# Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations

Alessandra Cutrì; Francesca Da Lio

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 3, page 484-502
- ISSN: 1292-8119

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topCutrì, Alessandra, and Da Lio, Francesca. "Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 484-502. <http://eudml.org/doc/249998>.

@article{Cutrì2007,

abstract = {
In this paper we prove a comparison result between semicontinuous
viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_t+H(x,Du) = 0$ in $\{\rm I\}\!\{\rm R\}^n\times(0,T)$ where the Hamiltonian H may be noncoercive in
the gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.
},

author = {Cutrì, Alessandra, Da Lio, Francesca},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Hamilton-Jacobi equations; sub-Riemannian metric; viscosity solution; comparison principle},

language = {eng},

month = {6},

number = {3},

pages = {484-502},

publisher = {EDP Sciences},

title = {Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations},

url = {http://eudml.org/doc/249998},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Cutrì, Alessandra

AU - Da Lio, Francesca

TI - Comparison and existence results for evolutive non-coercive first-order Hamilton-Jacobi equations

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/6//

PB - EDP Sciences

VL - 13

IS - 3

SP - 484

EP - 502

AB -
In this paper we prove a comparison result between semicontinuous
viscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_t+H(x,Du) = 0$ in ${\rm I}\!{\rm R}^n\times(0,T)$ where the Hamiltonian H may be noncoercive in
the gradient Du. As a consequence of the comparison result and the Perron's method we get the existence of a continuous solution of this equation.

LA - eng

KW - Hamilton-Jacobi equations; sub-Riemannian metric; viscosity solution; comparison principle

UR - http://eudml.org/doc/249998

ER -

## References

top- O. Alvarez, Bounded-from-below viscosity solutions of Hamilton-Jacobi equations. Differential Integral Equations10 (1997) 419–436.
- H. Attouch, Variational convergence for functions and operators. Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA (1984).
- M. Bardi and I. Capuzzo Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Birkhäuser, Boston (1997).
- M. Bardi and F. Da Lio, On the Bellman equation for some unbounded control problems. NoDEA4 (1997) 491–510.
- G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris (1994).
- E.N. Barron and R. Jensen, Generalized viscosity solutions for Hamilton-Jacobi equations with time-measurable Hamiltonians. J. Differential Equations68 (1987) 10–21.
- E.N. Barron and R. Jensen, Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex hamiltonians. Commun. Partial Differ. Equ.15 (1990) 1713–1742.
- A. Bellaiche and J.-J. Risler, Sub-Riemannian geometry, Progress in Mathematics144, Birkhäuser Verlag, Basel (1996).
- A. Bensoussan, Stochastic control by functional analysis methods, Studies in Mathematics and its Applications11, North-Holland Publishing Co., Amsterdam (1982)
- I. Birindelli and J. Wigniolle, Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Commun. Pure Appl. Anal.2 (2003) 461–479.
- R.W. Brockett, Control theory and singular Riemannian geometry, in: New Directions in Applied Mathematics (Cleveland, Ohio, 1980) Springer, New York-Berlin (1982) 11–27.
- R.W. Brockett, Pattern generation and the control of nonlinear systems. IEEE Trans. Automatic Control48 (2003) 1699–1711.
- P. Cannarsa and G. Da Prato, Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations. SIAM J. Control Optim.27 (1989) 861–875.
- I. Capuzzo Dolcetta, The Hopf solution of Hamilton-Jacobi equations. Elliptic and parabolic problems (Rolduc/Gaeta) (2001) 343–351.
- I. Capuzzo Dolcetta, Representations of solutions of Hamilton-Jacobi equations. Progr. Nonlinear Differential Equations Appl.54 (2003) 79–90.
- I. Capuzzo Dolcetta and H. Ishii, Hopf formulas for state-dependent Hamilton-Jacobi equations. Preprint.
- A. Cutrì, Problemi semilineari ed integro-differenziali per sublaplaciani. Ph.D. Thesis, Universitá di Roma Tor Vergata (1997).
- F. Da Lio and O. Ley, Uniqueness Results for Second Order Bellman-Isaacs Equations under Quadratic Growth Assumptions and Applications, Quaderno 8, Dipartimento di Matematica, Università di Torino (2004).
- F. Da Lio and W.M. McEneaney, Finite time-horizon risk-sensitive control and the robust limit under a quadratic growth assumption. SIAM J. Control Optim40 (2002) 1628–1661 (electronic).
- C.L. Fefferman and D.H. Phong, Subelliptic eigenvalue problems, in Conference on Harmonic Analysis in Honor of A. Zygmund, Wadsworth Math. Series2 (1983) 590–606 .
- L. Hörmander, Hypoelliptic second order differential equations. Acta Math.119 (1967) 147–171.
- H. Ishii, Perron's method for Hamilton-Jacobi equations. Duke Math. J.55 (1987) 369–384.
- H. Ishii, Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above. Appl. Anal.67 (1997) 357–372.
- D. Jerison and A. Sànchez-Calle, Subelliptic second order differential operator. Lect. Notes Math. Berlin-Heidelberg-New York 1277 (1987) 46–77.
- J.J. Manfredi and B. Stroffolini, A version of the Hopf-Lax formula in the Heisenberg group. Comm. Partial Differ. Equ.27 (2002) 1139–1159.
- R. Monti and F. Serra Cassano, Surface measures in Carnot Caratheodory spaces. Calc. Var. Partial Differ. Equ.13 (2001) 339–376.
- A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vector fields. I: Basic properties. Acta Math.155 (1985) 103–147.
- F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bellman equations with fast gradient-dependence. Indiana Univ. Math. J.49 (2000) 1043–1077.
- F. Rampazzo and H. Sussmann, 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.
- B. Stroffolini, Homogenization of Hamilton-Jacobi Equations in Carnot Groups. ESAIM: COCV13 (2007) 107–119.
- H.J. Sussmann, A general theorem on local controllability. SIAM J. Control. Optim.25 (1987) 158–194.

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