_{1}

^{*}

In this paper we propose a relaxation scheme for solving discrete HJB equations based on scheme II [1] of Lions and Mercier. The convergence of the new scheme has been established. Numerical example shows that the scheme is efficient.

Consider the following Hamilton-Jacobi-Bellman (HJB) equation:

where

Equation (1.1) can be discretized by finite difference method or finite element method. See [

where

[

Scheme I.

Step 1: Given

Step 2: Let

Step 3: If

Assume

That is: the lth row of matrix

Scheme II.

Step 1:

Step 2: For

Step 3: Compute

Step 4: If

In the last decade many numerical schemes have been given for solving (1.2). But the above schemes are still playing a very important role. See [

In this paper we propose, based on Scheme II above, a relaxation scheme with a parameter

We propose a new scheme which is an extension of Scheme II.

New Scheme II.

Step 1: Given

Step 2: For

Step 3: Compute

Step 4: Compute

Step 5: If

In [

Condition

In [

Theorem 2.1 If Condition

We have the following convergence theorem.

Theorem 2.2 Assume that Condition

Proof Since all

First, we prove

By (2.3) we have

which combining with (2.1) and (2.2) yields

Since

By (2.4) we obtain

By

and

which and (2.10) implies

Similarly, by (2.3) we derive

which combining with (2.2) and (2.6) implies

Hence we have

By (2.4), we have

By (2.12), (2.13) and

which combining with

By (2.11), (2.12) and (2.13) ,we get

which combining with (2.15) implies

It is easy to derive by induction that

and

It follows that (2.5) holds.

It follows from (2.2) and (2.3) that

Since the set

Therefore, we have

Then by (2.2) we obtain

which and (2.17) results in

From (2.4), (2.16) and (2.19) we have

It follows from (2.18), (2.19) and (2.20) that

which means

Finally, we prove the uniqueness of solution. Assume

It is easy to see from (2.21) and (2.22) that there exist

(2.23) and (2.26) implie

We use example 2 in [

where

The discretization of the above second order derivatives are:

where

We see that

We can see from

0.1 | 0.5 | 0.8 | 0.9 | 1.0 | |
---|---|---|---|---|---|

3.419e–004 | 2.099e–011 | 9.464e–012 | 6.861e–012 | 6.651e–012 | |

6.630e–003 | 1.784e–008 | 6.653e–011 | 6.062e–011 | 8.169e–006 |

1.1 | 1.3 | 1.5 | 1.8 | 1.9 | |
---|---|---|---|---|---|

3.440e–000 | 2.314e+001 | 4.670e+001 | 8.421e+001 | 9.730e–000 | |

1.667e–003 | 4.323e+001 | 1.754e+002 | 4.323e+001 | 2.089e+002 |

0.1 | 0.5 | 0.8 | 0.9 | 1.0 | |
---|---|---|---|---|---|

200 | 198 | 107 | 90 | 124 | |

600 | 495 | 282 | 258 | 400 |

0.1 | 0.5 | 0.8 | 0.9 | 1.0 | |
---|---|---|---|---|---|

1.091409800 | 1.086033962 | 1.082002083 | 1.080658123 | 1.079314164 | |

1.089751377 | 1.080022728 | 1.074891194 | 1.073533844 | 1.076283661 | |

1.088256293 | 1.075449958 | 1.072050161 | 1.071072814 | 1.073086733 | |

1.086758364 | 1.073060086 | 1.069451302 | 1.068586924 | 1.072407806 | |

Last | 1.065963994 | 1.065887109 | 1.065887109 | 1.065887109 | 1.065887109 |

0.1 | 0.5 | 0.8 | 0.9 | 1.0 | |
---|---|---|---|---|---|

1.077654026 | 1.073664734 | 1.070672766 | 1.069675443 | 1.068678121 | |

1.076493553 | 1.069008305 | 1.065427282 | 1.065027950 | 1.068036835 | |

1.075236529 | 1.065915940 | 1.063091196 | 1.062134520 | 1.066011200 | |

1.073996351 | 1.063479656 | 1.060857772 | 1.060476760 | 1.065563176 | |

Last | 1.054467308 | 1.054409847 | 1.054409847 | 1.054409847 | 1.054409847 |

This work was supported by Educational Commission of Guangdong Province, China (No. 2012LYM-0066) and the National Social Science Foundation of China (No. 14CJL016).