  In this paper we present a new algorithm for the solution of
Hamilton-Jacobi-Bellman equations related to optimal control problems. The key
idea is to divide the domain of computation into subdomains which are shaped by
the optimal dynamics of the underlying control problem. This can result in a
rather complex geometrical subdivision, but it has the advantage that every
subdomain is invariant with respect to the optimal dynamics, and then the
solution can be computed independently in each subdomain. The features of this
dynamics-dependent domain decomposition can be exploited to speed up the
computation and for an efficient parallelization, since the classical
transmission conditions at the boundaries of the subdomains can be avoided. For
their properties, the subdomains are patches in the sense introduced by Ancona
and Bressan [ESAIM Control Optim. Calc. Var., 4 (1999), pp. 445-471]. Several
examples in two and three dimensions illustrate the properties of the new
method.
