A Fully-Discrete Scheme for the Value
Function of Pursuit-Evasion Games with
State Constraints
E. Cristiani, M. Falcone
Dip. di Matematica
Università ”La Sapienza” - Roma
28 novembre 2006
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 1/28
Outline
Pursuit-Evasion games
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
vhk → vh
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
vhk → vh
vh → v ([Bardi, Koike, Soravia, 2000])
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
vhk → vh
vh → v ([Bardi, Koike, Soravia, 2000])
Capturability in Tag-Chase game if VP = VE
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
vhk → vh
vh → v ([Bardi, Koike, Soravia, 2000])
Capturability in Tag-Chase game if VP = VE
Some hints for the implementation
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
vhk → vh
vh → v ([Bardi, Koike, Soravia, 2000])
Capturability in Tag-Chase game if VP = VE
Some hints for the implementation
Interpolation in high dimension
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
vhk → vh
vh → v ([Bardi, Koike, Soravia, 2000])
Capturability in Tag-Chase game if VP = VE
Some hints for the implementation
Interpolation in high dimension
Reducing the size of the problem
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Outline
Pursuit-Evasion games
Convergence of the fully-discrete scheme with SC
vhk → vh
vh → v ([Bardi, Koike, Soravia, 2000])
Capturability in Tag-Chase game if VP = VE
Some hints for the implementation
Interpolation in high dimension
Reducing the size of the problem
Numerical tests
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 2/28
Pursuit-Evasion games
f (x, a, b) = f (xP , xE , a, b) =
µ
fP (xP , a)
fE (xE , b)
y = (y P , y E ) ,
¶
,
f P , f E ∈ Rn
x = (xP , xE )
v + min{−∇xE v · fE (xE , b)}+
b∈B
max{−∇xP v · fP (xP , a)} − 1 = 0
a∈A
x ∈ Ω\T
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 3/28
Time-discrete scheme for P-E games
with State Constraints
Ω = Ω1 × Ω 2 ,
©
y P ∈ Ω1 ,
y E ∈ Ω2
ª
Ah (x) := a ∈ A : xP + hfP (xP , a) ∈ Ω1 ,
ª
©
Bh (x) := b ∈ B : xE + hfE (xE , b) ∈ Ω2 ,

 vh (x) = max
x∈Ω
x ∈ Ω.
min {βvh (x + hf (x, a, b))} + 1 − β x ∈ Ω\T
b∈Bh (x) a∈Ah (x)

vh (x) = 0
x∈T
β = e−h .
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 4/28
Fully-discrete scheme
We build a regular triangulation of Ω denoting by X the set
of its nodes xi , i = 1, . . . , N and by S the set of simplices
Sj , j = 1, . . . , L. V (S) will denote the set of the vertices of
a simplex S and the space discretization step will be
denoted by k where k := maxj {diam(Sj )}.

ª
© k
k

vh (xi ) = max
min
βvh (xi + hf (xi , a, b)) + 1 − β



b∈B
(x
)
a∈A
(x
)
i
i
h
h


 k
vh (xi ) = 0






 v k (x) = P λ (x)v k (x ) , 0 ≤ λ (x) ≤ 1 , P λ (x) = 1
j
h
h j
j j
j j
xi ∈ (X\T )
xi ∈ T ∩ X
x∈Ω
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 5/28
Reachable set
R0 := T
Rn :=
(
x ∈ Ω\
n−1
[
Rj : for all bx ∈ Bh (x) there exists
j=0
âx (bx ) ∈ Ah (x) such that x+hf (x, âx (bx ), bx ) ∈ Rn−1
)
,
n ≥ 1.
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 6/28
Main theorem
Theorem Let Ω an open bounded set. Let f be continuous
and Lipschitz continuous w.r. to x. Assume Ω = ∪∞
j=0 Rj .
Moreover assume that minx,a,b |f (x, a, b| ≥ f0 > 0 and
0 < k ≤ f0 h. Then, for n ≥ 1
n
n
S
S
a) vh (x) ≤ vh (y) , for any x ∈
Rj , for any y ∈ Ω\
Rj ;
j=0
b) vh (x) = 1 − e−nh ,
c)
vhk (x)
=1−e
−nh
for any x
+ O(k)
n
P
j=0
∈ Rn ;
e−jh
for any x
∈ Rn ;
j=0
d) There exists a constant C > 0 such that
Ck
|vh (x) − vhk (x)| ≤ 1−e
for any x ∈ Rn .
−h ,
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 7/28
Convergence for P-E games
In order to obtain the convergence of vhk to the value
function v we can couple our result with that in [Bardi,
Koike, Soravia, 2000]. In addition to our and their
hypotheses, we have to assume that
fP (xP , A(xP )) and fE (xE , B(xE ))
are convex sets.
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 8/28
Convergence for differential games
We can generalize our convergence result to any kind of
differential games. To do this, we have to modify the
definition of admissible sets properly.
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 9/28
Tag-Chase game
We consider two boys P and E which run one after the
other in the same 2-dimensional domain, so that the game
2
is set in Ω = Ω1 ⊂ R4 where Ω1 is an open bounded set of
R2 . We denote by (xP , xE ) the coordinates of Ω where
xP , xE ∈ Ω1 . P and E can run in every direction with
velocity VP and VE respectively.
½
ẋP = VP a
a ∈ B2 (0, 1)
ẋE = VE b
b ∈ B2 (0, 1)
The case VP > VE was completely studied by Alziary de
Roquefort.
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 10/28
Capturability in Tag-Chase game
Proposition
Let the target be
T = {(xP , xE ) ∈ R2n : d(xP , xE ) ≤ ε} ,
ε ≥ 0.
and Ω1 an open bounded set. Then,
If VP > VE then the capture time
tc = T (xP , xE ) = − ln(1 − v(xP , xE )) is finite and
bounded by
|xP − xE |
tc ≤
.
VP − V E
If VP = VE , ε 6= 0 and Ω1 is convex then the capture
time tc is finite.
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 11/28
Interpolation in high dimension
P43
P23
1
P1
P83
P6 3
P22
x
P12
P21
P32
P33
P13
P42
P73
P53
x3
x2
x1
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 12/28
Error estimate
Theorem Let Qn := [a1 , b1 ] × . . . × [an , bn ] ⊂ Rn and
x = (x1 , . . . , xn ).
Assume f ∈ C 2 (Qn ; R) and let q(x), x ∈ Qn be the
approximate value of f (x) obtained by the n-dimensional
linear interpolation described above.
Then, the error E(x) := f (x) − q(x) is bounded by
|E(x)| ≤
n
X
∆2
i
i=1
where Mi =
∂ 2 f (x)
max | ∂x2 |
i
x∈Q
8
Mi ,
for all x
∈ Qn
and ∆i = bi − ai .
n
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 13/28
Reducing the size of the problem
As already noted by Pesch and Alziary de Roquefort, due
to the state constraints it seems not possible to use
reduced coordinates or a similar approach. In fact, using
reduced coordinates we loose every information about the
real positions of the two players, so that we can not detect
when they touch the boundary of the domain (and then
change the dynamics accordingly).
Nevertheless, we can make use of the symmetries of the
problem, if any.
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 14/28
Example in 1D
P
E
E
2
1.5
1
0.5
xE
P
Unidimensional Tag-Chase game. VP = 2, VE = 1.
Target
0
−0.5
−1
−1.5
−1.5
−1
−0.5
0
xP
0.5
1
1.5
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 15/28
Example in 2D
P
P
E
E
E
E
P
P
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 16/28
Test 1, VP > VE
ε = 10−3 , VP = 2, VE = 1, n = 50, nc = 48 + 1. Convergence
was reached in 85 iterations. The CPU time (IBM - 8 procs)
was 17h 36m 16s, the wallclock time was 2h 47m 37s.
2
1.5
1.5
1
1
0.5
0
0.5
−0.5
0
2
−1
1
2
1
0
−1.5
0
−1
−2
−1
−2
−2
−2
−1
0
1
2
Value function T (0, 0, xE , yE ).
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 17/28
Test 1, VP > VE
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
−1
0
1
2
−1
0
1
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 18/28
Test 2, VP > VE
architecture
IBM serial
IBM 2 procs
IBM 4 procs
IBM 8 procs
PC dual core, ser
PC dual core, par
wallclock time speed-up efficiency
26m 47s
14m 19s
1.87
0.93
8m 09s
3.29
0.82
4m 09s
6.45
0.81
1h 08m 44s
34m 51s
1.97
0.99
Tser
speed-up :=
Tpar
efficiency
:=
speed-up
np
.
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 19/28
Test 3, VP > VE
In this test the domain has a square hole in the center. The
side of the square is 1.06. ε = 10−4 , VP = 2, VE = 1,
n = 50, nc = 48 + 1. Convergence: 109 iterations. CPU
time: 1d 00h 34m 18s, wallclock time: 3h 54m 30s.
3
2.5
2
1.5
1
0.5
0
2
1
0
−1
−2
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Value function T (−1.5, −1.5, xE , yE ).
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 20/28
Test 3, VP > VE
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
−1
0
1
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 21/28
Test 4, VP > VE
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
−1
0
1
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 22/28
Test 5, VP = VE
v is discontinuous on ∂T . No convergence results but
v < 1 by our proposition.
ε = 10−3 , VP = 1, VE = 1, n = 50, nc = 36. Convergence
was reached in 66 iterations.
3.5
3
2.5
2
1.5
1
0.5
0
2
1
0
−1
−2
−1
0
1
2
Value function T (0, 0, xE , yE ).
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 23/28
Test 5, VP = VE
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
−1
0
1
2
−1
0
1
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 24/28
Test 6, VP = VE
The domain has a circular hole in the center. Non-convex
domain, then no guarantee capture occurs. v is equal to 1
in a large part of the domain. Strange behavior of some
optimal trajectories.
ε = 10−4 , VP = 1, VE = 1, n = 50, nc = 48 + 1.
Convergence: 94 iterations. CPU time: 1d 12h 05m 22s.
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2
−1
0
1
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 25/28
Test 7, VP < VE
v is discontinuous on ∂T . No guarantee capture occurs. v
is equal to 1 in a large part of the domain. ε = 10−3 , VP = 1,
VE = 1.25, n = 50, nc = 48 + 1. Convergence: 53 iterations.
CPU time: 12h 43m 02s, wallclock time: 2h 18h 06s.
2
1.5
14
1
12
0.5
10
8
0
6
−0.5
4
−1
2
0
−2
−1
0
1
2
−2
−1
0
1
2
−1.5
−2
−2
−1
0
1
2
Value function T (−1, −1, xE , yE ).
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 26/28
Test 7, VP < VE
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
−1
0
1
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 27/28
Test 8, VP < VE
ε = 10−4 , VP = 1, VE = 1.5, n = 50, nc = 36. Convergence:
65 iterations. CPU time: 15h 48m 46s, wallclock time: 2h
30m 19s.
2
2
1.5
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−1
0
1
2
−2
−2
−1
0
1
2
A Fully-Discrete Scheme for Differential Games with SC, Roma 2006 – p. 28/28