ON THE CONTROL OF HEAT CONDUCTION

Mathematical models of thermo control processes in a rectangular plate are considered. In the model under consideration, the temperature inside a plate is controlled by heat exchange through one boundary while the other three are insulated. The control parameter is a function that satisfies certain integral equations. Sufficient conditions for achieving the given projection of the temperature at a fixed point on the plate and given average temperature are studied. ABSTRAK: Model matematik bagi proses kawalan suhu dalam bekas segi empat tepat telah dipilih. Melalui model ini, suhu bekas dikawal dengan menukar haba melalui salah satu sisi bekas, manakala tiga sisi lain telah ditebat. Parameter kawalan ini ialah fungsi, di mana ia sesuai dengan persamaan sesetengah integral. Keadaan sesuai bagi mencapai suhu tetap bekas seperti cadangan dan suhu purata yang diberikan turut dikaji.


INTRODUCTION
Many physical processes and engineering problems are described by heat/diffusion equations and the study of the properties of the solutions is important.Methods of the solutions of various boundary value problems and problems with the initial conditions can be found in [1].For the first time, a detailed explanation of the problems of control of the system with distributed parameters, described by partial differential equations were given in [2].
In recent years, interest in the study of a system with distributed parameters has increased significantly.In works [3][4][5] by Il'in and Moiseev, the questions of boundary control by the various systems described by a wave equation are studied.Notable works [6,7], studied problems related to the process of control associated with the equations of parabolic type, particularly the heat transfer process.
It is known that the boundary condition (2) means µ(t) heat entry from the x = 0 side of the rectangle and zero temperature kept in other sides.In the present paper we study following problems: Let B > 0 be a given number.Find a temperature µ(t) such that problems (1), (3), (6) have a solution that satisfies (4).
Note, that boundary condition (6) means that heat flow µ(t) is coming from the boundary x = l1 other boundaries kept with zero temperature.

THE SOLUTION OF PROBLEM 1
In this section we solve problem 1.
Theorem 1: Let B > 0 be given number.At the time t a solution of the problem (1) - (3) satisfying the condition (4) if for the function µ(t) following equality is true: () Proof: Solution of the problem ( 1) -( 3) can be represented as following series: where and nm c is defined by (11).
According the Theorem a solution should satisfy condition (4) which means

 
In the second term, we change the order of integration and summation after evaluation to get Then from ( 14) and (10) Due to µ(0) = 0 the last integrals can be evaluated as follows

 
After evaluation of the series in third term and taking into account following notations Thus we obtain the following integral equation 0 ( , ) ( ) ( )

THE SOLUTION OF PROBLEM 2
In this section we study problem 2.
Theorem 2: Let B > 0 a given number.At time t a solution of the problem (1) -( 3) satisfies the condition (5)
Proof: It is known that a solution of problems ( 1) -( 3) has the form of (13).Let this solution satisfies condition (5).Then obtain the following equality From this we get (2 1) sin 2sin 4 () 21 . By evaluation of the series in the third term and using following notation .

THE SOLUTION OF PROBLEM 3
In this section we find the solution of problem 3.
Proof: We know that a solution of the problem (1), ( 3), (6) According to the conditions of theorem 3, a solution (17) must satisfy (4) which means

 
By evaluation of the series in the third term and by denoting