3_590_tomar.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 2, 2011, 117-128

ISSN 1307-5543 – www.ejpam.com

Hybridization of Neural Nets and Genetic Algorithms to

Compute the Boundary Control for Controlled Heat Equation

N. K. Tomar1∗, N. Sukavanam2, K. P. Singh3

1 Department of Mathematics, Indian Institute of Technology Patna-800013, India
2 Department of Mathematics, Indian Institute of Technology Roorkee-247667, India
3 Indian Institute of Information Technology Allahabad-211012, India

Abstract. This paper presents a computational method for boundary control of controlled heat equa-

tion. The proposed method relies upon the function approximation capability of feedforward neural

network and global optimization power of genetic algorithm.

2000 Mathematics Subject Classifications: 93C20, 92B20, 90C90

Key Words and Phrases: Control system governed by heat equation, Artificial Neural Networks, Ge-

netic Algorithms

1. Introduction

Let Ω be a nonempty, bounded, open and connected domain in Rn with a smooth boundary

Γ. Consider the time interval (0,τ) and the sets Q = Ω× (0,τ) and
∑
= Γ × (0,τ). The

controlled semilinear heat equation, with boundary control, in Q is

∂ ω(x , t)

∂ t
= c2∆ω(x , t) +φ(t,ω(x , t)); in Q

ω(x , 0) = ω0; in Ω

ω = C(σ, t); on Γ (1)

where, ω(x , t) denotes the temperature at x ∈ Ω and t ∈ [0,τ], ∆ denotes the Laplace oper-

ator, φ is a linear or nonlinear operator and c is a real constant. C(σ, t) represents boundary

control function, σ being boundary variable.

Now, the problem is to find a boundary control function C on Γ such that the solution ω

satisfies ω(x ,τ) =ωτ on Ω, where ωτ is any final required temperature.

∗Corresponding author.

Email addresses: nktomar�iitp.a
.in (N. Tomar), nsukvfma�iitr.ernet.in (N. Sukavanam),kpsinghiitr�gmail.
om (K. Singh)

http://www.ejpam.com 117 c© 2011 EJPAM All rights reserved.



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 118

Studies on the controlled heat equation have revealed many theoretical concepts for different

types of controllability problems, see for instance [19, 2, 9, 4, 14, 6, 17, 8, 16]. But there

is a gap in the literature for numerical solution for boundary control, which is one of the

important issues in the context of controlled heat equation model. The numerical computa-

tion of boundary control of a two dimensional linear heat equation has been shown in [2]

using finite difference, finite element and conjugate gradient method. In this paper, a method

for computation of boundary control for mathematical model (1), under the given initial, fi-

nal and boundary conditions, is proposed using developing less conventional Artificial Neural

Network (ANN) and Genetic Algorithm (GA) techniques.

In [11, 12] Neural Networks have been used for solving ordinary and partial differential equa-

tions with known initial and boundary conditions. Sukavanam and Panwar [15] presented a

numerical solution of boundary control for controlled heat equation using ANN. In all these

thee papers [11, 12, 15], the back propagation algorithm is applied to train a feed forward

neural network, which is based on steepest descent method.

In this paper a similar technique as that of [11, 12, 15] is used to approximate the state and

control terms, that is possible due to the capability of a feed forward neural network. After the

approximation, above problem is converted into an optimization problem, and then for result-

ing problem, a real coded GA is applied. Use of evolutionary techniques in control problems

is not an new idea. For example, Michalewicz et. at. [13] gave applications of genetic algo-

rithm to study some discrete-time optimal control problems. In the same year, Krishnakumar

and Goldberg [10] explored aerospace-related control system optimization problems using

genetic algorithms.

Chang [3] applied a real-coded genetic algorithm to the system identification and control for

a class of nonlinear systems. Some more applications of evolutionary algorithms in control

systems can be seen in [18, 1, 7].

The organization of this paper is as follows. Next section is concerned with some prelimi-

naries of a feedforward neural network and its derivatives formulations, which will be useful

in Section 3. Using ANN and GA, the proposed hybrid procedure is described in the Section

3. Numerical example, for two dimensional linear heat equation, illustrates the proposed

method in the Section 4. Finally, in the last, some concluding remarks are given.

2. Preliminaries

Consider a multilayer feedforward neural network, with n input units, one hidden layer

with H sigmoid units and a linear output unit. For a given input vector x , where

x = (x1, x2, . . . , xn), the output of the network is N(x) =
∑H

i=1 viσ(zi), where

zi =
∑n

j=1(wi j x j + bi). Here, wi j denotes the weight from the input unit j to the hidden unit

i , vi denotes the weight from the hidden unit i to the output, bi denotes the bias of hidden

unit i. σ(z) = 1/(1+ e−z) is the sigmoid transfer function. It is straightforward to show that

∂ kN

∂ x k
j

=

H∑

i=1

viw
k
i jσ

k
i



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 119

where σi = σ(zi) and σ(k) denote the kth-order derivative of the sigmoid function. Moreover,

∂ λ1

∂ x
λ1

1

·
∂ λ2

∂ x
λ2

2

. . .
∂ λn

∂ x
λn
n

N =

H∑

i=1

viρiσ
α
i (2)

where ρi =
∏n

k=1 w
λk

ik
and α =
∑n

i=1λi . Equation (2) indicates that the derivative of the

network N with respect to any of its inputs is equivalent to a feedforward neural network

with one hidden layer, having the same values for the weights, as N , wi j and thresholds bi

and each vi being replaced with viρi. The transfer function of each hidden unit is replaced

with the αth order derivative of the sigmoid.

One of the main uses of the feed forward neural network is the approximation of any function.

In the current effort, a feed forward neural network with three input units, one hidden layer

having six nodes (with bias at each node) and one output unit is used to approximate both

the temperature and the control functions separately.

3. Method of Computation of Boundary Control

First, we assume the discretization of domain Q = Ω× (0,τ) and its boundary∑
= Γ× (0,τ) into a set of points Q̂ = Ω× (t0 = 0, t1, t2, . . . , tk = τ) and∑̂
= Γ× (t0 = 0, t1, t2, . . . , tk = τ) where Ω and Γ denote the set of grid points in Ω and Γ

respectively and t0, t1, t2, . . . , tk are discrete points of the interval [0, τ]. Now the problem

(1) is transformed into the solution of following system of equations

∂ω(x , t)

∂ t
− c2∆ω(x , t)−φ ( t,ω(x , t)) = 0; ∀(x , t) ∈ Q̂ (3)

subject to the given initial and final conditions.

In the proposed approach, we construct a trial solution with adjustable parameters. This

trial solution is written as a sum of two parts. The first part is constructed to satisfy, the

known initial and final conditions, and the unknown boundary conditions in the form of a

feedforward neural network. The second part is also a feedforward neural network with

adjustable weights. It is constructed in a way so as not to affect the given conditions. If

ωtr(x , t) denotes the trial solution with adjustable weight vector p then it is written as

ωtr(x , t, p) = A(x , t, N1(x , t, p1)) + B(x , t, N2(x , t, p2)) (4)

where (x , t) ∈ Q̂ and p = [p1, p2].

The control term is approximated by an another feedforward neural network. A general de-

scription of the network approximating the boundary control function is not easy. It is clear

from the example given below that the form of network depends on the shape of boundary.

Now the problem of solving the system of equation (3) is transformed to the following mini-

mization problem

E =min
p

∑

(x ,t)∈Q̂

{
∂ωtr(x , t, p)

∂ t
− c2∆ωtr(x , t, p)−φ(t,ωtr(x , t, p))}2 (5)



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 120

The weight parameters are found by minimizing E[(p1, p2)] using GA.

In this study, the real coded genetic algorithm, MI-LXPM, for details see [5], has been used

to solve above single objective optimization problems. Computational steps of MI-LXPM algo-

rithm are as follows:

Step-1: Generate a suitably large initial set of random points within the domain, it may be 5 to

10 times of number of variables. Evaluate their fitness values.

Step-2: Check the stopping criteria? If satisfied stop else goto step 3.

Step-3: Apply tournament selection operator to decide which of these individuals are to be in

mating pool.

Step-4: Apply Laplace crossover operator to all individuals in mating pool with probability of

crossover Pc. Crossover operator produces new solution points in the search space.

Step-5: Apply Power mutation to all individuals in mating pool with probability of mutation Pm.

Mutation operator maintains diversity in the population, so it prevents the algorithm to

converge to local optima.

Step-6: Increase generation; goto step 2.

Parameter setting is one of the important aspect of genetic algorithms. In this study parame-

ters value have been taken as a = 0, b = 0.35, p = 4, Pc = 0.8, Pm = 0.005 with tournament

size three (details of the parameters a, b, and p can be seen in [5]). Moreover, 40,000 gener-

ation has been used as stopping criteria for a run of GA.

4. Numerical Result

Consider the controlled linear two-dimensional heat equation

∂ω(x , t)

∂ t
= ν
∂ 2ω(x , t)

∂ x2
+
∂ 2ω(x , t)

∂ y2
+ 3π3ν exp(2π2ν t)(sin(πx)+ sin(πy)) (6)

with initial condition

ω(x , y, 0) = π(sin(πx)+ sin(πy)) (7)

where t ∈ (0,1), ν = 1

2π2 and (x , y) ∈ Ω = (0,1)× (0,1).

on unit square boundary

Γ = {y = 0,0≤ x ≤ 1; x = 1,0≤ y ≤ 1; y = 1,0≤ x ≤ 1; x = 0,0≤ y ≤ 1}.

Now the problem is to find the unknown control functions, g,h, m, and n, defined on the given

boundary as

ω(0, y, t) = g(y, t)

ω(x , 0, t) = h(x , t)



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 121

ω(1, y, t) = m(y, t)

ω(x , 1, t) = n(x , t) (8)

such that at the final time t = 1 the solution ω(x , y, t) satisfies

ω(x , y, 1) = πexp(2π2ν)(sin(πx)+ sin(πy)) (9)

From equations (6)-(9) the following compatibility conditions on g, h, m and n can be ob-

tained

g(y, 0) = π sin(πy); g(y, 1) = πexp(2π2ν) sin(πy)

h(x , 0) = π sin(πx); h(x , 1) = πexp(2π2ν) sin(πx)

m(y, 0) = π sin(πy); m(y, 1) = πexp(2π2ν) sin(πy)

n(x , 0) = π sin(πx); n(x , 1) = πexp(2π2ν) sin(πx) (10)

Also

g(0, t) = h(0, t); g(1, t) = n(0, t); h(1, t) = m(0, t); m(1, t) = n(1, t) (11)

Hence, we consider the neural network approximation of g, h, m and n as

g(y, t) = [1− t + t exp(2π2ν)]π sin(πy) + t(1− t)[y(1− y)Ng + (1− y)N1 + yN2]

h(x , t) = [1− t + t exp(2π2ν)]π sin(πx)+ t(1− t)[x(1− x)Nh+ (1− x)N1+ xN3]

m(y, t) = [1− t + t exp(2π2ν)]π sin(πy) + t(1− t)[y(1− y)Nm+ (1− y)N3 + yN4]

n(x , t) = [1− t + t exp(2π2ν)]π sin(πx)+ t(1− t)[x(1− x)Nn+ (1− x)N2+ xN4]

(12)

where,

Ng = Ng(y, t, pg ) =
∑

vgσ(bg y + cg t + dg)

Nh = Ng(x , t, ph) =
∑

vhσ(ahx + ch t + dh)

Nm = Nm(y, t, pm) =
∑

vmσ(bm y + cm t + dm)

Nn = Nn(x , t, pn) =
∑

vnσ(an x + cn t + dn)

N1 = N1(t, p1) =
∑

v1σ(c1 t + d1)

N2 = N2(t, p2) =
∑

v2σ(c2 t + d2)

N3 = N3(t, p3) =
∑

v3σ(c3 t + d3)

N4 = N4(t, p4) =
∑

v4σ(c4 t + d4) (13)

where pα = [vα, cα, dα]
T , α = 1,2,3,4; pβ = [vβ , aβ , cβ , dβ]

T , β = h, n and

pγ = [vγ, bγ, cγ, dγ]
T , γ = g, m are the weight vectors, and σ is sigmoid function. The trial

solution of the system (6) with given conditions is



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 122

ωtr(x , y, t) = (1− x)g(y, t) + xm(y, t) + (1− y)h(x , t) + yn(x , t)

−(1− y)(1− x)h(0, t)− x(1− y)h(1, t)− y(1− x)n(0, t)− x yn(1, t)

+(1− t)[I(x , y)− {(1− x)g(y, 0) + xm(y, 0)+ (1− y)h(x , 0) + yn(x , 0)

−(1− y)(1− x)h(0,0)− x(1− y)h(1,0)− y(1− x)n(0,0)− x yn(1,0)}]

+t[F(x , y)− {(1− x)g(y, 1) + xm(y, 1) + (1− y)h(x , 1) + yn(x , 1)

−(1− y)(1− x)h(0,1)− x(1− y)h(1,1)− y(1− x)n(0,1)− x yn(1,1)}]

+t(1− t)x(1− x)y(1− y)Nω(x , y, t, p) (14)

where Nω(x , y, t, p) =
∑

vwσ(aw x + bw y + cw t + dw); p = [vw , aw , bw, cw , dw]
T is the weight

vector; I(x , y) = ω(x , y, 0) is the given initial condition and F(x , y) = ω(x , y, 1) is the final

required condition.

Using (10)-(11) the trial solution becomes

ωtr(x , y, t) = [1− t + t exp(2π2ν)]π(sin(πx)+ sin(πy)) + t(1− t)[(1− x)y(1− y)Ng

+(1− y)x(1− x)Nh+ x y(1− y)Nm+ y x(1− x)Nn+ (1− x)(1− y)N1 + y(1− x)N2

+x(1− y)N3 + x yN4] + t(1− t)x(1− x)y(1− y)Nω. (15)

It can be easily verified that the trial solution satisfies all initial and boundary conditions.

Let us consider the grid points in x , y and t as

x i = (0.1)i; i = 0,1,2, . . . , 9,10

y j = (0.1) j; j = 0,1,2, . . . , 9,10 and

tk = (0.1)k; k = 0,1,2, . . . , 9,10.

Then the error function is given by

E =
∑

i

∑

j

∑

k

[
∂ ωtr(xi ,y j ,tk)

∂ t
− ν{

∂ 2ωtr(xi ,y j ,tk)

∂ x2 +
∂ 2ωtr(xi ,y j ,tk)

∂ y2 }

−3π3ν exp(2π2ν tk)(sin(πx i) + sin(πy j))]
2 (16)

which is to be minimized with respect to all weight parameters p’s. Now, the real coded GA,

MI-LXPM [5], is used to find out the optimal weight parameters.

The exact solution of the above problem is given in [2] as follows: The (optimal) control

function is

πexp(2π2ν t) sin(πx); for 0< x < 1; y = 0,1

and

πexp(2π2ν t) sin(πy); for 0< y < 1; x = 0,1

and the temperature ω being defined by

ω(x , y, t) = πexp(2π2ν t)(sin(πx)+ sin(πy))



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 123

0
0.5

1

0

0.5

1
−0.02

−0.01

0

0.01

ty

er
ro

r

0
0.5

1

0

0.5

1
−0.02

0

0.02

tx

er
ro

r
0

0.5
1

0

0.5

1
−0.01

0

0.01

0.02

ty

er
ro

r

0
0.5

1

0

0.5

1
−0.01

0

0.01

tx
er

ro
r

(a) (b)

(d)(c)Figure 1: Di�eren
e between the exa
t and 
omputed values of 
ontrol at boundary (a)x = 0 (b)y = 0(
)x = 1 (d)y = 1

The figure 1 represents the error between exact and computed control functions at the bound-

ary. The figure 2 and tables 1-6 describe the pointwise error between exact and calculated

temperature distribution on the region Ω at some intermediate time points between 0 and 1.Table 1: Di�eren
e between the Exa
t and Computed values of Temperature Distribution at Time t = 0.1

y x 0.0 0.2 0.4 0.6 0.8 1.0

0.0 -0.00355 -0.00059 0.000221 0.001451 0.001674 -0.001

0.1 -0.00299 -0.00065 -0.00016 0.000301 0.000372 -0.00217

0.2 -0.00342 -0.00112 -0.00024 0.000177 0.000635 -0.0012

0.3 -0.00345 -0.0014 -0.00019 0.000189 0.000911 -0.00049

0.4 -0.00233 -0.00109 -4.6E-05 -6E-05 0.000532 -0.00109

0.5 -0.00124 -0.00085 -1.6E-05 -0.00038 0.000199 -0.00174

0.6 -0.0008 -0.00097 -0.00011 -0.00055 0.000404 -0.00155

0.7 -0.00043 -0.0011 -0.00029 -0.00089 0.00044 -0.00146

0.8 0.000385 -0.00051 0.000356 -0.00039 0.001404 -0.00023

0.9 0.001301 0.000415 0.001326 0.000316 0.002464 0.001163

1.0 0.000683 8.59E-05 0.001065 -0.00036 0.001959 0.000962



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 124

0
0.5

1
0

0.5
1

−5

0

5

x 10
−3

xy

er
ro

r

0
0.5

1
0

0.5
1

−0.01

0

0.01

xy

er
ro

r

0
0.5

1
0

0.5
1

−0.02

0

0.02

xy

er
ro

r

0
0.5

1
0

0.5
1

−0.02

0

0.02

xy

er
ro

r
0

0.5
1

0
0.5

1
−0.02

0

0.02

xy

er
ro

r

0
0.5

1
0

0.5
1

−0.02

0

0.02

xy
er

ro
r

0
0.5

1
0

0.5
1

−0.02

0

0.02

xy

er
ro

r

0
0.5

1
0

0.5
1

−0.01

0

0.01

xy

er
ro

rFigure 2: Di�eren
e between the exa
t and 
omputed values of temperature distribution at time (a)t = 0.1(b)t = 0.2 (
)t = 0.4 (d)t = 0.5 (e)t = 0.6 (f)t = 0.7 (g)t = 0.8 (h)t = 0.9

Table 2: Di�eren
e between the Exa
t and Computed values of Temperature Distribution at Time t = 0.2

y x 0.0 0.2 0.4 0.6 0.8 1.0

0.0 -0.00782 -1.6E-05 0.002893 0.00455 0.003965 -0.00073

0.1 -0.00617 -0.00022 0.001731 0.002088 0.001277 -0.00356

0.2 -0.00636 -0.00119 0.001062 0.0013 0.001247 -0.00251

0.3 -0.0059 -0.00178 0.000751 0.000895 0.00134 -0.00162

0.4 -0.00384 -0.00156 0.000579 0.000132 0.000401 -0.00274

0.5 -0.00179 -0.00144 0.000231 -0.0007 -0.00047 -0.00407

0.6 -0.00034 -0.00139 0.000213 -0.00069 0.000155 -0.00355

0.7 0.001082 -0.00125 0.000164 -0.00085 0.000678 -0.00313

0.8 0.002572 -0.00052 0.000891 -0.00018 0.002219 -0.00134

0.9 0.003587 0.000455 0.002139 0.001059 0.004434 0.001527

1.0 0.002104 -0.00057 0.001414 2.73E-05 0.004024 0.00172



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 125

Table 3: Di�eren
e between the Exa
t and Computed values of Temperature Distribution at Time t = 0.4

y x 0.0 0.2 0.4 0.6 0.8 1.0

0.0 -0.01087 -0.00241 0.000972 0.00394 0.00545 0.003469

0.1 -0.00685 -0.00213 -0.00056 0.000409 0.000656 -0.00354

0.2 -0.00526 -0.00284 -0.00147 -0.00095 -0.00031 -0.00467

0.3 -0.00326 -0.00314 -0.00179 -0.00165 -0.00062 -0.00474

0.4 -0.0005 -0.00317 -0.00218 -0.0028 -0.00202 -0.00631

0.5 0.002364 -0.00313 -0.00248 -0.00369 -0.00333 -0.00805

0.6 0.005879 -0.00184 -0.00105 -0.00222 -0.0017 -0.00675

0.7 0.009317 -0.00028 0.000641 -0.00034 0.000586 -0.00502

0.8 0.010905 8.36E-05 0.001309 0.000589 0.002272 -0.00379

0.9 0.010141 -0.00042 0.00198 0.002111 0.005338 -0.00027

1.0 0.006458 -0.00311 0.000629 0.001301 0.005781 0.000724

Table 4: Di�eren
e between the Exa
t and Computed values of Temperature Distribution at Time t = 0.6

y x 0.0 0.2 0.4 0.6 0.8 1.0

0.0 -0.00862 -0.00099 0.003168 0.006495 0.007819 0.007146

0.1 -0.00523 -0.00147 0.001031 0.002723 0.002478 -0.00131

0.2 -0.00287 -0.00213 -0.00024 0.000856 0.00028 -0.0049

0.3 -0.0004 -0.00258 -0.00103 -0.00026 -0.00075 -0.00647

0.4 0.001772 -0.00339 -0.00223 -0.00184 -0.00223 -0.0081

0.5 0.004502 -0.00357 -0.00277 -0.00281 -0.00322 -0.00903

0.6 0.008888 -0.00166 -0.00092 -0.00111 -0.00148 -0.00706

0.7 0.013116 0.000631 0.001627 0.00164 0.001465 -0.00395

0.8 0.014614 0.000564 0.00184 0.002137 0.002189 -0.00346

0.9 0.013394 -0.00113 0.001187 0.002427 0.003489 -0.00202

1.0 0.010967 -0.00373 -5.8E-05 0.002111 0.004252 -0.00118



N. Tomar, N. Sukavanam, K. Singh / Eur. J. Pure Appl. Math, 4 (2011), 117-128 126

Table 5: Di�eren
e between the Exa
t and Computed values of Temperature Distribution at Time t = 0.8

y x 0.0 0.2 0.4 0.6 0.8 1.0

0.0 -0.00359 -0.00246 -0.00166 -0.00034 0.001379 0.00572

0.1 -0.00107 -0.00189 -0.00185 -0.00114 -0.00046 0.001339

0.2 0.001704 -0.00088 -0.00148 -0.00129 -0.00133 -0.00129

0.3 0.004206 -9.6E-07 -0.001 -0.00117 -0.00164 -0.00268

0.4 0.0059 0.000105 -0.00118 -0.00161 -0.00235 -0.00404

0.5 0.007745 0.000439 -0.00109 -0.00172 -0.00266 -0.00473

0.6 0.010212 0.001763 0.000188 -0.00046 -0.00142 -0.00356

0.7 0.01196 0.003004 0.001789 0.001495 0.00079 -0.00111

0.8 0.011675 0.002412 0.001633 0.001702 0.001107 -0.0008

0.9 0.010012 0.000539 0.000527 0.001222 0.000816 -0.00134

1.0 0.009157 -0.00121 -1.4E-05 0.001639 0.001541 -0.00121

Table 6: Di�eren
e between the Exa
t and Computed values of Temperature Distribution at Time t = 0.9

y x 0.0 0.2 0.4 0.6 0.8 1.0

0.0 -0.00116 -0.00142 -0.00136 -0.00099 -0.0003 0.00255

0.1 -0.00037 -0.00128 -0.0014 -0.00101 -0.0006 0.001097

0.2 0.000944 -0.00064 -0.00109 -0.00085 -0.00072 0.000112

0.3 0.002166 5.63E-06 -0.00071 -0.00061 -0.00064 -0.00034

0.4 0.002943 0.000227 -0.00072 -0.00078 -0.00097 -0.00102

0.5 0.003754 0.000505 -0.00065 -0.00088 -0.00125 -0.00154

0.6 0.004704 0.001148 -0.0001 -0.00036 -0.0008 -0.00116

0.7 0.005135 0.001674 0.000597 0.00056 0.000276 6.22E-05

0.8 0.004602 0.001318 0.000459 0.000643 0.00043 0.000228

0.9 0.003754 0.000394 -0.00013 0.000315 5.77E-05 -0.00041

1.0 0.004129 -0.0002 -8.6E-05 0.000899 0.000611 -0.00043



REFERENCES 127

Conclusions

A numerical method has been presented for computation of boundary control of controlled

heat equation. It is hybridization of the function approximation capabilities of feed forward

neural network and global optimization capabilities of genetic algorithm. The solutions of

controlled heat equation problems obtained through this approach compare very well with the

exact solutions. Computational experience in using proposed ANN-GA algorithm in solving

this heat equation problem has shown that the total time requirement of this technique is less

then to compare with other local search techniques like gradient descent method, because

the time has been compared to those taken by conventional method used by Sukavanam and

Panwar [15]. Conventional search techniques depend on an initial guess, so these techniques

generally give local optimal solutions. However, initial guess is not required in proposed ANN-

GA technique. Since, GA works on population of almost all possible solutions, it gives, mostly,

global optimal solution.

The methodology discussed here can be applied to both linear and nonlinear state equation

and this technique may be further hybridized with local search techniques to find more precise

solutions. The operational range of the proposed, ANN-GA, technique can also be extended

for the optimal control problems by converting them into constrained optimization problems

via neural networks or any other discretization method.

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