Optimal Control for Filtration Agusto F. B.∗
Bamigbola, O. M.†
Layeni, O. P.‡
Abstract In this paper, the existence of optimal control for filtration at a decreasing rate given by Demchik (1998) is obtained using a cost function of the type: Z Z 1 α ξ 2 2 2 J(u) = |u − ud | dxdt + |u(T ) − ud (T )| dx + kvkL2 (QT ) 2 QT 2 Ω 2 Other properties of solution to the optimal control problem are discussed. Keywords: Optimal control, filtration, semigroup.
∗ Department of Pure and Applied Mathematics, Ladoke Akintola University of Technology Ogbomosho, Nigeria, folashade−
[email protected] † Department of Mathematics, University of Ilorin, Ilorin, Nigeria,
[email protected] ‡ Department of Mathematics, Obafemi Awolowo University, Ile - Ife, Nigeria, wn−
[email protected]
1
Introduction
Demchik [4] considered the filtration problem : uxt (x, t) + b(t)ut (x, t) − pu(x, t) = f (x, t) −1 u(x, 0) = c0 e−βv0 x u(0, t) = c0 (1 + γt)z b(t) = βv0−1 (1 + γt), p = βv0−1 γ(z − 1),
in QT in Ω in ΣT
(1.1)
z = a0 v0 γ −1
where β is the kinetic coefficient assumed to be constant [4]. We shall here consider the optimal control of the same problem with linearized boundary conditions: uxt (x, t) + b(t)ut (x, t) − pu(x, t) = f (x, t) in QT −1 u(x, 0) = −c0 βv0 x in Ω (1.2) u(0, t) = c0 (1 + zγt) in ΣT b(t) = βv0−1 (1 + γt), p = βv0−1 γ(z − 1), z = a0 v0 γ −1 subject to the afore given cost functional together with the condition u(·, t) = A(·) + B(·) exp−t . Let Ω ⊂ 0 (2.1)
Theorem 2.2 [8]: Let V and H be Hilbert spaces. Let (D∗ (M ), M ) be a linear operator from V to H with domain D∗ (M ) and range R(M ). Then, the following holds: 1. The inverse operator (D∗ (M ), M −1 ) exists if the null space N (M ) = {0}. 2. If the inverse operator exists, it is linear. Theorem 2.3: Problem 1.2 has a unique solution. To show the existence of solution of (1.2), we rearrange the problem as a standard evolution equation [9] i.e., ∂ )ut (x, t) − pu(x, t) = f (x, t) (b(t) + ∂x
2
Let M denote the corresponding Riesz map from H 1 (Ω) onto the dual H −1 (Ω). Let D be a subspace of V and L : D −→ V be an identity map. V = H 1 (Ω). We consider the problem find u, such that M ut (t) − Lu(t) = f (x, t),
t>0
(2.2)
∂ where L = p, M = b(t) + ∂x . Clearly M is invertible from Theorem 2.2. Since M is invertible implies M + L is surjective and f ∈ C 1 ([0, ∞), there exists a unique solution to problem (1.2).
3
Optimal Control of the Filtration Problem
We next show the existence of the optimal control of problem (1.2). We consider the cost functional: 1 J(u) = 2
Z QT
α |u − w| dxdt + 2 2
Z
|u(T ) − w(T )|2 dx +
Ω
ξ kvk2L2 (QT ) 2
(3.1)
The control space is given as U = L2 (QT ). Let Uad , the admissible space of control, be a closed convex subset of U . v is the control and the corresponding state u = u(v) is the solution of (3.2) below. w ∈ L2 (Ω) is a target state that we would like to obtain by controlling y, the second term is necessary in order to keep the solution close to w at time T and α, ξ > 0. The control problem thus becomes: min J(u) = 21 QT |u − w|2 dxdt + subject to uxt + b(t)ut − pu(x, t) = f + v u(x, 0) = −c0 βv0−1 x u(0, t) = c0 (1 + zγt) R
αR 2 Ω |u(T )
− w(T )|2 dx + 2ξ kvk2L2 (QT ) in QT in Ω in ΣT .
(3.2)
Our control problem can be put in the form: min J(Y ) = 0T |Y (s)|2Ug + δ|v(s)|2U ds + (Y (T ), Y (T ))Ug subject to ∂Y (t) ∂t = AY (t) + Bv(t) Y (0) = Y0 . R
"
In (3.3) above, δ > 0, Y = "
u ∂u ∂t
#
"
, A=
M −1
such that M =
∂ ∂x
(3.3)
#
+b 0 , B =A 0 −p
"
#
1 0
#
−c0 βv0−1 x and Y0 = . Ug = u ∈ L2 (QT )|ut − wt = 0 . q(x, t) was unspecified in Demchik’s q(x, 0) work, but we take it to have sufficient regularity in this article.
Theorem 3.1: There exists a unique solution to (3.3) To prove Theorem 3.1, we consider the following lemmas: Lemma 3.2: Operator A is an infinitesimal generator of a strongly continuous semigroup
3
Proof: The proof follows from Lumer-Phillips theorem [8]. We can easily check that A is dissipative since hAY, Y i < 0. Moreover, it is clear that the domain of A is dense in L2 (QT ). We claim that every positive λ is in the resolvent set of A. For this, we consider the problem ∂ + b(t) u = f1 , u+λ ∂x
∂u ∂u − λp = f2 ∂t ∂t
These two equations can be combined into one equation
(I − λp)ut + I + λ that is: ut + Ψu = Γ
where Ψ =
∂ +b(t)]) (I+λ[ ∂x
(I−λp)
∂ + b(t) ∂x
and Γ =
u = f1 + f2
f1 +f2 (I−λp)
The solvability of this equation follows from Theorem 2.1 Hence A is an infinitesimal generator of a strongly continuous semigroup. The optimal control can be obtained as a feedback control. For this purpose we describe the Dynamic Programming approach which consists of obtaining solution of the Riccati equation Lemma 3.3: Since lemma 3.2 holds, the Riccati equation 0
Q = A∗ Q + AQ − QBB ∗ Q + C ∗ C Q(0) = Q0
)
(3.4)
has a unique mild solution. Proof: In (3.4) Q ∈ C([0, ∞]; L(H)), B ∈ L(H, U ), B ∗ ∈ L(U, H), C ∈ L(H, Y ) and C ∗ ∈ L(Y, H), where H is the state space, U is the control space space. " and Y isthe observation # " The # adjoint ∂ −p 0 ∂x + b(t) of A denoted by A∗ is given by A∗ = ∂ −p , B = det1M ( ∂x +b(t)) 0 0 −p "
BB ∗
=
1 det M
#
−p 0 , CC ∗ = 0 0
"
1 0 0 1
#
The proof follows directly from Theorem 2.1. page 145 [2]. Lemma 3.4: Since lemma 3.2 holds, then equation 0
Y (t) = [A − BB ∗ Q(T − t)] Y (t), Y (0) = Y0
t ∈ [0, T ]
has a unique mild solution. The proof follows from [3] Proof of Theorem 3.1 Equation (3.3) has a unique solution by lemma 3.3 and lemma 3.4. The optimal control is related to the optimal state by the equation v ∗ (t) = −B ∗ Q(T − t)Y ∗ (t) 4
4
Conclusion
We have shown in the preceeding sectionsthe existence of a unique solution to the filtration problem and have similarly established the existence of the optimal control for the problem, giving the relationship between the optimal control and optimal state.
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Da Prato, G.: Linear Quadratic Control Theory for Infinite Dimensional Systems, ICTP Lecture Notes 2002.
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