Abstract. In this article we consider the Hamilton-Jacobi-Bellman (HJB) equation associated to the optimization problem with monotone controls. The problem deals with the infinite horizon case and costs with update coefficients. We study the numerical solution through the discretization in time by finite differences. Without the classical semiconcavity-like assumptions, we prove that the convergence in this problem is of order h γ in contrast with the order h γ 2 valid for general control problems. This difference arises from the simple and precise way the monotone controls can be approximated. We illustrate the result on a simple example.
In this article we study a finite horizon optimal control problem with monotone controls. We consider the associated Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function. We consider the totally discretized problem by using the finite element method to approximate the state space Ω. The obtained problem is equivalent to the resolution of a finite sequence of stopping-time problems. The convergence orders of these approximations are proved, which are in general (h + k √ h) γ where γ is the Hölder constant of the value function u. A special election of the relations between the parameters h and k allows to obtain a convergence of order k 2 3 γ , which is valid without semiconcavity hypotheses over the problem's data. We show also some numerical implementations in an example.
In this article we study a finite horizon optimal control problem with monotone controls. We consider the associated Hamilton-Jacobi-Bellman (HJB) equation which characterizes the value function.We consider the totally discretized problem by using the finite element method to approximate the state space Ω. The obtained problem is equivalent to the resolution of a finite sequence of stopping-time problems.The convergence orders of these approximations are proved, which are in general (h + k √ h ) γ where γ is the Hölder constant of the value function u. A special election of the relations between the parameters h and k allows to obtain a convergence of order k 2 3 γ , which is valid without semiconcavity hypotheses over the problem's data. We show also some numerical implementations in an example.
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