential inclusions. SIAM J. Control Optim. 32 (1994) 442-470. | MR | Zbl
[18] , , New necessary conditions for the generalized problem of Bolza. SIAM J. Control Optim. 34 (1996) 1496-1511. | MR | Zbl
[19] and , Bolza problems with general time constraints. SIAM J. Control Optim. 35 (1997) 2050-2069. | MR | Zbl
[20] and , On representation formulas for Hamilton Jacobi's equations related to calculus of variations problems. Topol. Methods Nonlinear Anal. 20 (2002) 85-118. | MR | Zbl
[21] , , On lipschitz regularity of minimizers of a calculus of variations problem with non locally bounded Lagrangians CR Math. 343 (2006) 69-74. | MR | Zbl
[22] , Equivalent subgradient versions of Hamiltonian and Euler - Lagrange equations in variational analysis. SIAM J. Control Optim. 34 (1996) 1300-1314. | MR | Zbl
[23] and , Variational Analysis. Springer-Verlag, Berlin (1998). | MR | Zbl
Cité par Sources :
