The value function representing Hamilton-Jacobi equation with hamiltonian depending on value of solution
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 3, p. 771-802
The full text of recent articles is available to journal subscribers only. See the article on the journal's website

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].

DOI : https://doi.org/10.1051/cocv/2013083
Classification:  49J52,  49L25,  35B37
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {20},
     number = {3},
     year = {2014},
     pages = {771-802},
     doi = {10.1051/cocv/2013083},
     mrnumber = {3264223},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 20 (2014) no. 3, pp. 771-802. doi : 10.1051/cocv/2013083. http://www.numdam.org/item/COCV_2014__20_3_771_0/

[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. | MR 1014576 | Zbl 0689.49025

[2] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). | MR 1484411 | Zbl 0890.49011

[3] G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Berlin Heidelberg (1994). | MR 1613876 | Zbl 0819.35002

[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. | MR 1080619 | Zbl 0732.35014

[5] L. Cesari, Optymization - theory and applications, problems with ordinary differential equations. Springer, New York (1983). | MR 688142 | Zbl 0506.49001

[6] F.H. Clarke, Optimization and nonsmooth analysis. Wiley, New York (1983). | MR 709590 | Zbl 0582.49001

[7] F.H. Clarke and P.D. Loewen, Variational problems with Lipschitzian minimizers. Ann. Inst. Henri Poincare, Anal. Nonlinaire 6 (1989) 185-209. | Numdam | MR 1019114 | Zbl 0677.49006

[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. | MR 779053 | Zbl 0563.49009

[9] M.G. Crandall and P.L. Lions, Viscosity solutions of Hamilton-Jacobi equations. Trans. Amer. Math. Soc. 277 (1983) 1-42. | MR 690039 | Zbl 0599.35024

[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. | MR 732102 | Zbl 0543.35011

[11] G. Dal Maso, H. Frankowska, Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities. ESAIM: COCV 5 (2000) 369-393. | Numdam | MR 1765430 | Zbl 0952.49024

[12] H. Frankowska, Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equations. SIAM J. Control Optim. 31 (1993) 257-272. | MR 1200233 | Zbl 0796.49024

[13] H. Frankowska, S. Plaskacz and T. Rzeʆuchowski, Measurable viability theorems and Hamilton-Jacobi-Bellman equation. J. Differ. Eqs. 116 (1995) 265-305. | MR 1318576 | Zbl 0836.34016

[14] G.H. Galbraith, Extended Hamilton - Jacobi characterization of value functions in optimal control. SIAM J. Control Optim. 39 (2000) 281-305. | MR 1780920 | Zbl 0971.49017

[15] G.H. Galbraith, Cosmically Lipschitz Set-Valued Mappings. Set-Valued Analysis 10 (2002) 331-360. | MR 1934750 | Zbl 1025.26020

[16] L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings. Springer (1999). | Zbl 1107.55001

[17] P.D. Loewen and R.T. Rockafellar, Optimal control of unbounded differential inclusions. SIAM J. Control Optim. 32 (1994) 442-470. | MR 1261148 | Zbl 0823.49016

[18] P.D. Loewen, R.T. Rockafellar, New necessary conditions for the generalized problem of Bolza. SIAM J. Control Optim. 34 (1996) 1496-1511. | MR 1404843 | Zbl 0871.49023

[19] P.D. Loewen and R.T. Rockafellar, Bolza problems with general time constraints. SIAM J. Control Optim. 35 (1997) 2050-2069. | MR 1478652 | Zbl 0904.49014

[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. | MR 1940532 | Zbl 1021.49024

[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. | MR 2241962 | Zbl 1093.49023

[22] R.T. Rockafellar, Equivalent subgradient versions of Hamiltonian and Euler - Lagrange equations in variational analysis. SIAM J. Control Optim. 34 (1996) 1300-1314. | MR 1395835 | Zbl 0878.49012

[23] R.T. Rockafellar and R.J.-B. Wets, Variational Analysis. Springer-Verlag, Berlin (1998). | MR 1491362 | Zbl 0888.49001