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An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

Adrian Petruşel, Jen-Chih Yao (2009)

Open Mathematics

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.

Analysis of Hamilton-Jacobi-Bellman equations arising in stochastic singular control

Ryan Hynd (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We study the partial differential equation         max{Lu − f, H(Du)} = 0 where u is the unknown function, L is a second-order elliptic operator, f is a given smooth function and H is a convex function. This is a model equation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. We establish the existence of a unique viscosity solution of the Dirichlet problem that has a Hölder continuous gradient. We also show that if H is uniformly convex, the gradient of this solution...

Approximation of control problems involving ordinary and impulsive controls

Fabio Camilli, Maurizio Falcone (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we study an approximation scheme for a class of control problems involving an ordinary control v, an impulsive control u and its derivative u ˙ . Adopting a space-time reparametrization of the problem which adds one variable to the state space we overcome some difficulties connected to the presence of u ˙ . We construct an approximation scheme for that augmented system, prove that it converges to the value function of the augmented problem and establish an error estimates in L∞ for this approximation....

Approximation of solutions of Hamilton-Jacobi equations on the Heisenberg group

Yves Achdou, Italo Capuzzo-Dolcetta (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like h where h is the mesh step. Such...

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