In this paper we prove a comparison result between semicontinuous viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form in where the hamiltonian may be noncoercive in the gradient As a consequence of the comparison result and the Perron’s method we get the existence of a continuous solution of this equation.
Mots-clés : Hamilton-Jacobi equations, sub-riemannian metric, viscosity solution, comparison principle
@article{COCV_2007__13_3_484_0, author = {Cutr{\`\i}, Alessandra and Lio, Francesca Da}, title = {Comparison and existence results for evolutive non-coercive first-order {Hamilton-Jacobi} equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {484--502}, publisher = {EDP-Sciences}, volume = {13}, number = {3}, year = {2007}, doi = {10.1051/cocv:2007021}, mrnumber = {2329172}, zbl = {1125.70013}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2007021/} }
TY - JOUR AU - Cutrì, Alessandra AU - Lio, Francesca Da TI - Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 SP - 484 EP - 502 VL - 13 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2007021/ DO - 10.1051/cocv:2007021 LA - en ID - COCV_2007__13_3_484_0 ER -
%0 Journal Article %A Cutrì, Alessandra %A Lio, Francesca Da %T Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 484-502 %V 13 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2007021/ %R 10.1051/cocv:2007021 %G en %F COCV_2007__13_3_484_0
Cutrì, Alessandra; Lio, Francesca Da. Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 13 (2007) no. 3, pp. 484-502. doi : 10.1051/cocv:2007021. http://archive.numdam.org/articles/10.1051/cocv:2007021/
[1] Bounded-from-below viscosity solutions of Hamilton-Jacobi equations. Differential Integral Equations 10 (1997) 419-436. | Zbl
,[2] Variational convergence for functions and operators. Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984). | MR | Zbl
,[3] Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997). | MR | Zbl
and ,[4] On the Bellman equation for some unbounded control problems. NoDEA 4 (1997) 491-510. | Zbl
and ,[5] Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris (1994). | MR | Zbl
,[6] Generalized viscosity solutions for Hamilton-Jacobi equations with time-measurable Hamiltonians. J. Differential Equations 68 (1987) 10-21. | Zbl
and ,[7] Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex hamiltonians. Commun. Partial Differ. Equ. 15 (1990) 1713-1742. | Zbl
and ,[8] Sub-Riemannian geometry, Progress in Mathematics 144, Birkhäuser Verlag, Basel (1996). | MR
and ,[9] Stochastic control by functional analysis methods, Studies in Mathematics and its Applications 11, North-Holland Publishing Co., Amsterdam (1982) | MR | Zbl
,[10] Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Commun. Pure Appl. Anal. 2 (2003) 461-479. | Zbl
and ,[11] Control theory and singular Riemannian geometry, in: New Directions in Applied Mathematics (Cleveland, Ohio, 1980) Springer, New York-Berlin (1982) 11-27. | Zbl
,[12] Pattern generation and the control of nonlinear systems. IEEE Trans. Automatic Control 48 (2003) 1699-1711.
,[13] Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations. SIAM J. Control Optim. 27 (1989) 861-875. | Zbl
and ,[14] The Hopf solution of Hamilton-Jacobi equations. Elliptic and parabolic problems (Rolduc/Gaeta) (2001) 343-351. | Zbl
,[15] Representations of solutions of Hamilton-Jacobi equations. Progr. Nonlinear Differential Equations Appl. 54 (2003) 79-90. | Zbl
,[16] Hopf formulas for state-dependent Hamilton-Jacobi equations. Preprint.
and ,[17] Problemi semilineari ed integro-differenziali per sublaplaciani. Ph.D. Thesis, Universitá di Roma Tor Vergata (1997).
,[18] Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications, Quaderno 8, Dipartimento di Matematica, Università di Torino (2004). | Zbl
and ,[19] Finite time-horizon risk-sensitive control and the robust limit under a quadratic growth assumption. SIAM J. Control Optim 40 (2002) 1628-1661 (electronic). | Zbl
and ,[20] Subelliptic eigenvalue problems, in Conference on Harmonic Analysis in Honor of A. Zygmund, Wadsworth Math. Series 2 (1983) 590-606 . | Zbl
and ,[21] Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147-171. | Zbl
,[22] Perron's method for Hamilton-Jacobi equations. Duke Math. J. 55 (1987) 369-384. | Zbl
,[23] Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above. Appl. Anal. 67 (1997) 357-372. | Zbl
,[24] Subelliptic second order differential operator. Lect. Notes Math. Berlin-Heidelberg-New York 1277 (1987) 46-77. | Zbl
and ,[25] A version of the Hopf-Lax formula in the Heisenberg group. Comm. Partial Differ. Equ. 27 (2002) 1139-1159. | Zbl
and ,[26] Surface measures in Carnot Caratheodory spaces. Calc. Var. Partial Differ. Equ. 13 (2001) 339-376. | Zbl
and ,[27] Balls and metrics defined by vector fields. I: Basic properties. Acta Math. 155 (1985) 103-147. | Zbl
, and ,[28] Hamilton-Jacobi-Bellman equations with fast gradient-dependence. Indiana Univ. Math. J. 49 (2000) 1043-1077. | Zbl
and ,[29] Set-valued differentials and a nonsmooth version of Chow's theorem, in Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, Florida (IEEE Publications, New York, 2001) 3 (2001) 2613-2618.
and ,[30] Homogenization of Hamilton-Jacobi Equations in Carnot Groups. ESAIM: COCV 13 (2007) 107-119. | Numdam | Zbl
,[31] A general theorem on local controllability. SIAM J. Control. Optim. 25 (1987) 158-194. | Zbl
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