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).