My talk will concern the existence of a solution u = u(t1 , ...., tN , x) to a
system of the form
∂u
+ H1 (x, Dx u) = 0
∂t1
...
(t1 , . . . , tN , x) ∈]0, T [N ×IRn (1)
...
∂u
∂tN + HN (x, Dx u) = 0
u(0, . . . , 0, x) = ψ(x) ∀x ∈ IRn . (2)
Notice that, for every i = 1, ..., N , the i−th equation in (1) is a Hamilton-Jacobi
equation in its own time ti . Since the system is over-determined —u being
scalar-valued— in general a solution fails to exist. Moreover, the differential
geometric conditions
{Hi , Hj } = 0 i, j = 1, ..., N ,
which would be suggested by the results in [BT] (or by local existence consider-
ations), do not even make sense when the Hamiltonians Hi are merely Lipschitz
continuous. Let us remark that, on one hand, Lipschitz continuity is a typical
hypothesis (for instance when the Hamiltonians are connected with minimum
problems). On the other hand, an approach based on regularizations of the Hi
seems to be not compatible with the arguments exploited in [BT].
In order to overcome this drawback we propose a method that consists in
the interpretation of the existence question in terms of commutativity of the
minimum problems —see [R]— that lie behind the Hamiltonians involved in
(1). Moreover, we prove a sufficient condition for such commutativity relying
on the notion of Lie bracket for nonsmooth vector-fields introduced in [RS].
References
[BT] G. Barles and A. Tourin, Commutation properties of semigroups for first–
order Hamilton–Jacobi equations and application to multi–time equations.
Indiana Univ. Math. Jour., 50, 2001, 1523–1542.
[MR] M.Motta and F. Rampazzo Nonsmooth multi-time Hamilton-Jacobi sys-
tems , Preprint, 2004, Dipartimento di Matematica Pura e Applicata dell’
Università di Padova.
[R] F. Rampazzo, Commutativity of control vector fields and inf-commutativity,
Preprint, 2004, Dipartimento di Matematica Pura e Applicata dell’ Uni-
versità di Padova.
[RS] F. Rampazzo and H.J. Sussmann, Set-valued differentials and a nonsmooth
version of Chow’s theorem Proceedings of the 40th IEEE Conference on
Decision and Control; Orlando, Florida, December 4 to 7, 2001 (IEEE
Publications, New York, 2001), Volume 3, pp. 2613-2618.
1
Converted from rampazzo05.pdf (PDF).