Controlability of Stochastic Parabolic Equation with
Multiplicative Noise
Aurel R¼
aşcanu
"Alexandru Ioan Cuza" University and "Octav Mayer" Mathematics Institute of Romanian Academy,
ROMANIA
May 9-19, 2011
CIMPA-UNESCO Conference
Monastir, Tunisie
LECTURE 2
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Object
The lecture is concerned with the controllability of the stochastic linear heat
equation with multiplicative noise:
8
dt Y (t, x )
>
>
>
>
>
<
∆x Y (t, x ) dt + a (t, x ) Y (t, x ) dt = [f (t, x ) + 1D0 (x )u (t, x )] dt
+Y (t, x ) σ(t, x ) dW (t ),
>
Y (ω, t, x ) = 0 on Ω ]0, T [ ∂D ,
>
>
>
>
:
Y (ω, 0, x ) = y0 (x ) in Ω D ,
in Ω ]0, T [ D ,
D0 cl (D0 ) D Rd is a bounded open subsets with a boundaries of
class C 2 ;
(Ω, F , P, fFt gt 0 ) is a stochastic basis;
fW (t ) = ( β i (t ))k 1 : t 0g is a Rk -dimensional Wiener process;
assumeFt = FtW .
a 2 L∞ 0, T ; H 1 (D) \ L∞ ((0, T ) D) , σ 2 C 2 [0, T ] D; Rk ;
k
σ (t, x ) dW (t ) =
∑ σi (t, x ) d βi (t ) ;
i =1
y0 2
L2 (D)
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Denote
Q =]0, T [ D ,
Q0 =]0, T [ D0 ,
Σ =]0, T [ ∂D ;
Σ0 =]0, T [ ∂D0 ,
1 D0 (x ) =
(
1,
0,
if x 2 D0 ,
if x 2
/ D0
H = L2 (D) and V = H01 (D). We have
V
and
H
V
∆ : V ! V is the duality mapping:
( ∆v , v )V
,V
= kv k2V = k ∆v k2V
We also have
( ∆v , u )V
Aurel R¼
aşcanu
,V
= (rv , ru )H .
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
SDE: existence
Let the forward stochastic di¤erential equation (FSDE))
8
dt Y ∆x Y dt + a (ω, t, x ) Y dt
>
>
>
<
= F (ω, t, x ) dt + Y σ(ω, t, x ) dt W (ω, t ) ,
u
> Y (ω, t, x ) = 0 on Ω Σ,
>
>
: u
Y (ω, 0, x ) = y0 (x ) in Ω D ,
where
8
i)
>
>
<
ii )
>
>
:
iii )
F 2 L0ad (Ω; L2 (0, T ; H )),
∞ ( Ω ]0, T [ D) ,
a 2 Lad
in Ω
Q
(1)
∞ Ω ]0, T [ D ; Rk ,
σ 2 Lad
y0 2 H = L2 (D) .
Under these assumptions the SDE (1) has a unique solution
Y 2 L0ad (Ω; C ([0, T ]; H)) \ L0ad (Ω; L2 (]0, T [; V)).
such that for all v 2 V = H01 (D) :
(Y (t ) , v )H +
= (y0 , v )H +
Z t
0
Z t
0
[(rx Y (s ) , rx v )H + (a (s ) Y (s ) , v )H ] ds
(F (s ) , v )H ds +
Aurel R¼
aşcanu
k Z t
∑
i =1 0
(Y (s ) σi (s ), v )H d β i (s ) .
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Continuous dependence
Moreover, there exists a constant C independent of y0 , F , u, v , D0 such that
#
"
RT
2
2
E sup jY (t )jH + 0 kY (t )kV dt
t 2[0,T ]
jy0 j2H + E
and
E
"
sup jY u (t )
t 2[0,T ]
RT
Y v (t )j2H +
h
C jY u (0 )
0
jF (t )j2H dt + E
RT
0
kY u (t )
Y v (0)j2H + E
Aurel R¼
aşcanu
R
R
Q0
u 2 (t ) dtdx
Y v (t )k2V dt
Q 0 (u (t, x )
#
i
v (t, x ))2 dtdx ,
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
If
F 2 L2ad (Ω
D),and y0 2 V , then
]0, T [
Y (ω, t ) 2 H01 (D) \ H 2 (D) ,
a)
b)
c)
Y 2
∆ Y 2 L2 ( Ω
Z t
0
a.e.,
Q ).
and in this case for all t 2 [0, T ]:
Y (t, x )
(ω, t )
L2 (Ω; L∞ (0, T ; V )),
∆x Y (s, x ) ds +
= y0 (x ) +
Z t
0
Z t
a (s, x ) Y (s, x )ds
0
F (s, x ) ds +
Aurel R¼
aşcanu
Z t
0
Y (s, x ) σ (s, x ) dW (s )
a.e. in Ω
D
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
BSDE: existence
We consider also the backward stochastic di¤erential equation (BSDE)
8
dt p ∆x p dt + a (ω, t, x )p dt
>
>
<
= [G (ω, t, x )+σ(ω, t, x ) q ] dt q dt W (ω, t ) in Ω Q,
>
>
:
p (ω, t, x ) = 0 on Ω Σ, and p (ω, T , x ) = pT (ω, x ) in Ω D ,
where
G 2 L2ad (Ω; L2 (0, T ; H)),
a2
∞
Lad
(Ω ]0, T [ D) ,
pT 2 L2 (Ω, FT , P; H),
σ2
∞
Lad
Ω ]0, T [
and
D ; Rk
Under these assumptions the BSDE has a unique solution (p, q ) (see e.g.
Pardoux &Rascanu ’99)
p 2 L2ad (Ω; C ([0, T ]; H)) \ L2ad (Ω; L2 (]0, T [; V),
q2
and
E
"
and
L2ad (Ω; L2 (]0, T [; Hk ))
sup
t 2[0,T ]
jp (t ) j2H
+
Z T
0
kp
(s )k2V ds
+
C E jpT j2H +
Aurel R¼
aşcanu
#
Z T
jq
Z T
jG (t )j2H dt .
0
0
(s )j2Hk
ds
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
h
i
C E jpT j2H jFt ,
and if G = 0, then jp (t )j2H
if pT 2 L2 (Ω, FT , P; V), then
a.s.
on Ω.
p 2 L2 (Ω ]0, T [; H01 (D) \ H 2 (D)) \ L∞ (0, T ; L2 (Ω; H01 (D)).
We note that by Ito’s formula
(Y (t ) , p (t ))H = (y0 , p0 )H +
+
Z t
0
Z t
0
[(F (s ) , p (s ))H
(Y (s ) , G (s ))H ] ds
(Y (s ) , (q (s ) + p (s ) σ (s )) dW (s ))H
and
(Y (t ) , p (t ))H = E [(Y (T ) , p (T ))H jFt ]
E
Z T
t
[(F (s ) , p (s ))H
Aurel R¼
aşcanu
(Y (s ) , G (s ))H ] ds jFt
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
In particular, if we consider the SDE
8
dt Y (ω, t, x ) ∆x Y (ω, t, x )dt + a (ω, t, x )Y (ω, t, x )dt = f (ω, t, x ) dt +
>
>
>
>
>
<
+1D0 (x )u (ω, t, x )dt + Y (ω, t, x )σ (ω, t, x ) dt W (ω, t ) in Ω Q,
>
Y (ω, t, x ) = 0 on Ω Σ,
>
>
>
>
:
Y (ω, 0, x ) = y0 (x ) in Ω
D,
(2)
and the dual BSDE
8
dt p (ω, t, x ) ∆x p (ω, t, x )dt + a (ω, t, x )p (ω, t, x )dt
>
>
<
= [g (ω, t, x ) + σ(ω, t, x ) q (ω, t, x )] dt q (ω, t, x ) dt W (ω, t ) , in Ω Q,
>
>
:
p (ω, t, x ) = 0 on Ω Σ, and p (ω, T , x ) = pT (ω, x ) in Ω D ,
(3)
we have the duality formula
R
R
E Q 0 u (t, x ) p (t, x ) dxdt + E Q [f (t, x ) p (t, x ) g (t, x ) Y (t, x )] dtdx
R
R
= E D Y (T , x ) p (T , x ) dx
D Y (0, x ) p (0, x ) dx
(4)
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
A control problem
Let the controlled SDE of parabolic type
8
< dY (t ) + AY (t ) dt = f (t, Y (t ) , v (t )) dt + σ (t, Y (t ))dW (t ) ,
t 2 [0, T ]
:
Y 0 = y0 2 H
(0.2)
and the minimizing problem of the cost function
J (v ) = E
Z T
0
g (t, Y (t ) , v (t )) dt + Eψ (Y (T ))
(0.1)
Here
V
H
fW (t ) : t
V
0g is a H0 cilidrical Wiener process
A 2 L(V , V ),
A = A and there exists λ, λ0 > 0 such that
(Av , v )V
,V
+ λ jv j2
Aurel R¼
aşcanu
λ0 kv k2 ,
8v 2 V ,
(H1 )
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
ψ : H ! R+ is continuously Fréchet di¤erentiable and convex on H;
σ : [0, T ] H ! L2 (H0 , H) is measurable in the …rst variable and
continuously Fréchet di¤erentiable in the second variable.
f : [0, T ] H U ! H and g : [0, T ] H U ! R+ are measurable in
the …rst variable, continuously Fréchet di¤erentiable in the second one and
Gâteaux di¤erentiable in the last one.
9 L > 0 such that, for all t 2 [0, T ], and for all y , z, h 2 H, u, v , w 2 U:
i)
ii )
iii )
iv )
jf (t, 0, 0)j + kσ(t, 0)k2 L,
fy0 (t, y , u )h + σy0 (t, y )h 2 L jh j ,
jfu0 (t, y , u )w j + jgu0 (t, y , 0)w j Ljw j,
gy0 (t, y , u ) gy0 (t, z, u ) + jgu0 (t, y , u ) gu0 (t, z, u )jU
+ jψ0 (y ) ψ0 (z )j L jy z j ,
jgu0 (t, y , u ) gu0 (t, y , v )jU Lju v jU .
Aurel R¼
aşcanu
(H2 )
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Denote AW the set of couples (u, Y u ), with
u 2 L2ad (Ω ]0, T [; U ),
and
Y u 2 L2ad (Ω ]0, T [; V ) \ L2ad (Ω; C ([0, T ]; H ))
is the associated solution of (2).
Theorem
Let the assumptions (H1 ) and (H2 ) be satis…ed, y0 2 H and we suppose that
(u, Y u ) 2 AW is an optimal pair. Then there exists a adjoint-couple
h
i
(p, k ) 2 L2W (Ω; C ([0, T ]; H )) \ L2W (Ω ]0, T [; V )
Λ2W ,
such that
8
dY (t ) + AY (t ) dt = f (t, Y (t ) , u (t )) dt + σ (t, Y (t ))dW (t ) ,
>
>
>
>
>
>
dp (t ) + Ap (t ) dt = fy0 (t, Y (t ) , u (t )) p (t ) + σy0 (t, Y (t )) k (t ) dt
>
>
<
gy0 (t, Y (t ) , u (t )) dt k (t ) dW (t ) ,
>
>
>
>
>
Y0 (ω ) = y0 2 H, and pT = ψ0 (Y (T )) ,
>
>
>
: 0
fu (t, Y (t ) , u (t )) p (t ) = gu0 (t, Y (t ) , u (t )) .
(5)
If f and σ are linear in (Y , u ) and g and ψ are convex quadratic in (Y , u ) ,
then the condition (5) is su¢ cient too and the optimal pair (u, Y u ) is unique.
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Particular case of control problem
Let the state SDE
dt Y (t, x )
∆Y (t, x ) dt +a (t, x ) Y (t, x ) dt
= 1D0 (x )v (t, x ) dt + Y (t, x ) hσ (t, x ) , dW (t )i
Y (ω, t, x ) = 0
on Ω
Y (ω, 0, x ) = y0 (x )
on Ω ]0, T ]
Σ,
in Ω
D,
D,
and a minimizing problem for
J (v ) =
1
E
2
Z
Q
v 2 (t, x ) dtdx +
Aurel R¼
aşcanu
Z
1
E
Y 2 (T , x ) dx .
2ε
D
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Then necessary and su¢ cient condition for (uε , Yε ) to be an optimal pair is
8
dYε (t, x ) ∆Yε (t, x ) dt +a (t, x ) Yε (t, x ) dt = 1D0 (x )uε (t, x ) dt
>
>
>
>
>
>
+Yε (t, x ) σ (t, x ) dW (t ) in Ω Q
>
>
>
>
>
>
Yε (ω, t, x ) = 0 on Ω Σ,
>
>
>
>
>
>
>
Yε (ω, 0, x ) = y0 (x ) , in Ω D
>
>
>
<
dpε (t, x ) ∆pε (t, x ) dt + a (t, x ) pε (t, x ) = σ (t, x ) qε (t, x ) dt
>
>
>
>
>
>
>
>
>
>
pε (ω, t, x ) = 0 on Ω Σ,
>
>
>
>
>
1
>
>
pε (ω, T , x ) =
Yε (ω, T , x ) , in Ω D
>
>
>
ε
>
>
:
1D0 (x )pε (t, x ) = uε (t, x ) , in Ω Q.
Aurel R¼
aşcanu
qε (t, x ) dW (
in Ω
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
By (4) we have
1
E
2
Hence
E
Z
Z
Q
Z
1
E Y 2 (T , x ) dx
2ε D ε
Z
Z
1
1
= E uε (t, x ) pε (t, x ) dtdx
E pε (T , x ) Yε (T , x ) dx
2 Q0
2 D
1R
=
y0 (x ) pε (0, x ) dx .
2 D
uε2 (t, x ) dtdx +
Z
Z
Z
1
1
Yε2 (T , x ) dx = E
uε2 (t, x ) dtdx + E
Yε2 (T , x ) dx
pε2 (t, x ) dtdx + E
ε
ε
Q
D
D
Q0
2 jy0 jH jpε (0)jH
!!!!!!
and here we have need of Carleman’s estimates.
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
A Carleman estimate
Lemma
Let p 2 L2 (Q ) a solution of the backward parabolic equation
8
∂p
<
∆p + a1 p + b1 rp = g in Q = (0, T )
∂t
:
p = 0,
on Σ = (0, T ) ∂D
D
where a1 2 L∞ (Q ) , b1 2 L∞ Q; Rk and g 2 L2 (Q ) . Then there exists a
constant CT such that
jp (0)j2H
CT
Moreover
CT
Z
Q0
jp (t, x )j2 dtdx +
Z
Q
jg (t, x )j2 dtdx .
h
i
m
C ka1 km
L ∞ (Q ) + k b 1 k L ∞ (Q )
for some C > 0 and m 2 N independent of a1 , b1 and g .
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Reduction to a random di¤.eq.
Denote
ξ (t, x ) = exp [ σ (t, x ) W (t )]
and
Ŷ (t ) = Y (t ) ξ (t )
By Itô’s formula in H for Ŷ (t ) , we obtain that
8
∂Ŷ
>
>
<
∆Ŷ + γŶ + µ rŶ = 1D0 û in Ω
∂t
>
>
on Ω Σ and Ŷ (0) = y0 ,
: Ŷ = 0,
Q
in Ω
(6)
D
where
γ (t, x ) = a (t, x ) +
∂σ (t, x )
W (t ) + 12 jσ (t, x )j2
∂t
jr [σ (t, x ) W (t )]j2
∆ [σ (t, x ) W (t )] ,
µ (t, x ) =
2 r [σ (t, x ) W (t )]
û (t, x ) = u (t, x ) ξ (t, x )
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Approximating control problem
If we consider now for the state system (6) with the cost function
Ĵ (v̂ ) =
1
2
Z
Q
v̂ 2 (t, x ) dtdx +
Z
1
Ŷ 2 (T , x ) dx .
2ε D
then the optimal pair
Ŷε (t ) , ûε (t ) = (Yε (t ) ξ (t ) , uε (t ) ξ (t ))
is characterized for almost each ω 2 Ω, by
8
∂Ŷε
>
>
∆Ŷε + γŶε + µ rŶε = 1D0 v in Ω Q,
>
>
>
∂t
>
>
>
>
>
Ŷ = 0,
on Ω Σ, and Ŷ (0) = y0 , in Ω
>
>
>
>
>
>
∂p̂ε
<
∆p̂ε + γp̂ε div (µ rp̂ε ) = 0, in Q,
∂t
>
>
> p̂ε = 0,
on Ω Σ,
>
>
>
>
>
>
1
>
>
Ŷε (T ) , in Ω D
p̂ε (T ) =
>
>
>
ε
>
>
:
1D0 (x )p̂ε (t, x ) = ûε (t, x ) , in Ω ]0, T ] D .
Aurel R¼
aşcanu
D,
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
We have
1D0 (x )p̂ε (t, x ) = ûε (t, x ) = uε (t ) ξ (t ) = 1D0 (x )pε (t, x ) ξ (t )
In same manner we obtain
Z
Q0
p̂ε2 (t, x ) dtdx +
Z
1
Ŷ 2 (T , x ) dx =
ε D ε
Z
Q
ûε2 (t, x ) dtdx +
2 jy0 jH jp̂ε (0)jH .
Z
1
Ŷ 2 (T , x ) dx
ε D ε
Now using the Carleman estimate we have
jp̂ε (0)j2H
Z
CT
Q0
jp̂ε (t, x )j2 dtdx .
Hence
2 jy0 jH jp̂ε (0)jH
2 jy0 jH
1
2
and consequently P
1
2
Z
Q0
Z
Q0
a.s. ω 2 Ω :
p̂ε2 (t, x ) dtdx +
p
CT
Z
Q0
jp̂ε (t, x )j2 dtdx
1/2
2
jp̂ε (t, x )j dtdx + 4 jy0 j2H CT
Z
1
Ŷ 2 (T , x ) dx
ε D ε
Aurel R¼
aşcanu
4 jy0 j2H CT (ω )
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Therefore
1
E
2
Z
Q
uε2 (t, x ) ξ 2 (t ) dtdx =
1
E
2
Z
pε2 (t, x ) ξ 2
Q0
4 jy0 j2H E (CT )
(t ) dtdx
<∞
and
E
Z
D
Yε2 (T , x ) ξ 2 (T ) dx
4ε jy0 j2H ECT .
Hence on a subsequence ε n ! 0,
ξuε n ! ξu weakly in L2ad (Ω
ξYε n ! ξY weakly in
Y ε n (T ) ξ (T ) ! 0
in
L2ad
L2
Q)
Ω; L2 (]0, T [; V )
(Ω
(boundedness estimates on the SPDE)
D) .
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
By Mazur’s theorem there are
mn
(uen , yen , yen (T )) = ∑ ri (uε i , yε i , yε i (T )) 2 conv f(uε i , yε i , yε i (T )) : i 2 N g ,
i =1
ri
such that
ξe
un ! ξu, strongly in L2ad (Ω
ξe
yn ! ξY , strongly in L2ad (Ω
yen (T ) ξ (T ) ! 0 in L2 (Ω
n
0, ∑m
i =1 ri = 1
Q)
[0, T ]; V )
D)
and by linearity (u
en , yen ) is a solution of the equation (2) with yen (0) = y0 .
Moreover
Z
1
E
ξ2u
en2 dtdx 4 jy0 j2H E (CT ) .
2
Q
Since
RT
E sup jξ (t ) yen (t ) ξ (t ) yem (t ) j2H + E 0 kξ (s ) yen (s ) ξ (s ) yem (s )k2V ds
t 2[0,T ]
R
C E Q ( ξ (s ) u
en (s ) ξ (s ) u
em (s ))2 dtdx
then ξe
yn ! ξY , in L2ad (Ω; C ([0, T ]; H )), and yen ! Y , in
q
Lad (Ω; C ([0, T ]; H )) for all 1 q < 2. Passing to limit in (2) we see that
(Y , u ) is a solution of the equation (2) with y (0) = y0 and Y (T ) = 0. This
completes the proof.
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
We proved the following theorem
Theorem
Let the SPDE
∆Y (t, x ) dt +a (t, x ) Y (t, x ) dt = 1D0 (x )u (t, x ) dt
dt Y (t, x )
Y (ω, t, x ) = 0
on Ω
Y (ω, 0, x ) = y0 (x )
+Y (t, x ) hσ (t, x ) , dW (t )i
Σ,
in Ω
E
Q
Q,
D,
Then there exists a control u 2 L0ad (Ω
Z
on Ω
Q ) such that
u 2 (t, x ) exp [ σ (t, x ) W (t )] dtdx
C
Z
D
y02 (x ) dx
and the corresponding state
Y u (ω, T , x ) = 0
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
V. Barbu : Exact controllability of stochastic parabolic equations with
multiplicative noise, arXiv:1104.4603v1 [math.AP], (Submitted on 24 Apr
2011).
V. Barbu, A. R¼
aşcanu, G. Tessitore: Carleman estimates and
controllability of linear stochasti heat equations, Appl. Math. Optim.
47:97-1209, (2003).
J. Bismut : On optimal control of linear stochastic equations with a
linear-quadratic criterion, Siam J. Control and Optim., vol.15, no. 1, p. 1 4, 1977.
G.Da Prato and J. Zabczyk : Stochastic Equations in In…nite Dimensions,
Cambridge University Press, 1992.
A.V. Fursikov, O. Yu Imanuvilov : Controllability of Evolution
Equations, Lecture Notes Series, 34, (1996), Research Institute of
Mathematics, Seoul National University, Korea.
D. Goreac: Approximate controllability for linear stochastic di¤erential,
equations in in…nite dimensions, Appl. Math. Optimiz., 105-132 (2009).
E. Pardoux : Equations aux dérivées partielles stochastiques nonlinéaires
monotones. Etude de solutions fortes du type Itô, Thèse, Paris-Sud, Orsay,
1975.
E. Pardoux and A. R¼
aşcanu :Backward stochastic variational inequalities,
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise
Thank you for your attention !
and
I invite you to ITN School and Conference in Iasi, Romania
Deterministic and Stochastic Controlled Systems
School: June 18-30, 2012
Conference: July 2-7, 2012
http://www.math.uaic.ro/~ITN2012/
Available Post-Doc Positions in ITN-Marie Curie Project of Iasi
http://www.math.uaic.ro/~ITN_Marie_Curie/recruitment.php
Iasi: 6 months.
Milano: 6 months.
Manchester: 6 months.
Marrakech: 6 months.
Aurel R¼
aşcanu
Controlability of Stochastic Parabolic Equation with Multiplicative Noise