** **

** **

**Control Operator for the Two-Dimensional Energized Wave Equation**

** **

Victor Onomza WAZIRI^{*}, Sunday
Augustus REJU

*Mathematics/Computer
Science Department, Federal University of Technology, Minna 920003, Niger
State, **Nigeria**; ^{*}corresponding
author*

*National
Open University of **Nigeria**, **Victoria Island**, **Lagos**, **Nigeria*

dronomzawaziri@yahoo.com, sreju@nou.edu.ng

**Abstract**

This paper studies the analytical model for the construction of the two-dimensional Energized wave equation. The control operator is given in term of space and time t independent variables. The integral quadratic objective cost functional is subject to the constraint of two-dimensional Energized diffusion, Heat and a source. The operator that shall be obtained extends the Conjugate Gradient method (ECGM) as developed by Hestenes et al (1952, [1]). The new operator enables the computation of the penalty cost, optimal controls and state trajectories of the two-dimensional energized wave equation when apply to the Conjugate Gradient methods in (Waziri & Reju, LEJPT & LJS, Issues 9, 2006, [2-4]) to appear in this series.

**Keywords**

Conjugate Gradient method (CGM), Extended Conjugate Gradient Method (ECGM), Convolution integral, Control operator, Hilbert space, Integral quadratic objective function

**Introduction**

The energized wave equation is a euphemism of wave with diffusion and source effects. The constraint equation is generally a combination of wave and heat (energy) with a source. It has the understated physical configuration:

_{} (1)

where *c ^{2}* and

_{} (2)

while the energy or diffusion effect configuration is:

_{} (3)

The term *u(x,y,t)* is the source (negative
source), which controls some inflows at some control demand; say into the
medium of propagation.

This research is geared towards obtaining the operator that would enable complete the stability, controllability and observability of the dynamical system of equation (1).

In this work, results retain their partial differential forms and we intent to get them solved analytically in their time and space variables. In other words, the control operator retains three independent variables (time t and the two space variables x and y).

**Modelling the Problem**

** **

*Problem P1*

Our optimization problem P1 is modelled in according with the formulation of Waziri [5]. The model of the two-dimensional energized wave equation is:

_{} (4)

subject to the dynamical constraint:

_{} (5)

with initial and boundary conditions:

_{}

These initial conditions shall for usefulness in the construction of the inverse Laplace transform in the derivation of the ECGM control operator.

** **

*Problem P2*

To optimize the integral quadratic
problem P1, we have to construct an unconstrained equation by penalizing the
dynamical constraint equation (4) with a cost functional *μ(x,y,t)
≥ 0*. The unconstraint equation is defined hereunder as:

_{} (6)

Equation (6) in bilinear Hermitian form yields:

* *

*Problem P3*

** _{} **(7)

where *w = (w _{1}, w_{2})* and equation (7) denotes an inner product space expressible as:

_{} (8)

It is important to note that the control operator associated with the penalized functional (8) is equivalent to that associated with its Bilinear Hermitian form (7); and equivalence is preserved under the following equivalent laws:

*z _{1} = z_{2} = z; z_{1t}
= z_{2t} = z_{t}; z_{1tt} = z_{2tt} = z_{tt};
z_{1xx} = z_{2xx} = z_{xx}; z_{1yy} = z_{2yy}
= z_{y}; u_{1} = u_{2} = u*

The Bilinear Hermitian form (7) is conveniently written compactly as:

_{} (9)

The H appearing in equation (9)
symbolizes a Hilbert space which is defined as: *H = w _{2}^{2}[0,1][0,1]*l_{2}^{2}[0,1][0,1]*,
where “*” is a multiplicative operator. The term

The operator *A* in equation (9)
is associated with equation (8) which is a squared symmetric positive definite,
linear inner product operator defined equivalently:

_{}

In the computational processes, the
control operator A is not fixable for our integral quadratic objective
functional posed by problem *P1*. We need a new operator that would be
compatible to our problem. Hence, we must redesign the new control operator in
line of the development from [6]. Let the new control operator be recognized as
B which has the same Hilbert space property as defined on the control operator
A and satisfies the Bilinear Hermitian form.

**Derivation of
the ECGM Fundamental First Fourth Order PDE**

** **

In constructing the new control operator (otherwise known as the ECGM operator that would replace the old operator A), we replace A with B in equation (8) such that:

_{} (10)

In a more compact format, equation (10) can be rewritten as:

_{} (11)

We obtain *B _{11}* and

_{}

Thus with u_{2} = 0,
equation (10) reads:

_{} (12)

Upon simplifying, equation (12), we have:

_{} (13)

On the strength of Gelfand and Fomin Lemma ([7], which we reproduce here in term of our dimensional case), equation (13) can be solved.

** **

*Lemma (Gelfand-Fomin)
*

If *φ(x,y,t)* and *ψ(x,y,t)*
are continuous functions in closed region *[a,b][c,d]*:

_{} (14)

In view of Lemma, reconsider equation (14) such that:

γ = z_{2}; α = μz_{2tt}
+ μz_{2t} - μz_{2xx} - μz_{2yy}; β
= -α

We rewrite equation (14) and with
the boundary assumption *[x,y,t] = [1,1,1]* as inherited from the posed
problem *P1*:

_{} (15)

From equation (14), set:

_{} (16)

Invoking the Gelfand-Fomin (Lemma,
ibid [7]), we must therefore have that the two embedded factors in equation (15);
(α-_{}),
(β-_{}),
and (β-_{})
are continuous functions on the interval [0,x][0,y] and are continuously
differentiable on the interval [0,1][0,1]; also they are equipped with a normed
space such that:

_{}

where *y ^{(n)}* is the n-th differential
of y = y(ζ) = [x,y,t] Î [0,1][0,1]

Thus (α-_{}), (β-_{}), and
(β-_{})
are continuously partial differentiable functions such that:

_{} (17)

Hence (17):

_{} (18)

Equation (18) resolves into the form:

_{} (19)

But α = - β by symmetric definition; therefore equation (19) is rewritten as:

_{} (20)

which is expressible uniquely as:

_{} (21)

Thus, it is not difficult to see that equation (21) yields:

_{} (22)

*The
derivation of the ECGM fundamental second fourth order PDE*

The second ECGM fourth order partial
differential equation is obtained by setting *z _{2} = 0* in
equation (13) such that:

_{} (23)

By putting z_{2} = 0, equation (10) becomes:

_{} (24)

Let α - μu_{2tt}(x,y,t) and β - μu_{2yy}(x,y,t)
be continuous and continuously differentiable over the space region [0,1][0,1].
Therefore invoking the Gelfand-Fomin algorithm and after some due analytical
simplification processes as in previous section, we obtained the second fourth
order partial differential equation:

_{} (25)

From equation (23), we deduced the element operator *B _{12}*:

B_{12} = (1+μ)u_{2}(x,y,t) (26)

We will in the next section solve
equations (22) and (26) using the initial and boundary conditions as inherited
from problem *P1*.

** **

*Solving the
first and second fourth order PDE*

We now solve the fourth order
partial differential equation (22). By taking the Laplace transform of equation (22) with respect to t-variable and
considering the initial conditions as stipulated from the original problem *P1*,
we have the transformation:

_{} (27)

Taking the Laplace transform further with respect to *y* and *x* variables,
equation (27) becomes:

_{} (28)

Considering the initial and boundary conditions, equation (5.2) is expressible as:

_{} (29)

Dividing all terms in equation (29)
by λ^{2}ρ^{4}s^{4} yields:

_{} (30)

The inverse Laplace transforms with respect to each of the energized wave independent space variables and time t are considered. Nonetheless, worthy to consider are some vital propositions credited to (Reju, 1995 [8]); shall form useful tools in the derivation of the inverse Laplace transform of equation (30).

The prepositions:

·
*Preposition 1*

_{}

·
*Preposition 2*

_{}

·
*Preposition 3*

_{}

·
*Preposition 4*

_{}

Using the four stated preposition
appropriately, the inverse Laplace transform to equation (30), and then differentiating
equation with respect to t-variable and simplifying results, we obtain the
element control operator *B _{11}* defined hereunder (

_{} (31)

_{} (31)

**Solution of
the Second Fourth Order PDE**

** **

Reconsider equation (25) and taking the Laplace transform of each independent variables x, y and t of equation (25) and differentiating the result with respect to t, yields:

_{} (32)

By re-arranging equation (32) component-wise
for the control operator *B _{22}*, we obtained:

_{} (33)

**Summary for the
Control Operator**

** **

We give summary of the analytical control operator hereunder:

_{} (34)

wherein *w _{2} = (z_{2}, u_{2})*.

The summary of the control operator
components are obtainable by referring to the following listed equations.
Equations (16), (26), (31) and (33) constitute the control operator when
substituted into equation (34). The substituted components completely define
the control operator that would replace the analytical operator *A* in the
implementation of the ECGM algorithm. In fact, the control operator *B*
differs remarkably from the operator *A* obtained in section *Modelling
the Problem* in that while the former is completely expressed in matrix term
of state variables and time, the latter is a matrix constant.

** **

** **

**Conclusion**

** **

The control operator has been derived for the two-dimensional energized wave equation. The operator shall form the framework in the derivation of the optimal control and state trajectories numerical values using the conjugate gradient algorithm as formulated.

**References**

** **

[1] Hestenes M. R., Stiefel E,
*Method of conjugate Gradient Method for solving linear systems*, J. Res.
Nat. Bureau Standards, 49, p. 409-36, 1952.

[2] Waziri V. O., Reju S.
A., *Implementation of the Extended Conjugate Gradient Method for the
Two-Dimensional Energized Wave equation,* Leonardo Electronic Journal of Practices
and Technologies, Romania; ISSN 1583-1078, Issue 9, July-December, p. 33-42, 2006.

[3] Waziri V. O., Reju S. A*, **The
Analysis of the Two-dimensional Diffusion Equation With a Source*, Leonardo Electronic
Journal of Practices and Technologies, Romania, ISSN 1583-1078. Issue 9,
July-December, p.43-54, 2006.

[4] Waziri V. O., Reju S. A., *The Penalty
Cost Functional for the Two-dimensional Energized Wave Equation*. Leonardo
Journal of Science, Romania, ISSN 1583-0233, Issue 9, July-December, p. 45-52,
2006.

[5] Waziri, V. O., *Optimal
Control of Energized Wave equations using the Extended Conjugate Gradient
Method (ECGM)*, Ph.D. Thesis, Federal Univ. of Technology, Minna, Nigeria, 2004.

[6] Ibiejugba M. A., Onumanyi
P., *On control operator and some of its applications*, J. Math. Analy. Applic.,
103, p. 31-7, 1984.

[7] Gelfand I. M., Fomin S. F.,
*Calculus of Variations*, Prentice Hall Inc., Eaglewood Cliffs, New Jersey,
1963.

[8] Reju S. A., *Computational
Optimization in Mathematical Physics*, Ph.D. Thesis, Univ. of Ilorin, Ilorin, Nigeria, 1995.