Optimality conditions for parabolic systems with variable coefficients involving Schrödinger operators

A.H. Qamlo

doi:10.1016/j.jksus.2013.05.005

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ORIGINAL ARTICLE

26 (

); 107-112

doi:

10.1016/j.jksus.2013.05.005

Optimality conditions for parabolic systems with variable coefficients involving Schrödinger operators

A.H. Qamlo^*

Mathematics Department, Faculty of Applied Sciences, Umm AL-Qura University, P.O. Box 1348, Makkah 21955, Saudi Arabia

*Tel.: +966 553541240 drahqamlo@yahoo.com (A.H. Qamlo)

Received: 2013-2-25, Accepted: 2013-5-9, Published: 2013-04-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Available online 17 May 2013

Abstract

In this paper, we study the existence of a solution to n × n parabolic systems with variable coefficients involving Schrödinger operators defined on an unbounded domain of Rⁿ. We then discuss the necessary and sufficient conditions of optimality for these systems.

Keywords

Parabolic systems

Optimal control

Schrödinger operator

Weight function

Variable coefficients

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1 Introduction

Today, optimal control problems of distributed systems, that include partial differential equations have many mechanical and technical sources and a variety of technological and scientific applications.

Indeed, many optimal control problems of elliptic systems involving Schrödinger operators of the distributed type have been studied, as in Serag (2000, 2004) and Serag and Qamlo (2005). Whereas some of these problems had positive weight functions (Serag, 2004; Serag and Qamlo, 2005), others had constant coefficients, e.g., (Serag, 2000).

The necessary and sufficient conditions of optimality for 2 × 2 parabolic and hyperbolic systems involving Schrödinger operators have already been discussed in (Bahaa, 2006; Qamlo, 2013).

In addition, optimal control problems for systems involving parabolic and hyperbolic operators with an infinite number of variables have been introduced in (Kotarski et al., 2002; Serag, 2007; Qamlo, 2008, 2009; Bahaa and El-Shatery, 2013 ).

Furthermore, time-optimal control of infinite order parabolic and hyperbolic systems has been studied in (Kowalewski and Krakowiak, 2008; Kowalewski, 2009).

Here, we discuss the following n × n parabolic systems with variable coefficients involving Schrödinger operators that are defined on an unbounded domain of Rⁿ:

(1)

\{\begin{matrix} \frac{\partial}{\partial t} Y + L_{q} Y = A (x) Y + F (x, t) & in Q, \\ Y (x) \to 0 as | x | \to \infty, \\ Y (x) = 0 & on \sum, \\ y_{i} (x, 0) = y_{i, 0} (x) \forall i = 1, \dots n, & in Ω, \end{matrix}

Y and F are column matrices with elements y_i and f_i, repectively, In addition, Q = Ω × (0,T) with boundary Σ = ∂ Ω × (0,T) and Ω is an unbounded domain of Rⁿ with boundary ∂Ω. and L_q is a n × n diagonal matrix of the Schrödinger operator (−Δ + q), the potential q is a positive function that tends to ∞ at infinity, and A(x) is an n × n matrix of variable coefficients a_ij(x) (1 ⩽ i,j ⩽ n) that satisfy the following conditions:

there exist r > 1 and k > 0 such that

(2)

a_{ij} (x) \in (0, \frac{k}{{(1 + | x |^{2})}^{r}}) \forall i, j = 1, 2, \dots, n, \forall x \in Ω,

(3)

a_{ij} (x) ⩽ \sqrt{a_{ii} (x) a_{jj} (x)} \forall i, j = 1, 2, \dots, n, \forall x \in Ω .

We first prove the existence and uniqueness of the state for system (1), and we then introduce the necessary and sufficient conditions of optimality for this system by a set of equations and inequalities.

2 Some facts and results

To prove our theorems, we recall certain results that are introduced in Djellit and Yechoui (1997) regarding the existence of the principal eigenvalue $λ_{q}^{+}$ of the following problem:

(4)

\{\begin{matrix} (- Δ + q) y = λ g (x) y & in Ω, \\ y \to 0 as | x | \to \infty, \\ y = 0 & on Γ, \end{matrix}

where Ω is an unbounded connected open subset of Rⁿ with boundary ∂Ω and both the potential q and the weight function g(x) are measurable functions that tend to zero at infinity.

For n > 2, if ∃α > 0, β ⩾ 1,α > β, ∃k > 0, c > 0 such that

(5)

0 < g (x) ⩽ \frac{k}{{(1 + | x |^{2})}^{α}}, 0 < q (x) ⩽ \frac{c}{{(1 + | x |^{2})}^{β}},

where the eigenvalue problem (4) has a positive principal eigenvalue

λ_{1}^{+}

that is simple and associated with a positive eigenfunction φ_q in V₊. Moreover

λ_{1}^{+}

is characterized by:

(6)

λ_{1}^{+} \int_{Ω} g (x) | y |^{2} ⩽ \int_{Ω} (| \nabla y |^{2} + q | y |^{2}),

where

$V_{+} = {y \in V (Ω) : \int_{Ω} g | u |^{2} dx > 0}$ and $V (Ω) = {y \in D^{'} (Ω) : p_{1} y \in L^{2} (Ω), \nabla y \in L^{2} (Ω)}; p_{α} = ρ^{2 α} (x), α > 0, ρ (x) = {(1 + | x |^{2})}^{- 1 / 2},$

Furthermore, V(Ω) is a Hilbert space with the inner product ${(y, ψ)}_{V} = \int_{Ω} (\nabla y \cdot \nabla ψ + p_{1} y \cdot ψ) dx$ and the corresponding norm $‖ y ‖_{V} = {(\int_{Ω} (| \nabla y |^{2} + p_{1} | y |^{2}) dx)}^{1 / 2}$ which is equivalent to $‖ y ‖_{q} = {(\int_{Ω} (| \nabla y |^{2} + q | y |^{2}) dx)}^{1 / 2}$ .

Now, to study our system (1), we recall the introduced by Serag (2000): $L_{g}^{2} (Ω) = {y : Ω \to R : \int_{Ω} g (x) y^{2} dx < \infty},$ with an inner product ${(y, ψ)}_{g} = \int_{Ω} g (x) y ψ dx$ . We then have the following embeddings: $\begin{matrix} V (Ω) \subseteq L_{g}^{2} (Ω) \subseteq V^{'} (Ω), \\ V (Ω) \subseteq L^{2} (Ω) \subseteq V^{'} (Ω), \end{matrix}$ and we introduce the space L²(0,T;V(Ω)) of measurable functions t → f(t) which is defined on the open interval (0,T), as the variable t ∈ (0,T) denotes the time, where T < ∞.

L²(0,T;V(Ω)) is a Hilbert space with the scalar product ${(f (t), g (t))}_{L^{2} (0, T; V (Ω))} = \int_{(0, T)} {(f (t), g (t))}_{V (Ω)} dt,$ and the norm $‖ f (t) ‖_{L^{2} (0, T; V (Ω))} = {(\int_{(0, T)} ‖ f (t) ‖_{V (Ω)}^{2} dt)}^{1 / 2} < \infty$ .

Analogously, we can define the space $L^{2} (0, T; L_{g}^{2} (Ω)) = L^{2} (Q)$ ,

with the following scalar product: ${(f (t), g (t))}_{L^{2} (Q)} = \int_{(0, T)} {(f (t), g (t))}_{L_{g}^{2} (Ω)} dt = \int_{Q} f (t) g (t) dx dt .$ Then we have the following chain: $L^{2} (0, T; V (Ω)) \subseteq L^{2} (Q) \subseteq L^{2} (0, T; V^{'} (Ω)),$ and by the Cartesian product, we have ${(L^{2} (0, T; V (Ω)))}^{n} \subseteq {(L^{2} (Q))}^{n} \subseteq {(L^{2} (0, T; V^{'} (Ω)))}^{n} .$ In addition, we will use the following definition of M-matrices (Bermann and Plemmons, 1979; Serag, 2004).

Definition 1

A nonsingular matrix β = (b_ij) is an M-matrix if b_ij < 0 for i ≠ j,b_ii > 0 and if the principal minors extracted from β are positive.

3 The scalar case

In this section, we consider the scalar case (i.e., a system that consists of one equation):

(7)

\{\begin{matrix} \frac{\partial y (x)}{\partial t} + (- Δ + q) y (x) = g (x) y (x) + f (x, t) & in Q, \\ y (x) \to 0 as | x | \to \infty \\ y (x) = 0 & on \sum, \\ y (x, 0) = y_{0} (x) & in Ω \end{matrix}

Proposition 1

For f ∈ L²(0,T;V′(Ω)) and y₀(x) ∈ V(Ω), there exists a unique solution y ∈ L²(0,T;V(Ω)) for system (7) if $1 < λ_{1}^{+}$ .

Proof

The continuous bilinear form:

(8)

π (t; y, φ) = \int_{Ω} (\nabla y \nabla φ + qy φ) dx - \int_{Ω} g (x) y φ dx

is obviously coercive on V(Ω).

In fact, we have: $π (t; y, y) = \int_{Ω} (| \nabla y |^{2} + q | y |^{2}) dx - \int_{Ω} g (x) y^{2} dx = \int_{Ω} (| \nabla y |^{2} + (q + mg) | y |^{2}) dx - (1 + m) \int_{Ω} g (x) y^{2} dx, m > 0,$ Then, from (6): $π (t; y, y) ⩾ (1 - \frac{1 + m}{λ_{1}^{+} + m}) \int_{Ω} (| \nabla y |^{2} + (q + mg) | y |^{2}) dx,$ that is,

(9)

π (t; y, y) ⩾ (1 - \frac{1 + m}{λ_{1}^{+} + m}) ‖ y ‖_{q}^{2},

which proves the coerciveness condition of the bilinear form (8) on V(Ω). Then, by the Lax-Milgram lemma, there exists a unique solution y ∈ L²(0,T;V(Ω)) for the system (7). Now, we can formulate the optimal control problem for system (7) as follows:

The space L²(Q) is the space of controls. For a control u ∈ L²(Q), the state y(u) ∈ L²(0,T;V(Ω)) of the system is given by the solution of the following problem:

(10)

\{\begin{matrix} \frac{\partial y (u)}{\partial t} + (- Δ + q) y (u) = g (x) y (u) + f (x, t) + u & in Q, \\ y \to 0 as | x | \to \infty \\ y (u) = 0 & on \sum, \\ y (x, 0, u) = y_{0} (x) & in Ω \end{matrix}

where

y (u) \in L^{2} (0, T; V (Ω)), \frac{\partial y (u)}{\partial t} \in L^{2} (0, T; V^{'} (Ω))

The observation equation is given by z(u) = y(u).

For a given z_d ∈ L²(Q), the cost function is given by

(11)

J (v) = ‖ y (v) - z_{d} ‖_{L^{2} (Q)}^{2} + M ‖ v ‖_{L^{2} (Q)}^{2},

where M is a positive constant.

The control problem is to find u ∈ U_ad such that J(u) ⩽ J(v),

where U_ad is a closed convex subset of L²(Q).

The cost function (11) can be written as was performed by Lions (1971): $J (v) = a (v, v) - 2 L (v) + ‖ y (0) - z_{d} ‖_{L^{2} (Q)}^{2},$ where a(v,v) is a continuous coercive bilinear form and L(v) is a continuous linear form on L²(0,T;V(Ω)). Then using the general theory of Lions (1971), there exists a unique optimal control u ∈ U_ad such that J(u) = inf J(v) for all v ∈ U_ad. Moreover, we have the following proposition that gives the necessary and sufficient conditions of optimality: □

Proposition 2

Assume that (9) holds. If the cost function is given by (11), the optimal control u ∈ L²(Q) is then characterized by:

(12)

\{\begin{matrix} \frac{- \partial p (u)}{\partial t} + (- Δ + q) p (u) - g (x) p (u) = y (u) - z_{d} & in Q, \\ p \to 0 as | x | \to \infty, \\ p (u) = 0 & on \sum, \\ p (x, T, u) = 0 & in Ω \end{matrix}

where

p (u) \in L^{2} (0, T; V (Ω)), \frac{\partial p (u)}{\partial t} \in L^{2} (0, T; V^{'} (Ω))

. Furthermore, we have the inequality

(13)

{(p (u) + Mu, v - u)}_{L^{2} Q} ⩾ 0 \forall v \in U_{ad},

together with (10), where p(u) is the adjoint state.

4 The case of systems

4.1

4.1 Operator equation

In this section, we prove the existence and uniqueness of a solution to system (1), which can be written as follows: $\{\begin{matrix} \frac{\partial y_{i} (x)}{\partial t} + (- Δ + q) y_{i} (x) = \sum_{j = 1}^{n} a_{ij} (x) y_{j} + f_{i} (x, t) & in Q, \\ y_{i} \to 0 as | x | \to \infty, \\ y_{i} (x) = 0 & on \sum, \\ y_{i} (x, 0) = y_{i, 0} (x) \forall i = 1, 2, 3 \dots n . & in Ω, \end{matrix}$ We introduce the continuous bilinear form π(t;y,φ):(V(Ω))ⁿ × (V(Ω))ⁿ → R as follows:

(14)

π (t; y, φ) = \sum_{i = 1}^{n} \int_{Ω} (\nabla y_{i} \nabla φ_{i} + {qy}_{i} φ_{i}) dx - \sum_{j \neq i}^{n} \int_{Ω} a_{ij} (x) y_{j} φ_{i} dx - \sum_{i = 1}^{n} \int_{Ω} a_{ii} (x) y_{i} φ_{i} dx,

Proposition 3

If conditions (2) and (3) hold, then the bilinear form (14) is coercive on (V(Ω))ⁿ if the matrix:

(15)

(Λ_{1}^{+} (a_{ii}) - I) = [\begin{matrix} λ_{1}^{+} (a_{11}) - 1 & - 1 & \dots & - 1 \\ - 1 & λ_{1}^{+} (a_{22}) - 1 & \dots & - 1 \\ : & : & : & ⋮ \\ - 1 & - 1 & \dots & λ_{1}^{+} (a_{nn}) - 1 \end{matrix}]

is a nonsingular M-matrix (15); it is assumed that

Λ_{1}^{+} (a_{ii})

is a diagonal matrix with elements

λ_{1}^{+} (a_{ii})

$λ_{1}^{+} (a_{ii})$ is the principal eigenvalue for the eigenvalue problem (4) when we replace the function g(x) with a_ii (x) in (4).

Proof

$\begin{matrix} π (t; y, y) = \sum_{i = 1}^{n} \int_{Ω} (| \nabla y_{i} |^{2} + q | y_{i} |^{2}) - \sum_{j \neq i}^{n} \int_{Ω} a_{ij} (x) y_{i} y_{j} - \sum_{i = 1}^{n} \int_{Ω} a_{ii} (x) | y_{i} |^{2} \\ π (t; y, y) = \sum_{i = 1}^{n} \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) - \sum_{j \neq i}^{n} \int_{Ω} a_{ij} (x) y_{i} y_{j} - 2 \sum_{i = 1}^{n} \int_{Ω} a_{ii} (x) | y_{i} |^{2} . \end{matrix}$ Consider the following variational characterization of $λ_{1}^{+} (a_{ii})$ :

(16)

λ_{1}^{+} (a_{ii}) \int_{Ω} a_{ii} (x) | y |^{2} ⩽ \int_{Ω} (| \nabla y |^{2} + q | y |^{2}),

By employing this characterization, we obtain the following:

π (t; y, y) ⩾ \sum_{i = 1}^{n} \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) - \sum_{j \neq i}^{n} \int_{Ω} a_{ij} (x) y_{i} y_{j} - 2 \sum_{i = 1}^{n} \frac{1}{λ_{1}^{+} (a_{ii}) + 1} \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) .

Using (3), we obtain the following:

π (t; y, y) ⩾ \sum_{i = 1}^{n} \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) - 2 \sum_{\binom{i = 1}{j = i + 1}}^{n} \int_{Ω} \sqrt{a_{ii} (x) a_{jj} (x)} y_{i} y_{j} - 2 \sum_{i = 1}^{n} \frac{1}{λ_{1}^{+} (a_{ii}) + 1} \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) .

By the Cauchy-Schwartz inequality and (16), we deduce:

\begin{matrix} π (t; y, y) ⩾ \sum_{i = 1}^{n} (1 - \frac{2}{λ_{1}^{+} (a_{ii}) + 1}) \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) \\ - \sum_{\binom{i = 1}{j = i + 1}}^{n} \frac{2}{\sqrt{λ_{1}^{+} (a_{ii}) + 1} \sqrt{λ_{1}^{+} (a_{jj}) + 1}} . \end{matrix}

\begin{matrix} {(\int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}))}^{1 / 2} \cdot {(\int_{Ω} (| \nabla y_{j} |^{2} + (q + a_{jj} (x)) | y_{j} |^{2}))}^{1 / 2} \\ = \sum_{i = 1}^{n} (\frac{λ_{1}^{+} (a_{ii}) - 1}{λ_{1}^{+} (a_{ii}) + 1}) \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) \\ - \sum_{\binom{i = 1}{j = i + 1}}^{n} (\frac{\int_{Ω} | \nabla y_{i} |^{2} + (q + a_{ii}) | y_{i} |^{2}}{λ_{1}^{+} (a_{ii}) + 1} + \frac{\int_{Ω} | \nabla y_{j} |^{2} + (q + a_{jj}) | y_{j} |^{2}}{λ_{1}^{+} (a_{jj}) + 1}) \\ + \sum_{\binom{i = 1}{j = i + 1}}^{n} {(\sqrt{\frac{\int_{Ω} | \nabla y_{i} |^{2} + (q + a_{ii}) | y_{i} |^{2}}{λ_{1}^{+} (a_{ii}) + 1}} - \sqrt{\frac{\int_{Ω} | \nabla y_{j} |^{2} + (q + a_{jj}) | y_{j} |^{2}}{λ_{1}^{+} (a_{jj}) + 1}})}^{2} . \end{matrix}

We then have the following:

\begin{matrix} ⩾ \sum_{i = 1}^{n} (\frac{λ_{1}^{+} (a_{ii}) - 1}{λ_{1}^{+} (a_{ii}) + 1}) \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) \\ - \sum_{\binom{i = 1}{j = i + 1}}^{n} (\frac{\int_{Ω} | \nabla y_{i} |^{2} + (q + a_{ii}) | y_{i} |^{2}}{λ_{1}^{+} (a_{ii}) + 1} + \frac{\int_{Ω} | \nabla y_{j} |^{2} + (q + a_{jj}) | y_{j} |^{2}}{λ_{1}^{+} (a_{jj}) + 1}) \\ = \sum_{i = 1}^{n} (1 - \frac{n + 1}{λ_{1}^{+} (a_{ii}) + 1}) \int_{Ω} (| \nabla y_{i} |^{2} + (q + a_{ii} (x)) | y_{i} |^{2}) . \end{matrix}

From (15), we deduce that

⩾ \sum_{i = 1}^{n} (1 - \frac{n + 1}{λ_{1}^{+} (a_{ii}) + 1}) \int_{Ω} (| \nabla y_{i} |^{2} + q | y_{i} |^{2}) .

Hence

(17)

π (t; y, y) ⩾ C \sum_{i = 1}^{n} ‖ y_{i} ‖_{q}^{2},

which proves the coerciveness condition of the bilinear form (14) on (V(Ω))ⁿ. Thus, by the Lax-Milgram lemma, there exists a unique solution y = {y₁,y₂,…,y_n} ∈ (L²(0,T;V(Ω)))ⁿ such that:

(\frac{\partial y}{\partial t}, φ) + π (t; y, φ) = L (φ) \forall φ \in {(L^{2} (0, T; V (Ω)))}^{n},

where L(φ) is a continuous linear form on (L²(0,T;V(Ω)))ⁿ that takes the following form:

L (φ) = \sum_{i = 1}^{n} \int_{Q} f_{i} (x, t) φ_{i} (x) dxdt + \sum_{i = 1}^{n} \int_{Ω} y_{i, 0} (x) φ_{i} (x, 0) dx,

□

As a result, we have the following theorem:

Theorem 1

Under hypotheses (2), (3) and (15), for a given f = {f₁,f₂,…, f_n} ∈ (L²(0,T;V′(Ω)))ⁿ and y_i,0(x) ∈ V(Ω), there exists a unique solution y = {y₁,y₂,…,y_n} ∈ (L²(0,T;V(Ω)))ⁿ to system (1).

4.2

4.2 The control problem

In this section, using the theory of Lions (1971), we discuss the existence and characterization of the optimal control for system (1).

The space (L²(Q))ⁿ is the space of controls. For a control u = {u₁,u₂,…,u_n} ∈ (L²(Q))ⁿ, the state y(u) = {y₁(u),y₂(u),…,y_n(u)} ∈ (L²(0,T;V(Ω)))ⁿ of system (1) is given by the solution of:

(18)

\{\begin{matrix} \frac{\partial y_{i} (u)}{\partial t} + (- Δ + q) y_{i} (u) = \sum_{j = 1}^{n} a_{ij} (x) y_{j} (u) + f_{i} (x, t) + u_{i} & in Q, \\ y_{i} \to 0 as | x | \to \infty, \\ y_{i} (u) = 0 & on \sum, \\ y_{i} (x, 0, u) = y_{i, 0} (u) \forall i = 1, 2, 3, \dots, n & in Ω, \end{matrix}

with

y (u) \in {(L^{2} (0, T; V (Ω)))}^{n}, \frac{\partial y (u)}{\partial t} \in {(L^{2} (0, T; V^{'} (Ω)))}^{n}

The observation equation is given by z(u) = {z₁(u),z₂(u),…,z_n(u)} = y(u) = {y₁(u),y₂ (u),…,y_n(u)}.

For a given z_d = {z_d1,z_d2,…,z_dn} in (L²(Q))ⁿ, the cost function is given by:

(19)

J (v) = \sum_{i = 1}^{n} \int_{(0, T)} {(y_{i} (v) - z_{di}, y_{i} (v) - z_{di})}_{L_{g}^{2} (Ω Ω)} dt + M ‖ v ‖_{{(L^{2} (Q))}^{n}}^{2}, where M is a positive constant.

Thus, the control problem is to find inf J(v) over a closed convex subset U_ad of (L²(Q))ⁿ.

The cost function (19) can be written as in Lions (1971): $J (v) = a (v, v) - 2 L (v) + ‖ y (0) - z_{d} ‖_{{(L^{2} (Q))}^{n}}^{2},$ where a(v,v) is a continuous coercive bilinear form and L(v) is a continuous linear form on (L²(0,T;V(Ω)))ⁿ. Then using the general theory of Lions (1971), there exists a unique optimal control u ∈ U_ad such that J(u) = inf J(v) for all v ∈ U_ad. Moreover, we have the following theorem which gives the necessary and sufficient conditions of optimality.

Theorem 2

Assume that (2), (3) and (15) hold. If the cost function is given by (19), there exists a unique optimal control u = {u₁,u₂, … u_n} ∈ (L²(Q))ⁿ such that J(u) ⩽ J(v) ∀v ∈ U_ad.

Moreover, this control is characterized by the following equations and inequalities:

(20)

\{\begin{matrix} \frac{- \partial p_{i} (u)}{\partial t} + (- Δ + q) p_{i} (u) - \sum_{j = 1}^{n} a_{ji} (x) p_{j} (u) = y_{i} (u) - z_{di} & in Q, \\ p_{i} \to 0 as | x | \to \infty, \\ p_{i} (u) = 0 & on \sum, \\ p_{i} (x, T, u) = 0 \forall i = 1, 2, 3, \dots, n & in Ω, \end{matrix}

where

p (u) \in {(L^{2} (0, T; V (Ω)))}^{n}, \frac{\partial p (u)}{\partial t} \in {(L^{2} (0, T; V^{'} (Ω)))}^{n}

(21)

\sum_{i = 1}^{n} \int_{(0, T)} {(p_{i} (u), v_{i} - u_{i})}_{L_{g}^{2} (Ω)} dt + M {(u, v - u)}_{{(L^{2} (Q))}^{n}} ⩾ 0 \forall v = (v_{1}, v_{2}, \dots v_{n}) \in U_{ad},

The above equation can be combined with (18), where p(u) = {p₁(u),p₂ (u), … p_n(u)} is the adjoint state.

Proof

The optimal control u = (u₁,u₂, … u_n) ∈ (L²(Q))ⁿ is characterized by Lions (1971): $\sum_{i = 1}^{n} J^{'} (u) (v_{i} - u_{i}) ⩾ 0 \forall v = (v_{1}, v_{2}, \dots, v_{n}) in U_{ad},$ which is equivalent to $\sum_{i = 1}^{n} {(y_{i} (u) - z_{di}, y_{i} (v) - y_{i} (u))}_{L^{2} (Q)} + M {(u, v - u)}_{{(L^{2} (Q))}^{n}} ⩾ 0 .$ This inequality can be written as follows:

(22)

\sum_{i = 1}^{n} \int_{(0, T)} {(y_{i} (u) - z_{di}, y_{i} (v) - y_{i} (u))}_{L_{g}^{2} (Ω)} dt + M {(u, v - u)}_{{(L^{2} (Q))}^{n}} ⩾ 0 .

Now,

{(p, Ay)}_{{(L^{2} (Q))}^{n}} = \sum_{i = 1}^{n} \int_{(0, T)} {(p_{i} (u), \frac{\partial y_{i} (u)}{\partial t} + (- Δ + q) y_{i} (u) - \sum_{j = 1}^{n} a_{ij} (x) y_{j} (u))}_{L_{g}^{2} (Ω)} dt,

where

\begin{matrix} Ay (u) = A {y_{1} (u), y_{2} (u), \dots, y_{n} (u)} = \{\frac{\partial y_{1} (u)}{\partial t} + (- Δ + q) y_{1} (u) - \sum_{j = 1}^{n} a_{1 j} (x) y_{j} (u), \\ \frac{\partial y_{2} (u)}{\partial t} + (- Δ + q) y_{2} (u) - \sum_{j = 1}^{n} a_{2 j} (x) y_{j} (u), \\ ⋮ \\ \frac{\partial y_{n} (u)}{\partial t} + (- Δ + q) y_{n} (u) - \sum_{j = 1}^{n} a_{nj} (x) y_{j} (u)\} . \end{matrix}

By using Green’s formula

{(p, Ay)}_{{(L^{2} (Q))}^{n}} = \sum_{i = 1}^{n} \int_{(0, T)} {(\frac{- \partial p_{i} (u)}{\partial t} + (- Δ + q) p_{i} (u) - \sum_{j = 1}^{n} a_{ji} (x) p_{j} (u), y_{i})}_{L_{g}^{2} (Ω)} dt = {(A^{*} p, y)}_{{(L^{2} (Q))}^{n}} .

Hence, we have

(23)

\begin{matrix} A^{*} p (u) = A^{*} {p_{1} (u), p_{2} (u), \dots p_{n} (u)} \\ = \{\frac{- \partial p_{1} (u)}{\partial t} + (- Δ + q) p_{1} (u) - \sum_{j = 1}^{n} a_{j 1} (x) p_{j} (u), \\ \frac{- \partial p_{2} (u)}{\partial t} + (- Δ + q) p_{2} (u) - \sum_{j = 1}^{n} a_{j 2} (x) p_{j} (u), \\ ⋮ \\ \frac{- \partial p_{n} (u)}{\partial t} + (- Δ + q) p_{n} (u) - \sum_{j = 1}^{n} a_{jn} (x) p_{j} (u)\} . \end{matrix}

inasmuch as the adjoint equation takes the following form:

\frac{- \partial p (u)}{\partial t} + A^{*} p (u) = y (u) - z_{d} .

Therefore, from Theorem 1, we obtain a unique solution that satisfies p(u) ∈ (L²(0,T;V(Ω)))ⁿ.

This result proves Eq. (20).

Now, Eq. (22) can be written as: $\sum_{i = 1}^{n} \int_{(0, T)} {(\frac{- \partial p_{i} (u)}{\partial t} + (- Δ + q) p_{i} (u) - \sum_{j = 1}^{n} a_{ji} (x) p_{j} (u), y_{i} (v) - y_{i} (u))}_{L_{g}^{2} (Ω)} dt + M {(u, v - u)}_{{(L^{2} (Q))}^{n}} ⩾ 0 .$ Using Green’s formula, we obtain: $\sum_{i = 1}^{n} \int_{(0, T)} {(p_{i} (u), (\frac{\partial}{\partial t} + (- Δ + q) - \sum_{j = 1}^{n} a_{ij}) y_{i} (v) - y_{i} (u))}_{L_{g}^{2} (Ω)} dt + M {(u, v - u)}_{{(L^{2} (Q))}^{n}} ⩾ 0 .$ Furthermore, using (18), we have that $\sum_{i = 1}^{n} \int_{(0, T)} {(p_{i} (u), v_{i} - u_{i})}_{L_{g}^{2} (Ω)} dt + M {(u, v - u)}_{{(L^{2} (Q))}^{n}} ⩾ 0,$ which proves (21). □

Remarks 1

If q = 0 in system (1), we have some existence results for the elliptic operator that were discussed in Serag (1998b).
If q = 0 and n = 2 in system (1), we obtain some results for the elliptic operator that were introduced in Serag (1998a).
If a_ij(x) = g(x) in system (1), we obtain some existence results for the elliptic operator that were obtained in Serag and Qamlo (2008).
If a_ij(x) = g(x) and n = 2 in system (1), we obtain some results for the elliptic operator that were obtained in Gali and Serag (1995).

References

Bahaa G.M., . Boundary control for cooperative parabolic systems governed by Schrodinger operator. Differ. Equ. Control Processes. 2006;1:79-88.
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Bahaa G.M., El-Shatery F., . Optimal control for n × n coupled parabolic systems with control-constrained and infinite number of variables. Int. J. Math. Computat.. 2013;19(2):106-118.
[Google Scholar]
Bermann A., Plemmons R.J., . Nonnegative Matrices in the Mathematical Sciences. New York: Academic Press; 1979.
Djellit A., Yechoui A., . Existence and non-existence of a principal eigenvalue for some boundary value problems. Maghreb Math. Rev.. 1997;6:29-37.
[Google Scholar]
Gali I.M., Serag H.M., . Optimal control of cooperative elliptic systems defined on Rⁿ. J. Egypt. Math. Soc.. 1995;3:33-39.
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Kowalewski A., . Time-optimal control of infinite order hyperbolic systems with time delays. Int. J. Appl. Math. Comput. Sci.. 2009;19(4):597-608.
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Kotarski W., El-Saify H.A., Bahaa G.M., . Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay. IMA J. Math. Control Inf.. 2002;19:461-476.
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Kowalewski A., Krakowiak A., . Time-optimal boundary control of an infinite order parabolic system with time lags. Int. J. Appl. Math. Comput. Sci.. 2008;18(2):189-198.
[Google Scholar]
Lions J.L., . Optimal Control of Systems Governed by Partial Differential Equations. Vol vol. 170. Springer–Verlag; 1971.
Qamlo A.H., . Boundary control for cooperative systems involving parabolic operators with an infinite number of variables. Adv. Differ. Equ. Control Processes. 2008;2(2):135-151.
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Qamlo A.H., . Distributed control of cooperative systems involving hyperbolic operators with an infinite number of variables. Int. J. Funct. Anal. Oper. Theory Appl.. 2009;1(2):115-128.
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Qamlo A.H., . Distributed control for cooperative hyperbolic systems involving Schrödinger operator. Int. J. Dyn. Control. 2013;1:54-59.
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Serag H.M., . On optimal control for elliptic system with variable coefficients. Rev. Mat. Apl.. 1998;19:37-41.
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Serag H.M., . On maximum principle and existence of solutions for N × N elliptic systems with variable coefficients. J. Egypt. Math. Soc.. 1998;6(1):91-97.
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Serag H.M., . Distributed control for cooperative systems governed by Schrödinger operator. J. Discrete Math. Sci. Cryptogr.. 2000;3(1–3):227-234.
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Serag H.M., . Optimal control of systems involving Schrödinger operators. Int. J. Control Intell. Sys.. 2004;32(3):154-157.
[Google Scholar]
Serag H.M., . Distributed control for cooperative systems involving parabolic operators with an infinite number of variables. IMA J. Math. Control Inf.. 2007;24(2):149-161.
[Google Scholar]
Serag H.M., Qamlo A.H., . On elliptic systems involving Schrödinger operators. Mediterranean J. Meas. Control. 2005;1(2):91-96.
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Introduction

Some facts and results

The scalar case

The case of systems

Operator equation
The control problem

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[1] Bahaa G.M., . Boundary control for cooperative parabolic systems governed by Schrodinger operator. Differ. Equ. Control Processes. 2006;1:79-88.
[Google Scholar]

[2] Bahaa G.M., El-Shatery F., . Optimal control for n × n coupled parabolic systems with control-constrained and infinite number of variables. Int. J. Math. Computat.. 2013;19(2):106-118.
[Google Scholar]

[3] Bermann A., Plemmons R.J., . Nonnegative Matrices in the Mathematical Sciences. New York: Academic Press; 1979.

[4] Djellit A., Yechoui A., . Existence and non-existence of a principal eigenvalue for some boundary value problems. Maghreb Math. Rev.. 1997;6:29-37.
[Google Scholar]

[5] Gali I.M., Serag H.M., . Optimal control of cooperative elliptic systems defined on Rⁿ. J. Egypt. Math. Soc.. 1995;3:33-39.
[Google Scholar]

[6] Kowalewski A., . Time-optimal control of infinite order hyperbolic systems with time delays. Int. J. Appl. Math. Comput. Sci.. 2009;19(4):597-608.
[Google Scholar]

[7] Kotarski W., El-Saify H.A., Bahaa G.M., . Optimal control of parabolic equation with an infinite number of variables for non-standard functional and time delay. IMA J. Math. Control Inf.. 2002;19:461-476.
[Google Scholar]

[8] Kowalewski A., Krakowiak A., . Time-optimal boundary control of an infinite order parabolic system with time lags. Int. J. Appl. Math. Comput. Sci.. 2008;18(2):189-198.
[Google Scholar]

[9] Lions J.L., . Optimal Control of Systems Governed by Partial Differential Equations. Vol vol. 170. Springer–Verlag; 1971.

[10] Qamlo A.H., . Boundary control for cooperative systems involving parabolic operators with an infinite number of variables. Adv. Differ. Equ. Control Processes. 2008;2(2):135-151.
[Google Scholar]

[11] Qamlo A.H., . Distributed control of cooperative systems involving hyperbolic operators with an infinite number of variables. Int. J. Funct. Anal. Oper. Theory Appl.. 2009;1(2):115-128.
[Google Scholar]

[12] Qamlo A.H., . Distributed control for cooperative hyperbolic systems involving Schrödinger operator. Int. J. Dyn. Control. 2013;1:54-59.
[Google Scholar]

[13] Serag H.M., . On optimal control for elliptic system with variable coefficients. Rev. Mat. Apl.. 1998;19:37-41.
[Google Scholar]

[14] Serag H.M., . On maximum principle and existence of solutions for N × N elliptic systems with variable coefficients. J. Egypt. Math. Soc.. 1998;6(1):91-97.
[Google Scholar]

[15] Serag H.M., . Distributed control for cooperative systems governed by Schrödinger operator. J. Discrete Math. Sci. Cryptogr.. 2000;3(1–3):227-234.
[Google Scholar]

[16] Serag H.M., . Optimal control of systems involving Schrödinger operators. Int. J. Control Intell. Sys.. 2004;32(3):154-157.
[Google Scholar]

[17] Serag H.M., . Distributed control for cooperative systems involving parabolic operators with an infinite number of variables. IMA J. Math. Control Inf.. 2007;24(2):149-161.
[Google Scholar]

[18] Serag H.M., Qamlo A.H., . On elliptic systems involving Schrödinger operators. Mediterranean J. Meas. Control. 2005;1(2):91-96.
[Google Scholar]

[19] Serag H.M., Qamlo A.H., . Maximum principle and existence of solutions for non necessarily cooperative systems involving Schrödinger operators. Math. Slovaca. 2008;58(5):567-580.
[Google Scholar]