Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-22T22:18:59.706Z Has data issue: false hasContentIssue false

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

Published online by Cambridge University Press:  05 June 2007

Alessandra Cutrì
Affiliation:
Dipartimento di Matematica, Università ”Tor Vergata”, via della Ricerca Scientifica, 00133 Roma, Italy; [email protected]
Francesca Da Lio
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università di Padova, via Belzoni 7, 35131 Padova, Italy.
Get access

Abstract

In this paper we prove a comparison result between semicontinuousviscosity subsolutions and supersolutions to Hamilton-Jacobi equations of the form $u_t+H(x,Du) = 0$ in ${\rm I}\!{\rmR}^n\times(0,T)$ where the Hamiltonian H may be noncoercive inthe 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alvarez, O., Bounded-from-below viscosity solutions of Hamilton-Jacobi equations. Differential Integral Equations 10 (1997) 419436.
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).
Bardi, M. and Da Lio, F., On the Bellman equation for some unbounded control problems. NoDEA 4 (1997) 491510. CrossRef
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris (1994).
Barron, E.N. and Jensen, R., Generalized viscosity solutions for Hamilton-Jacobi equations with time-measurable Hamiltonians. J. Differential Equations 68 (1987) 1021. CrossRef
Barron, E.N. and Jensen, R., Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex hamiltonians. Commun. Partial Differ. Equ. 15 (1990) 17131742.
A. Bellaiche and J.-J. Risler, Sub-Riemannian geometry, Progress in Mathematics 144, Birkhäuser Verlag, Basel (1996).
A. Bensoussan, Stochastic control by functional analysis methods, Studies in Mathematics and its Applications 11, North-Holland Publishing Co., Amsterdam (1982)
Birindelli, I. and Wigniolle, J., Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Commun. Pure Appl. Anal. 2 (2003) 461479.
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.
Brockett, R.W., Pattern generation and the control of nonlinear systems. IEEE Trans. Automatic Control 48 (2003) 16991711. CrossRef
Cannarsa, P. and Da Prato, G., Nonlinear optimal control with infinite horizon for distributed parameter systems and stationary Hamilton-Jacobi equations. SIAM J. Control Optim. 27 (1989) 861875. CrossRef
I. Capuzzo Dolcetta, The Hopf solution of Hamilton-Jacobi equations. Elliptic and parabolic problems (Rolduc/Gaeta) (2001) 343–351.
Capuzzo Dolcetta, I., Representations of solutions of Hamilton-Jacobi equations. Progr. Nonlinear Differential Equations Appl. 54 (2003) 7990.
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).
Da Lio, F. and McEneaney, W.M., Finite time-horizon risk-sensitive control and the robust limit under a quadratic growth assumption. SIAM J. Control Optim 40 (2002) 16281661 (electronic). CrossRef
C.L. Fefferman and D.H. Phong, Subelliptic eigenvalue problems, in Conference on Harmonic Analysis in Honor of A. Zygmund, Wadsworth Math. Series 2 (1983) 590–606 .
Hörmander, L., Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147171. CrossRef
Ishii, H., Perron's method for Hamilton-Jacobi equations. Duke Math. J. 55 (1987) 369384. CrossRef
Ishii, H., Comparison results for Hamilton-Jacobi equations without growth condition on solutions from above. Appl. Anal. 67 (1997) 357372. CrossRef
D. Jerison and A. Sànchez-Calle, Subelliptic second order differential operator. Lect. Notes Math. Berlin-Heidelberg-New York 1277 (1987) 46–77.
Manfredi, J.J. and Stroffolini, B., A version of the Hopf-Lax formula in the Heisenberg group. Comm. Partial Differ. Equ. 27 (2002) 11391159. CrossRef
Monti, R. and Serra Cassano, F., Surface measures in Carnot Caratheodory spaces. Calc. Var. Partial Differ. Equ. 13 (2001) 339376. CrossRef
Nagel, A., Stein, E.M. and Wainger, S., Balls and metrics defined by vector fields. I: Basic properties. Acta Math. 155 (1985) 103147. CrossRef
Rampazzo, F. and Sartori, C., Hamilton-Jacobi-Bellman equations with fast gradient-dependence. Indiana Univ. Math. J. 49 (2000) 10431077.
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.
Stroffolini, B., Homogenization of Hamilton-Jacobi Equations in Carnot Groups. ESAIM: COCV 13 (2007) 107119. CrossRef
Sussmann, H.J., A general theorem on local controllability. SIAM J. Control. Optim. 25 (1987) 158194. CrossRef