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Pursuit-Evasion Games with Multi-Pursuer:a decomposition approach.
Adriano Festa
(join work with Richard B. Vinter)
EEE Department, IC London.
27th November 2012
G. Castelnuovo, Sapienza Universita di Roma
Festa-Vinter Multi-Pursuer Differential Games
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Outline of the talk
1 Starting example: Surge Tank Control
2 Pursuit-Evasion games
3 A decomposition technique vs high dimensionality
4 Numerical Tests
5 Concluding Remarks
Festa-Vinter Multi-Pursuer Differential Games
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The Surge Tank Control Problem
Falugi, Kountouriotis, Vinter. Differential Games Controllers ThatConfine a System to a Safe Region in the State Space, WithApplications to Surge Tank Control. IEEE Trans. Automat. Contr.57(11): 2778-2788 (2012)
Festa-Vinter Multi-Pursuer Differential Games
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The Surge Tank Control Problem
dx(t)dt
= f (x(t),a(t)) + σ(x(t))b(t)
=
[0 10 0
]x(t) +
[01
](−a(t) + b(t)) .
The constraints that the surge tank must neither overflow orempty are expressible (in normalized units) as
−1 < x1(t) < +1 .
so x ∈ Ω := (−1,1)× R.Permitted tolerances on the Max Rate of Change of Outflow(MROC) index are captured by the additional constraint on theoutflow:
−1 ≤ a(t) ≤ +1 .
Festa-Vinter Multi-Pursuer Differential Games
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Surge Tank as a Differential Game
A := {a(.) : [0,∞)→ R | a(t) ∈ [−1,1]} .B := {b(.) : [0,∞)→ R} .
The space Φ of closed loop controls for the a player is
Φ := {non-anticipative mappings φ(.) : B → A} .
The Differential game is: find
v(x) = supφ∈Φ
infb∈B
J (x , φ(b(.)),a(.))
where the payoff function is
J(x ,a,b) :=∫ τx
0
(12|b(t)|2 + θ
)dt .
with θ ≥ 0 (design parameter) and τx first exit time from Ω.
Festa-Vinter Multi-Pursuer Differential Games
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Safe Region for the Surge Tank
Festa-Vinter Multi-Pursuer Differential Games
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Safe Region for the Surge Tank
Festa-Vinter Multi-Pursuer Differential Games
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Safe Region for the Surge Tank
Festa-Vinter Multi-Pursuer Differential Games
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Associated Optimal Control Problems
The set Ω = {x | − 1 < h(x) < +1} can be represented
Ω = Ω1 ∩ Ω2where
Ω1 = {x |h(x) < +1}, Ω2 = {x | − 1 < h(x)}
For every x ∈ Ωj , j = 1,2, consider the optimal control problem
(P jx )
vj(x) = infb∈B
∫ τx0 (
12 |b(t)|
2 + θ)dtẏ(t) = f (y(t),aj) + σ(y(t))b(t) a.e. on [0, τx )y(0) = x , hj(y(τx )) = 0 .
aj is frozen at value of a, driving away from boundary of Ωj
v(x) = min{v1(x), v2(x)} ∀x ∈ Ω
Festa-Vinter Multi-Pursuer Differential Games
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Comparison with min-max methods
Figure: Value function using a minmax technique (left) anddecomposition procedure (right), θ = 10.
Festa-Vinter Multi-Pursuer Differential Games
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Pursuit-Evasion games
The dynamic system is modelled as{y ′(t) = −g(y(t))a(t) + h(y(t))b(t) + l(y(t))y(0) = x
(1)
where y(t) ∈ Ω ⊂ RN is the state, and a and b are the controls.
We assumeg : Ω→ RNh : Ω→ RN are continuous
A,B are compact metric spaces(2)
and, for some constant L,
|g(x)− g(y)|+ |h(x)− h(y)| ≤ L|x − y | ∀x , y ∈ RN . (3)
Festa-Vinter Multi-Pursuer Differential Games
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Pursuit-Evasion games
We take as admissible control
A := {a : [0,+∞[→ A measurable } (4)
B := {b : [0,+∞[→ B measurable } (5)
and we consider only a ∈ A, b ∈ B.We are given a closed set T ⊆ RN , and define
tx (a,b) :={
min{t : yx (t ; a,b) ∈ T }+∞ if yx (t ; a,b) /∈ T ∀t .
(6)
the first player ”a” wants to minimize the time of hitting, and thesecond player ”b” wants to maximize the same cost.
Festa-Vinter Multi-Pursuer Differential Games
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PE games
We re-normalize these costs by the nonlinear transformation
ψ(u) :={
1− e−u if u < +∞1 if u = +∞. (7)
and consider the discounted cost functional
J(x ,a,b) = ψ(tx (a,b)) =∫ tx
0e−sds. (8)
Festa-Vinter Multi-Pursuer Differential Games
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PE games
We need the notion of nonanticipating strategy; for the firstplayer is
Γ := {α : B → A : t > 0,b(s) = b̃(s) for all s ≤ timplies α[b](s) = α[b̃](s) for all s ≤ t} (9)
for the second player is defined in the analogous way is ∆.
The lower and the upper values for the game are
v(x) := supβ∈∆
infa∈A
J(x ,a, β[a]) = infα∈Γ
supb∈B
J(α[b],b) (10)
the fact that lower and upper value are coincident is due to thenature of the dynamics and the cost functional.
Festa-Vinter Multi-Pursuer Differential Games
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PE games
the value function v(x) is the viscosity solution of the followingHamilton-Jacobi-Isaacs equation{
v(x) + H(x ,Dv(x)) = 0 x ∈ Ω \ Tv(x) = 0 x ∈ ∂T (11)
where
H(x ,p) := maxa∈A
minb∈B{−(g(x)a− h(x)b + l(x)) · p} − 1
= minb∈B
maxa∈A{−(g(x)a− h(x)b + l(x)) · p} − 1. (12)
Festa-Vinter Multi-Pursuer Differential Games
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Why the value function?
Solving this equation, and therefore getting the value functionof the game, we can get the optimal behavior for every playerfrom the starting point x0 as
a(t) = S(yx0(t))S(z) ∈ argmaxa∈A minb∈B
{−(−g(x)a + h(x)b + l(x)) · Dv(x)}
(13)b(t) = W (yx0(t))W (z) ∈ argminb∈B maxa∈A
{−(−g(x)a + h(x)b + l(x)) · Dv(x)}.
(14)
Festa-Vinter Multi-Pursuer Differential Games
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Example 1
We consider the pursuit-evasion game with two pursuers p1,p2and one evader e where all the agents are free to move in the1D space with various velocities.
p′1 =23a1
p′2 = a2e′ = b2p1(0) = p01p2(0) = p02e(0) = e0
(15)
where a1,a2,b ∈ B(0,1) = [−1,1], p1,p1,e ∈ R.
Capture happens when mini∈{1,2} |pi − e| ≤ r with somer ≥ 0.We underline that this problem is in a space of dimensionthree.
Festa-Vinter Multi-Pursuer Differential Games
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Example 1- reduced dynamics
We get the reduced dynamicsy ′1 = −
23a1 +
b2
y ′2 = −a2 +b2
y1(0) = p01 − e0
y2(0) = p02 − e0
(16)
here we have a1,a2,b ∈ B(0,1), y1, y2 ∈ [0,+∞] and
T := {(y1, y2) ∈ R2 : mini∈{1,2}
|yi | ≤ r}. (17)
The HJI equation associated to the problem isv(x) + max
a1,a2∈Aminb∈B
{−(−23a1 +
b2 ,−a2 +
b2 ) · Dv(x)
}= 1
x ∈ [0,+∞]2 \ Tv(0) = 0 x ∈ ∂T
(18)Festa-Vinter Multi-Pursuer Differential Games
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A decomposition technique vs hightdimensionality
I need some additional Hypotheses:
We assumeA = Bn(0, ρa1)× Bn(0, ρa2)× ...× Bn(0, ρaM ),B = [Bn(0, ρb)]m,g(x)ρa − h(x)ρb − |l(x)| > 0, ∀x ∈ Ω.
(19)
where with [Bn(0, ρb)]m we mean the space
{(b1,b2, ...,bn︸ ︷︷ ︸,b1, ...,bn, ...,b1, ...,bn)︸ ︷︷ ︸m times
∈ RN : |(b1, ...,bn)| = ρb}
Festa-Vinter Multi-Pursuer Differential Games
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Under these Hypotheses we can show
PropositionH(x ,p) is convex with respect to the variable p.
H(x ,p) = g(x) maxa∈A{a · p} − h(x) max
b∈B{−b · p} − l(x) · p − 1
Figure: in this case N = 2,m = 2,n = 1.
Festa-Vinter Multi-Pursuer Differential Games
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Decomposition
We consider the following class of problems, withi ∈ I := {1, ...m} ⊂ N,{
u(xi) + H (x ,Dui(x)) = 0 x ∈ Ω \ Tiui(x) = 0 x ∈ Ti
(20)
where T := ∪iTi .
Theorem
Let assume the standard Hypotheses and (19).Then we have that
u(x) := min{ui ; i ∈ I} (21)
is the unique value function of the P-E game.
Festa-Vinter Multi-Pursuer Differential Games
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Useful? Example 1
T := T1 ∪ T2 with Ti := {(x1, x2) ∈ R2 : |yi | ≤ r};
the second equation of the dynamics doesn’t affect thedecomposed problem.
v(x1, x2) + min
a1∈Amaxb∈B
{−(23a1 −
b2 ) ·
∂∂x1
v(x1, x2)}
= 1
x1 ∈ (r ,+∞]v(0) = 0 x2 ∈ [0, r ]∂∂x2
v(x1, x2) = 0(22)
this is rather easy to solve. We get
v(x1, x2) = re−6(x1−r)−e−6(x1−r)+1 and u(x1, x2) = −6(x − r)(1− r)(23)
Festa-Vinter Multi-Pursuer Differential Games
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Useful? Example 1
in the same way if we consider T2
v(x1, x2) = re−2(x1−r)−e−2(x1−r)+1 and u(x1, x2) = −2(x − r)(1− r).(24)
the Hypotheses of the Theorem are satisfied:g(x)ρa − h(x)ρb =
(23 −
12 ,1−
12
)= (16 ,
12) > 0
u(x1, x2) ={−6(x − r)(1− r) if x1 ≤ 13x2 +
23 r
−2(x2 − r)(1− r) if x1 > 13x2 +23 r .
(25)
Festa-Vinter Multi-Pursuer Differential Games
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Example 1 - value function
In this way, solving two simpler problems (of dimention 1).We get the solution of the original one.
Festa-Vinter Multi-Pursuer Differential Games
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Sketches of the proof I
DefinitionWe say that v : Ω→ R is semiconcave on the open convex setω if there exists a constant C ≥ 0 such that
λv(x) + (1− λ)v(y) ≤ v (λx + (1− λ)y) + 12
Cλ(1− λ)|x − y |2
(26)for all x , y ∈ Ω and λ ∈ [0,1].
We can show that in our case, every solution of a decomposedproblem is semicancave.Moreover we have the following property of s.c. functions. Said
D∗v(x) ={
p ∈ RN : p = limn→+∞
Dv(xn), xn → x}
we have D+v(x) = coD∗v(x).Festa-Vinter Multi-Pursuer Differential Games
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Sketches of the proof II
We want to show that u = min{ui ; i ∈ I} is the viscosity solutionof the HJI equation associated to the PE game.
The minimum of a family of supersolution is alwayssupersolution.
We prove that u is subsolution, too.We know now that the propriety of semiconcavity is preservedtaking the minimum of a class of semiconcave functions, so
D+u(x) = coD∗u(x) ⊆ co {D∗ui(x)|i ∈ I}⊆ co {coD∗ui(x)|i ∈ I} = co
{D+ui(x)|i ∈ I
}(27)
Festa-Vinter Multi-Pursuer Differential Games
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Sketches of the proof III
This implies, that a p ∈ D+u(x) and Λ = {λi , i ∈ Is.t .∑
i λi = 1}
p = (λ1, λ2...)·(p1,p2, ...) =∑
i
λipi (λ1, λ2...) ∈ Λ,pi ∈ D+ui(x)
(28)we know that for every pi ∈ D+ui(x) ,H(x ,pi) ≤ 0 for everyi ∈ I.
H(x ,p) = H
(x ,∑
i
λipi
)≤∑
i
λiH(x ,pi) ≤ 0. (29)
This shows that u(x) is a subsolution and conclude the proof.
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Decomposition technique for PE games
TheoremWe call I := {1,2, ...,m} and T := {yi ∈ Rn : min |yi | ≤ r}.Said vi : RN × I → R, xi = (xni , ..., xn(i+1)−1),
vi(xi) + max
ai∈Aminb∈B
{−fi(xi ,ai ,b) · ∂∂xi vi(xi)
}= 1
xi ∈ [0,+∞]n \ Tivi(xi) = 0 xi ∈ Ti∂∂x vi(x) = 0
(30)with fi(xi ,ai ,b) = −g(xi)ai(t) + h(xi)b(t) + l(xi).We have that the value function of the PE game is
u(x) := log(
1−mini∈I
vi(x)).
Festa-Vinter Multi-Pursuer Differential Games
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Tag-Chase: example 1
(case1 c = 0.95)
Festa-Vinter Multi-Pursuer Differential Games
mov1.mpgMedia File (video/mpeg)
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Tag-Chase: example 2
(case2 c = 0.95)
Festa-Vinter Multi-Pursuer Differential Games
mov2.mpgMedia File (video/mpeg)
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Tag-Chase: example 3
(case3 c = 0.95)
Festa-Vinter Multi-Pursuer Differential Games
mov3.mpgMedia File (video/mpeg)
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Remarks
The key idea to preserve convexity of the Hamiltonian is:
there is a “specular” behavior of the two players
the player who minimize “dominates” the other one
There is a generalization of this result, but until now, we needthe semiconcavity of the decomposed problems.(it’s a strong and inconvenient assumption)
Thank you.
Festa-Vinter Multi-Pursuer Differential Games
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Remarks
The key idea to preserve convexity of the Hamiltonian is:
there is a “specular” behavior of the two players
the player who minimize “dominates” the other one
There is a generalization of this result, but until now, we needthe semiconcavity of the decomposed problems.(it’s a strong and inconvenient assumption)
Thank you.
Festa-Vinter Multi-Pursuer Differential Games
-
Remarks
The key idea to preserve convexity of the Hamiltonian is:
there is a “specular” behavior of the two players
the player who minimize “dominates” the other one
There is a generalization of this result, but until now, we needthe semiconcavity of the decomposed problems.(it’s a strong and inconvenient assumption)
Thank you.
Festa-Vinter Multi-Pursuer Differential Games