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A Semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with Dirichlet boundary conditions

Elisabetta Carlini, Athena Picarelli, Francisco J. Silva

Abstract

We study the numerical approximation of time-dependent, possibly degenerate, second-order Hamilton-Jacobi-Bellman equations in bounded domains with nonhomogeneous Dirichlet boundary conditions. It is well known that convergence towards the exact solution of the equation, considered here in the viscosity sense, holds if the scheme is monotone, consistent, and stable. While standard finite difference schemes are, in general, not monotone, the so-called semi-Lagrangian schemes are monotone by construction. On the other hand, these schemes make use of a wide stencil and, when the equation is set in a bounded domain, this typically causes an overstepping of the boundary and hence the loss of consistency. We propose here a semi-Lagrangian scheme defined on an unstructured mesh, with a suitable treatment at grid points near the boundary to preserve consistency, and show its convergence for problems where the viscosity solution can even be discontinuous. We illustrate the numerical convergence in several tests, including degenerate and first-order equations.

A Semi-Lagrangian scheme for Hamilton-Jacobi-Bellman equations with Dirichlet boundary conditions

Abstract

We study the numerical approximation of time-dependent, possibly degenerate, second-order Hamilton-Jacobi-Bellman equations in bounded domains with nonhomogeneous Dirichlet boundary conditions. It is well known that convergence towards the exact solution of the equation, considered here in the viscosity sense, holds if the scheme is monotone, consistent, and stable. While standard finite difference schemes are, in general, not monotone, the so-called semi-Lagrangian schemes are monotone by construction. On the other hand, these schemes make use of a wide stencil and, when the equation is set in a bounded domain, this typically causes an overstepping of the boundary and hence the loss of consistency. We propose here a semi-Lagrangian scheme defined on an unstructured mesh, with a suitable treatment at grid points near the boundary to preserve consistency, and show its convergence for problems where the viscosity solution can even be discontinuous. We illustrate the numerical convergence in several tests, including degenerate and first-order equations.

Paper Structure

This paper contains 14 sections, 12 theorems, 142 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on the parabolic HJB equation
  3. Revisiting the Milstein-Tretyakov scheme in the linear case
  4. The classical SL scheme in $[0,T]\times{\mathbb R}^d$
  5. A variation of the Milstein-Tretyakov scheme in $Q_T$
  6. A fully discrete linear SL scheme in $Q_T$
  7. The fully discrete scheme for the HJB equation
  8. Convergence in the case where the boundary conditions hold in the strong sense
  9. Convergence in the case where the boundary conditions hold in the weak sense
  10. Numerical tests
  11. Test 1: One-dimensional problem with a varying diffusion term
  12. Test 2: Two-dimensional problem on a circular domain with a smooth solution.
  13. Test 3: Two-dimensional problem on a square domain with a nonsmooth solution
  14. Test 4: Stochastic minimum time problem of exiting from a room with two doors

Key Result

Proposition 2.1

Assume that (H1) hold and that $\partial \Omega$ is of class $C^2$. Let $v_{1}\colon\overline{Q}_{T}\to{\mathbb R}$ be upper semicontinuous and let $v_{2}\colon\overline{Q}_{T}\to{\mathbb R}$ be lower semicontinuous. Suppose that $v_1$ and $v_2$ are, respectively, viscosity sub- and supersolutions t

Figures (6)

  • Figure 1: Numerical solutions $V_{\Delta t,\Delta x}(0,\cdot)$ of Test 1 computed with $\nu=1,\,0.1,\,0.01,\,0$. The abscissa and the ordinate represent, respectively, the space variable and the approximated value function at the initial time.
  • Figure 2: On the left, we display the numerical solution $V_{\Delta t,\Delta x}(0,\cdot)$ of Test 2 computed with ${\Delta x}=0.125$ and $\Delta t=\Delta x/2$. The $x_1x_2$ plane and the $x_3$-axis represent, respectively, the space variable and the approximated value function at the initial time. On the right, we present the projection of the numerical solution onto the $x_1x_2$ plane together with the computational mesh.
  • Figure 3: Numerical solution $V_{{\Delta t},{\Delta x}}$ of Test 3 at different times. (A) $t = 1.5$, (B) $t = 1.2$, (C) $t = 0.9$, (D) $t = 0.3$, (E) $t = 0.1$, (F) $t = 0$
  • Figure 4: Test 4: approximate viscosity solution at time $t=0$ with $\Delta t = \Delta x = \sqrt{2}/50$ (left). Contour lines on the $x_1x_2$-plane and simulations of approximate optimal trajectories (right).
  • Figure 5: Test 4: contour lines on the $x_1x_2$-plane and simulations of approximate optimal trajectories.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Example 2.1
  • Definition 2.3
  • Definition 2.4
  • ...and 20 more