Under usual assumptions on the Hamiltonian, we prove that any viscosity solution of the corresponding Hamilton-Jacobi equation on the manifold M is locally semiconcave and C loc outside the closure of its singular set (which is nowhere dense in M). Moreover, we prove that, under additional assumptions and in low dimension, any viscosity solution of that Hamilton-Jacobi equation satisfies a gene...

For the numerical solution of complex eigenvalue problems, arising with gyrotropic materials in resonators, the Jacobi-Davidson method is considered. In this paper the correction equation, which has to be solved within the Jacobi-Davidson method, is simplified and several preconditioning strategies, including also a multigrid scheme, are compared for the approximate solution of this correction ...

The paper gives the numerical stencil for the two-dimensional convection diffusion equation and the technique of elimination, and builds up the new iterative scheme to solve the implicit difference equation. The scheme’s convergence and its higher rate of convergence than the Jacobi iteration are proved. And the numerical example indicates that the new scheme has the same parallelism and a high...

We consider a class of ergodic Hamilton-Jacobi-Bellman (HJB) equations, related to large time asymptotics of non-smooth multiplicative functional of diffusion processes. Under suitable ergodicity assumptions on the underlying diffusion, we show existence of these asymptotics, and that they solve the related HJB equation in the viscosity sense.

Starting from the tri-Hamiltonian formulation of the Lagrange top in a six-dimensional phase space, we discuss the reduction of the vector field and of the Poisson tensors. We show explicitly that, after the reduction on each one of the symplectic leaves, the vector field of the Lagrange top is separable in the sense of Hamilton–Jacobi.

This paper studies the H 1 control problem for a nonlinear, unbounded, innnite dimensional system with state constraints. We characterize the solvability of the problem by means of a Hamilton-Jacobi-Isaacs equation, proving that the H 1 problem can be solved if and only if the HJI equation has a nonnegative viscosity supersolution, vanishing at the origin.

This paper gives comparison principles for first-order PDEs of the Hamilton-JacobiBellman type that arise in the problem of filtering under unknown disturbances with set-membership bounds on the uncertainty. The exact solutions of this problem, given in set-theoretic terms as “information sets,” are expressed as level sets to the solutions of some specific types of the HJB equation. But these s...

We consider a (microscopic) car-following model in traffic flow that can be seen as a semidiscrete scheme (discretization in space only) of a (macroscopic) Hamilton-Jacobi equation. For this discrete model, and for general velocity laws satisfying a strict chord inequality, we construct travelling solutions that are naturally associated to ”travelling shocks” for the conservation law derived fr...