2005-Oujda International Conference on Nonlinear Analysis.
Electronic Journal of Differential Equations, Conference 14, 2006, pp. 135–147.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
OPTIMAL CONTROLS FOR A CLASS OF NONLINEAR
EVOLUTION SYSTEMS
ABDELHAQ BENBRIK, MOHAMMED BERRAJAA, SAMIR LAHRECH
Abstract. We consider the abstract nonlinear evolution equation ż + Az =
R
uBz + f . Viewing u as control, we seek to minimize J(u) = 0T L(z(t), u(t)) dt.
Under suitable hypotheses, it is shown that there exists an optimal control u
and that it satisfies the appropriate optimality system. An example involving
the p-Laplacian operator demonstrates the applicability of our results.
1. Introduction
In this paper, we investigate the optimal control problem governed by the abstract non linear evolution equation
ż + Az = uBz + f
(1.1)
These systems with linear operators A and B are called bilinear systems (see.
[2, 3, 11]). They appear in many mathematical models from physical processes,
for example, models involving the p Laplacian operator (see [4]). Our aim is to
investigate the case where A is not linear.
We organize this work as follows: After formulating the problem, we address the
question of existence and uniqueness of solutions to these systems. In section 3, we
prove the existence theorem of optimal control and we give the necessary conditions
of optimality. Finally, we present an example involving the p-Laplacian operator
which illustrates the applications of the abstract framework and the results of the
theory developed in the previous sections.
2. Setting of the problem
Throughout the paper, H denotes a separable Hilbert space and V a subspace
of H having the structure of a reflexive Banach space which is continuously and
densely embedded in H.
Identifying H with its dual H 0 , we have the Gelfand triplet V ,→ H ,→ V 0 where
V 0 is the dual of V . We suppose that these embeddings are compact. Let h., .i be
the duality pairing between V and V 0 as well as the inner product on H. Let k · k,
2000 Mathematics Subject Classification. 49J20, 49K20.
Key words and phrases. Optimal control; monotone operator; compact embedding;
p-Laplacian; bilinear system.
c
2006
Texas State University - San Marcos.
Published September 20, 2006.
135
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A. BENBRIK, M. BERRAJAA, S. LAHRECH
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| · | and k · kV 0 denote the norms on V , H and V 0 respectively. Given a fixed real
number T > 0 and 2 < p < +∞ we introduce the spaces:
0
0
Lp (H) = Lp (0, T ; H), Lp (V 0 ) = Lp (0, T ; V 0 ),
0
where ( p1 + p10 = 1) and W = w ∈ Lp (V ) : ẇ ∈ Lp (V 0 ) . Here the derivative is
understood in the sense of vector valued distributions.
It is well known that every w ∈ W is after eventual modification on a set of
measure zero, continuous from [0, T ] in H and the embedding W ,→ C([0, T ]; H)
is continuous [6, 7]. Furthermore, if V ,→ H compactly, then also W ,→ Lp (H)
compactly.
We study the control problem
Lp (V ) = Lp (0, T ; V ),
inf J(u)
(2.1)
u
subject to the state equation
ż + Az(t) = u(t)Bz(t) + f (t)
z(0) = z0 ,
where the cost functional is
Z
J(u) =
T
L(z(t), u(t)) dt.
0
Our aim is to provide conditions under which the optimal solutions (2.1) exist. By
an optimal solution we mean a control u on which the infimum is attained.
For problem (2.1) we need the following hypotheses:
(H1) A : V → V 0 is such that:
(i) kAϕkV 0 ≤ α1 kϕkp−1 with α1 > 0.
(ii) hAϕ, ϕi ≥ α2 kϕkp with α2 > 0.
(iii) hAϕ1 − Aϕ2 , ϕ1 − ϕ2 i ≥ α3 kϕk2 with β > 0.
(iv) ϕ → A(ϕ) is continuously Frechet differentiable.
(H2) B : H → H is linear and continuous with |Bϕ| ≤ b|ϕ|, b > 0.
(H3) u ∈ Lr (0, T ) with r = p/(p − 2).
0
(H4) f ∈ Lp (V 0 ).
(H5) z0 ∈ H.
(H6) L : H × R → R is a integrand convex such that:
RT
(i) L is coercive: limkukLr (0,T ) →∞ 0 L(z(t), u(t)) dt = +∞
(ii) (x, u) → L(x, u) is continuously Frechet differentiable.
(iii) for every x ∈ C([0, T ], H) and every u ∈ Lr (0, T ), J(u) is finite.
0
Remark 2.1. A(ϕ) ∈ Lp (V 0 ) if ϕ ∈ Lp (V ) and then
p−1
kAϕkLp0 (V 0 ) ≤ α1 kϕkL
p (V ) .
0
For u ∈ Lr (0, T ) and ϕ ∈ Lp (V ) we have uBϕ ∈ Lp (V 0 ) and
kuBϕkLp0 (V 0 ) ≤ β1 kukLr (0,T ) kϕkLp (V ) ,
where β1 > 0. Therefore, the choice of control space is compatible with the equation.
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137
3. Results on the evolution problem
We consider the evolution problem
ż(t) + Az(t) = u(t)Bz(t) + f
(3.1)
z(0) = z0
We recall that by a solution to the above problem, we mean a function z ∈ W that
satisfies (3.1).
Theorem 3.1. Under hypothesis (H1)(i), (H1)(ii), (H1)(iii), (H2), (H3), (H4)
and (H5), equation (3.1) admits a unique solution z such that z ∈ L∞ (H) and
z ∈ W.
Proof. Uniqueness: if z1 and z2 are solutions of (3.1), then z = z1 − z2 satisfies
ż + Az1 − Az2 = uBz
(3.2)
z(0) = 0
and for t ∈ [0, T ],
Z t
1
2
|z(t)| ≤ b
|u(τ )||z(τ )|2 dτ .
2
0
Using the Gronwall lemma, we obtain z1 = z2 .
The existence follows from a standard application of the Galerkin method [6]
and the a priori estimates given in Lemma 3.2. We remark that by Theorem 3.1,
z ∈ C([0, T ]; H).
Lemma 3.2. Under the hypothesis of Theorem 3.1, if z is a solution of (3.1) then
1/p
0
kzkLp (V ) ≤ K1 |z0 |2 + kukrLr (0,T ) + kf kpLp0 (V 0 )
(3.3)
2
0
1/2
kzkL∞ (H) ≤ K2 |z0 | + kukrLr (0,T ) + kf kpLp0 (V 0 )
(3.4)
p−1/p−2
0
kżkLp0 (V 0 ) ≤ K3 kzkp−1
(3.5)
Lp (V ) + kukLr (0,T ) + kf kLp (V 0 )
Proof. Let z be a solution of (3.1), then
Z T
Z T
Z
hż(t), z(t)i dt +
hAz(t), z(t)i dt =
0
0
T
Z
T
hu(t)Bz(t), z(t)i dt +
0
hf (t), z(t)i dt.
0
Using (H1)(ii), (H2) and the continuity of the embedding V ,→ H, we have
Z T
1
1
2
2
|z(T )| − |z0 | + α2
kz(τ )kpV dτ
2
2
0
Z T
Z T
0
2
≤ K1
|u(τ )|kz(τ )k dτ +
kf (τ )kV 0 kz(τ )k dτ .
0
0
1
r
2
p
By the Young inequality [10], for + = 1, we have
Z T
α2
K10
|u(τ )|kz(τ )k2 dτ ≤
kzkpLp (V ) + K20 kukrLr (0,T ) ,
4
0
Z T
0
α2
kf (τ )kV kz(τ )k dτ ≤
kzkpLp (V ) + K30 kf kpLp0 (V 0 ) .
4
0
Hence
0
α2
1
kzkpLp (V ) ≤ |z0 |2 + K20 kukrLr (0,T ) + K30 kf kpLp0 (V 0 ) ,
2
2
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from which, we deduce then (3.3).
Multiplying (3.1) by z and integrating on [0, t] we obtain
Z
0
1
1
α2 t
2
2
kz(τ )kp dτ ≤ K200 kukrLr (0,T ) + K 00 kf kpLp0 (V 0 )
|z(t)| − |z0 | +
2
2
2 0
and then
1/2
0
|z(t)|H ≤ K2 |z0 |2 + kukrLr (0,T ) + kf kpLp0 (V 0 )
which implies 3.4.
Multiplying 3.2 by ξ ∈ Lp (V ), we have
Z T
hż(t), ξ(t)i dt|
|
0
Z
T
Z
T
0
Z
T
hf (t), ξ(t)i|.
hu(t)Bz(t), ξ(t)i| + |
hAz(t), ξ(t)i dt| + |
≤|
0
0
Hence
Z
T
0
r
p
p
hż(t), ξ(t)i dt ≤ α1 kzkp−1
Lp (V ) + β1 kukL (0,T ) kzkL (V ) + kf kLp (V 0 ) kξkL (V ) ,
0
which by Young inequality implies (3.5).
4. Optimal controls
The aim of this section is to prove the existence of optimal controls for problem
(2.1). The differentiability of the mapping u 7→ z permits the characterization of
the optimal control u by necessary conditions corresponding to J 0 (u) = 0.
Existence theorem for the control problem.
Theorem 4.1. If (H1), (H2), (H3), (H4), (H5) and (H6) hold, then (2.1) admits
an optimal solution.
Proof. Let (un )n be a minimizing sequence for (2.1), i.e. the pairs (zn , un ) are
admissible for (2.1) and
lim J(un ) = J.
n
From (H6) we have kun kLr (0,T ) ≤ M .
And from Lemma 3.2, we know that (zn )n belongs to a bounded subset of
L∞ (H) ∩ W . By passing to a subsequence if necessary, we may assume that
un * u
w − Lr (0, T )
zn * z
w ∗ −L∞ (H)
zn * z
w − Lp (V )
0
Azn * χ w − Lp (V 0 )
0
un Bzn * Ψ w − Lp (V 0 )
żn * Λ
0
w − Lp (V 0 )
1. Using the convergence σ(D(0, T ; V ); D0 (0, T ; V 0 )) we obtain Λ = ż.
2. V ,→ H compactly implies that
zn → z
s − Lp (H)
zn (t) → z(t) s − H
for all t ∈ [0, T ].
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139
For ϕ ∈ Lp (V ), we have
Z T
hun (t)Bzn (t), ϕ(t)i dt
0
Z
T
Z
T
hun (t)B(zn (t) − z(t); ϕ(t)i dt +
=
0
hun (t)Bz(t), ϕ(t)i dt.
0
Note that
Z T
hun (t)B(zn (t) − z(t); ϕ(t)i dt ≤ K1 kun kLr (0,T ) kzn − zkLp (H) kϕkLp (H)
0
and
T
Z
Z
un (t)hBz(t), ϕ(t)i dt →
T
u(t)hBz(t), ϕ(t)i dt
0
0
0
because hBz, ϕi ∈ Lr (0, T ). We deduce that Ψ = uBz.
3. For y ∈ Lp (V ), we set
Z T
Xm =
hAzm (t) − Ay(t); zm (t) − y(t)i dt .
0
We have
Z
Xm =
T
T
Z
hAzm (t); zm (t)i dt −
T
Z
hAzm (t); y(t)i dt −
0
0
hAy(t); zm (t) − y(t)i dt
0
and
T
Z
hAzm (t), zm (t)i dt
0
=
But
Z
1
1
|zm,0 |2 − |zm (T )|2 +
2
2
Z
T
Z
hum Bzm (t), zm (t)i dt +
T
hf (t), zm (t)i dt.
0
0
T
hum (t)Bzm (t), zm (t)i − hu(t)Bz(t), z(t)i dt
0
Z
T
Z
T
hum (t)Bzm (t), zm (t) − z(t)i dt +
=
0
hum (t)Bzm (t) − u(t)Bz(t), z(t)i dt
0
The first integral in the right-hand side approaches zero because zm → z (s −
Lp (H)). The second integral approaches zero because um Bzm * uBz (w −
0
Lp (V 0 )). We deduce that
Z T
lim sup
hAzm (t), zm (t)i dt
m
0
1
1
≤ |z0 |2 − |z(T )|2 +
2
2
Since z satisfies
Z
T
Z
hu(t)Bz(t), z(t)i dt +
0
T
hf (t), z(t)i dt .
0
ż + χ = uBz + f
z(0) = z0
it follows that
Z T
Z T
Z T
1
1
2
2
|z0 | − |z(T )| +
hu(t)Bz(t), z(t)i dt +
hf (t), z(t)i dt =
hχ(t), z(t)i dt.
2
2
0
0
0
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Hence
Z
T
0 ≤ lim sup Xm ≤
m
hχ(t) − Ay(t), z(t) − y(t)i dt for all y ∈ Lp (V )
0
Using the continuity of the operator A we obtain χ = Az. We deduce that (z, u) is
admissible for (2.1). From (H6) we have
Z T
Z T
L(zm (t), um (t)) dt = J
L(z(t), u(t)) dt ≤ lim inf
m
0
0
Hence u is an optimal control.
Optimality conditions. Before proceeding with investigation of the mapping
Θ : u 7→ z, where z is defined by (3.1), we introduce a technical lemma generalizing the Gronwall inequality.
Lemma 4.2. Let T > 0 and c ≥ 0. Assume that λ and m are integrable in [0, T ]
with positive values. Let ϕ : [0, T ] → R+ be such that:
(a) λϕ and λϕ2 are integrable on [0, T ].
Rt
Rt
(b) 21 ϕ2 (t) ≤ 12 c2 + 0 λ(s)ϕ(s) ds + 0 m(s)ϕ2 (s) ds for t ≥ 0.
Then
Z t
h
i
Z t
ϕ(t) ≤ c +
λ(s) ds exp
m(s) ds .
0
0
Proof. Set
h
Z
2
Ψ(t) = c + 2
t
Z
λ(s)ϕ(s) ds + 2
0
t
m(s)ϕ2 (s) ds
i1/2
.
0
We have that ϕ(t) ≤ Ψ(t) and Ψ̇ ≤ λ(t) + m(t)Ψ(t). Then
Z t
Z t
Z s
i
dh
Ψ(t) exp −
m(s) ds −
λ(s) exp −
m(τ ) dτ ≤ 0.
dt
0
0
0
Hence
Z t
Z t
h
Z τ
i
Ψ(t) ≤ exp
m(τ ) dτ c +
λ(τ ) exp −
m(s) ds dτ ,
0
0
0
which completes the proof.
Lemma 4.3. Suppose the hypothesis (H1), (H2), (H3), (H4) and (H5) hold, then
the mapping Θ : Lr (0, T ) → L∞ (H) ∩ L2 (V ), u 7→ z is locally Lipschitz.
Proof. Let u and h be in Lr (0, T ) with khkLr (0,T ) ≤ 1. Set z = Θ(u), zh = Θ(u+h)
and z = zh − z. Then z satisfies
ż + Azh − Az = uBz + hBzh
z(0) = 0
Multiplying by z and integrating on [0, t] we have
Z t
Z t
Z t
1
2
2
2
|z(t)| + β
kz(τ )kV dτ ≤ b
|u(τ )||z(τ )| dτ + b
|h(τ )||zh (τ )||z(τ )| dτ .
2
0
0
0
Invoking the Lemma 4.2, we have
Z t
h Z t
i
|h(τ )||zh (τ )| dτ
|z(t)|H ≤ exp b
|u(τ )| dτ b
0
0
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but
Z
141
t
|h(τ )||zh (τ )| dτ ≤ KkhkLr (0,T ) kzh kL∞ (H)
0
and
h
i1/2
0
kzh kL∞ (H) k ≤ K1 |z0 |2 + ku + hkrLr (0,T ) + kf kpLp0 (V 0 )
≤ K0
where K 0 is a positive constant depending on z0 , u and f (because khk ≤ 1). We
obtain
kzkL∞ (H) ≤ K10 khkLr (0,T ) ,
kzkL2 (V ) ≤ K20 khkLr (0,T )
Theorem 4.4. Suppose that:
(i) The hypothesis of Lemma 4.3 are satisfied with f = 0.
(ii) For ϕ and Ψ in C([0, T ); H) with kΨkC([0,T ];H) ≤ 1 we have
kA0 (ϕ(t) + Ψ(t)) − A0 (ϕ(t))kL(H) ≤ γ(t)|Ψ(t)|H
where γ ∈ L1 (0, T ).
Then the mapping Θ : Lr (0, T ) → L∞ (H) ∩ L2 (V ) is Fréchet differentiable and the
derivative Θu0 .h is a solution of
0
ẏ(t) + Az(t)
y(t) = u(t)By(t) + h(t)Bz(t)
y(0) = 0
(4.1)
where z = Θ(u).
Proof. 1. Since A is strongly monotone. For λ > 0, ϕ and Ψ in V , we have
1
(A(ϕ + λΨ) − A(ϕ)), Ψ ≥ βkΨk2V .
λ
Hence hA0ϕ Ψ, Ψi ≥ βkΨk2V
2. For u ∈ Lr (0, T ), the mapping h 7→ y defined by (4.1) is linear. Multiplying
(4.1) by y and integrating on [0, t] we obtain
Z t
Z t
1
|y(t)|2 + β
ky(τ )k2V dτ ≤ b
|u(τ )||y(τ )|2 dτ + |h(τ )||z(τ )||y(τ )| dτ
2
0
0
By Lemma 4.3,
Z
|y(t)| ≤ b
0
t
h Z t
i
|h(τ )||z(τ )| dτ exp b
|u(τ )| ,
0
but
1/2
kzkL∞ (H) ≤ K1 |z0 |2 + kukrLr (0,T )
.
Then
kykL∞ (H) ≤ K10 khkLr (0,T ) ,
kykL2 (V ) ≤ K20 khkLr (0,T ) ,
where Ki0 are positive constants depending on z0 and u. Hence the mapping h 7→ y
is continuous.
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3. Set z = Θ(u), zh = Θ(u + h), z = zh − z and w = z − y where y is a solution of
(4.1). We have
ẇ(t) + A0z(t) w(t) = u(t)Bw(t) + h(t)Bz(t) + g(t)
w(0) = 0
(4.2)
where
0
z(t) − Azh (t) − Az(t)
g(t) = Az(t)
Z 1
=
[A0 z(t) − A0 (z(t) + sz(t))]z(t) ds.
0
Then
Z
1
γ(t)
|z(t)|2
2
0
On the other hand, multiplying (4.2) by w and integrating on [0, t] we obtain
Z t
1
|w(t)|2 + β
kw(τ )k2V dτ
2
0
Z
Z t
Z t
1 t
2
γ(τ )|z(τ )|2 |w(τ )| dτ
≤b
|u(τ )||w(τ )| dτ + b
|h(τ )||z(τ )||w(τ )| dτ +
2
0
0
0
|g(t)|H ≤
γ(t)|sz(t)||z(t)| ds =
Then Lemma 4.3 gives
Z
Z t
i
h Z t
1 t
|w(t)| ≤ exp b
|u(τ )| dτ b
γ(τ )|z(τ )|2 dτ ,
|h(τ )||z(τ )| dτ +
2 0
0
0
but kzkL∞ (H) ≤ KkhkLr (0,T ) then
kwkL∞ (H) ≤ K1 khk2Lr (0,T ) ,
kwkL2 (V ) ≤ K2 khk2Lr (0,T ) .
It follows that Θ is fréchet differentiable from Lr (0, T ) on L∞ (H)∩L2 (V ) and Θu0 .h
is a solution of (4.1).
Theorem 4.5. Assume the hypotheses of Theorem 4.4 and (H6) hold. Then an
optimal control u, its corresponding state z, and its adjoint state p are necessarily
tied by the optimality system:
(1) ż + Az = uBz z(0) = z0
∗
(2) −ṗ + A0 z p = uB ∗ p + ∂1 L(z(t), u(t)) p(T ) = 0
(3) hBz(t), p(t)i + ∂2 L(z(t), u(t)) = 0 a.e. in [0, T ]
Proof. Since L is Fréchet differentiable, we deduce that the functional
Z T
J(u) =
L(z(t), u(t)) dt
0
r
is Fréchet differentiable on L (0, T ). Since u is a minimum point for J,
Ju0 .h = 0,
∀h ∈ Lr (0, T )
but
Ju0 .h
Z
T
Z
h∂1 L(z(t), u(t), y(t)i dt +
=
0
T
h∂2 L(z(t), u(t), h(t)i dt
0
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OPTIMAL CONTROLS OF NONLINEAR EVOLUTION SYSTEMS
143
where y = Θ0u .h. We define p by (2), then
J 0 (u).h =
Z
T
∗
h−ṗ(t) + A0 z(t) p(t) − u(t)B ∗ p(t), y(t)i dt +
Z
T
hp(t), ẏ(t) +
=
0
Az(t)
y(t)
Z
Z
T
− u(t)By(t)i dt +
0
=
h(t)∂2 L(z(t), u(t)) dt
0
0
Z
T
h(t)∂2 L(z(t), u(t)) dt
0
T
hp(t), Bz(t)i + ∂2 L(z(t), u(t)) h(t) dt
0
Hence part (3) of the theorem is consequence of the above equality.
5. Example
In this section, we present an example which illustrates the application of the
results of the theory developed in the previous sections. Let Ω be a bounded domain
in RN with smooth boundary Γ = ∂Ω. We consider the control problem (2.1) with
Z
|z(x, t)|4 dx dt +
J(u) =
T
Z
Q
0
|u(t)|2RN dt
Where z satisfies the nonlinear evolution equation
N
X
∂z
∂z
ui (t)
− div(|∇z|2 ∇z) =
∂t
∂x
i
i=1
in Q = Ω×]0, T [
(5.1)
z = 0 in Σ = Γ×]0, T [
z(x, 0) = z0 (x)
Setting V = W01,4 (Ω), H = L2 (Ω) and V 0 = W −1,4/3 (Ω) we have V ,→ H ,→ V 0
continuously and densely. Furthermore V ,→ H compactly.
The equation (5.1) can be written in the form
ż(t) + Az(t) = u(t)Bz(t)
z(0) = z0
where
(1) A : V → V 0 , ϕ 7→ − div(|∇ϕ|2 ∇ϕ) which satisfies (H1) (see [6]).
(2) B = (B1 , . . . , BN ) with Bi : V → H, ϕ 7→ Bi ϕ = ϕxi . Hence kBi ϕkH ≤
bi kϕkV , bi > 0 and kBϕkH N ≤ bkϕkV .
(3) u = (u1 , . . . , uN ) ∈ U = L2 (0, T ; RN ) . Here
u(t)Bz(t) =
N
X
i=1
ui (t)Bi z(t) =
N
X
i=1
ui (t)
∂z(t)
.
∂xi
The cost function becomes
J(u) = kuk2L2 (0,T ;RN ) + kzk4L4 (0,T ;Q)
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A. BENBRIK, M. BERRAJAA, S. LAHRECH
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R
Since Ω u(t)Bz(x, t)z(x, t) dx = 0, the a priori estimates given by Lemma 3.2
become
kzkL4 (V ) ≤ K1 |z0 |1/2 ,
kżkL4/3 (V 0 )
kzkL∞ (H) ≤ K2 |z0 |,
3/2
3/2
≤ K3 kzkL4 (V ) + kukL2 (0,T ;RN )
Corollary 5.1. For z0 in L2 (Ω) and u in L2 (0, T ; RN ), the equation (3.1) with
f = 0 admits a unique solution which satisfies
z ∈ L∞ (0, T ; L2 (Ω)) ∩ L4 (0, T ; W01,4 (Ω)),
ż ∈ L4/3 (0, T ; W −1,4/3 (Ω))
Proposition 5.2. The mapping Θ : U → C([0, T ]; H), u 7→ z, with z the solution
of (3.1) with f = 0. is differentiable in the sense of Fréchet, and Θ0u .h satisfies
0
ẏ + Az(t)
y(t) = u(t)By(t) + h(t)Bz(t)
y(0) = 0,
where z = (Θ(u))(t) and
N
X
|∇ϕ|2 hxi + 2h∇ϕ, ∇hi1 ϕxi xi
=−
A0ϕ .h
i=1
with h∇ϕ, ∇hi1 =
PN
i=1
ϕxi hxi (ϕ and h ∈ V ).
Proof. 1. For ϕ ∈ V , the mapping, A0ϕ : V → V 0 ,
h 7→ A0ϕ h = −
N
X
|∇ϕ|2 hxi + 2h∇ϕ, ∇hi1 ϕxi xi
i=1
is linear and for v ∈ V we have
hA0ϕ h, viV 0 ,V =
N Z
X
i=1
fi vxi dx
Ω
with fi = |∇ϕ|2 hxi + 2h∇ϕ, ∇hi1 ϕxi . Furthermore,
Z
hZ
i
4/3
kfi kL4/3 (Ω) ≤ K1
|∇ϕ|8/3 |hxi |4/3 dx +
|h∇ϕ, ∇hi1 |4/3 |ϕxi |4/3 dx
Ω
Ω
h Z
1/4 Z
1/3 i
≤ 2K1
|∇ϕ|4
|∇h|4
.
Ω
Ω
Then
kfi kL4/3 (Ω) ≤ Kkϕk2W 1,4 (Ω) khkW 1,4 (Ω) .
0
0
Using the norm in V 0 it follows that A0ϕ ∈ L(V, V 0 ).
For ϕ and h in V , we have
A(ϕ + h) − A(ϕ) − A0ϕ (h) = F (ϕ, h)
where
F (ϕ, h) = −
N
X
i=1
[|∇h|2 hxi + |∇h|2 ϕxi + 2h∇ϕ, ∇hi1 hxi ]xi .
(5.2)
EJDE/CONF/14
OPTIMAL CONTROLS OF NONLINEAR EVOLUTION SYSTEMS
145
For v ∈ V ,
hF, viV 0 ,V =
N Z
X
fi vxi dx,
Ω
i=1
where
fi = |∇h|2 ϕxi + |∇h|2 hxi + 2h∇ϕ, ∇hi1 hxi .
Then
4/3
kfi kL4/3 (Ω) ≤ K 0
hZ
|∇ϕ|4/3 |∇h|8/3 dx +
Z
|∇ϕ|4/3 |∇h|8/3 dx +
Ω
Ω
i
h
4/3
8/3
≤ K 00 kϕkW 1,4 (Ω) khkW 1,4 (Ω) + khk4W 1,4 (Ω) .
0
Z
|∇h|4 dx
i
Ω
0
0
We deduce that
kA(ϕ + h) − A(ϕ) − A0ϕ (h)kV 0 ≤ K[kϕkV khk2V + khk3V k]
Hence A is differentiable in the sense of Frechet.
2. The equation (5.2) admits a unique solution satisfying
y ∈ L2 (V ) ∩ L∞ (H),
ẏ ∈ L2 (V 0 ).
The existence follows from a standard application of the Galerkin method and the
a priori estimates obtained for (4.1). We remark that y ∈ C([0, T ]; H).
3. The mapping A0 : V → L(V, V 0 ), ϕ 7→ A0ϕ is locally Lipschitz. Let ϕ and ψ be
in V with ψ in neighbourhood of 0. For h in V , we have
A0ϕ+ψ h = −
N
X
|∇(ϕ + ψ)|2 hxi + 2h∇(ϕ + ψ), ∇hi1 ((ϕ + ψ))xi xi
i=1
and (A0ϕ+ψ − A0ϕ )h = F , where
F =−
N h
X
|∇ψ|2 hxi + 2h∇ϕ, ∇ψi1 hxi + 2h∇ϕ, ∇hi1 ψxi
i=1
+ 2h∇ψ, ∇hi1 ϕxi + 2h∇ψ, ∇hi1 ψxi
i
.
xi
Then for v ∈ V ,
hF, viV 0 ,V =
N Z
X
i=1
fi vxi dx,
Ω
where
fi = |∇ψ|2 hxi +2h∇ϕ, ∇ψi1 hxi +2h∇ψ, ∇hi1 ϕxi +2h∇ψ, ∇hi1 ψxi +2h∇ϕ, ∇hi1 ψxi .
Hence
kfi kLp0 (Ω) ≤ K kψk2V + kψkV kϕkV khkV .
Since ψ is in neighbourhood of 0,
k(A0ϕ+ψ − A0ϕ )hkV 0 ≤ K 0 kϕkV kψkV khkV .
Hence
kA0ϕ+ψ − A0ϕ kL(V,V 0 ) ≤ K 00 kψkV .
It follows by theorem 4.4. that Θ is Frechet differentiable and its derivative Θ0u .h
satisfies (5.2)
146
A. BENBRIK, M. BERRAJAA, S. LAHRECH
EJDE/CONF/14
RT
Now the functional J can be written as J(u) = 0 L(z(t), u(t)) dt with L satisfying (H6).
The differentiability of Θ and the norm ensures the differentiability of J and the
expression of derivative is
Z
Z T
hu(t), h(t)iRN dt
dJ(u).h = 4
|z(x, t)|2 z(x, t)y(x, t) dx dt + 2
0
Q
where y = Θ0u h.
From Theorems 3.1, 4.4 and 4.5, we get the following result.
Corollary 5.3. An optimal control u, its corresponding state z, and its adjoint
state p are necessarily tied by the optimality system: For 1 ≤ i ≤ N and t ∈ [0, T ],
Z
∂z
ui (t) = −2
p(x, t)
(x, t) dx
∂xi
Ω
N
X
∂z
∂z
− div(|∇z|2 ∇z =
ui (t)
∂t
∂xi
i=1
z(x, t) = 0
z(x, 0) = z0 (x)
−
in Q
in Σ
in Ω
N
X
∂p
∂p
0
p=−
ui (t)
+ Az(t)
+ |z(x, t)|2 z(x, t)
∂t
∂x
i
i=1
p(x, t) = 0
in Σ
p(x, T ) = 0
in Ω
in Q
References
[1] V. Barbu and Th. Precupanu, Convexity and Optimization in Banach spaces. Mathematics
and its applications, Reidel publishing Company, 19xx.
[2] A. Benbrik, A. Addou Existence and uniqueness of optimal contrôl for distributed-parameter
bilinear system. Journal of dynamical and control system, vol 8, number 2, pp1 41-152, April
2002.
[3] C. Bruni, G. Dipillo and G. Koch, Bilinear systems. An application class of “nearly linear”
systems in theory and applications. IEEE, transactions on automatic control, vol. AC 19,
no. 4, August 1974.
[4] J. Colinge and J. Rappaz, A strongly non linear problem arising in glaciology. R.A.I.R.O.,
33, 396–406, 1999.
[5] D. Dan Tiba, Optimal control of nonsmooth distributed parameter systems. Lecture notes in
Mathematics, vol 1459, Springer-Verlag.
[6] J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod,
1969.
[7] J. L. Lions and E. Magénes, Problèmes aux limites non homgènes et applications. Volumes
1, 2 and 3, Dunod, 1968.
[8] J. L. Lions, Contrôle optimal des systèmes gouvernés par des équations aux dérivées partielles. Dunod, 1968.
[9] S. Migorski, Variational stability analysis of optimal control problems for systems governed
by nonlinear second order evolution equations. J. Math. systems, estimation and control, 6,
no. 4, 1–24, (1996).
[10] D.S. Mitrinović, Analytic Inequalities. Springer-Verlag, Berlin,1970.
[11] R.R. Mohler, Bilinear control processes. Academic Press, 1975.
EJDE/CONF/14
OPTIMAL CONTROLS OF NONLINEAR EVOLUTION SYSTEMS
147
Abdelhaq Benbrik
Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc
E-mail address: benbrik@sciences.univ-oujda.ac.ma
Mohammed Berrajaa
Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc
E-mail address: berrajaa@sciences.univ-oujda.ac.ma
Samir Lahrech
Département de Mathématiques et Informatique, Faculté des Sciences, Université Mohammed 1er, Oujda, Maroc
E-mail address: lahrech@sciences.univ-oujda.ac.ma