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The problems of weak and strong invariance of a constant multivalued mapping with respect to the heat conductivity equation with time lag are studied. Sufficient conditions of weak and strong invariance of a given multivalued mapping are obtained.

There are many theoretical and practical problems in control problems with distributed parameters where known methods do not work to solve them. The typical examples of such problems are conservation of temperature of a volume within admissible bounds and deviation from undesirable states.

Note that the works [

In the paper [

However, all the works mentioned above relate to control systems with concentrated parameters. In the papers [

The present paper deals with the problems of weak and strong invariance of given multivalued mapping for the 3rd heat conductivity boundary value problem with time lag. In the equation of this problem, the control parameter appears on the right hand side. We obtain conditions which can be easily checked to determine the invariance of the given constant multivalued mapping.

First of all, we recall some definitions. A bounded region

Let

Define the operator

The number

Since problem (

there exist a countable set of eigenvalues

for each eigenvalue

the set of all eigenvalues

where the equality is understood in the following sense:

The Fourier coefficients

We now define inner product and norm in the spaces

Denote by

Consider the following heat exchange control problem with lag:

Further, we use the same letter

The problem (

In the control problem (

Assume that the problem (

The following assumption will be needed throughout the paper.

Let functions

For any function

Indeed, for

Let

To prove the theorem we use the formal representation (

We use the Cauchy-Schwartz inequality to obtain the following chain of relations:

We now show that the function

By (

We now show that the problem (

Consider heat exchange control problem with lag (

A set

A set

Further, strong and weak invariant sets of the form

If

Let (

Then

The same reasoning can be applied for the time interval

We now turn to the proof of the second part of the theorem. Let the set

Let

Let

We now turn to the problem (

The set

The set

For any function

The proof follows from the following chain of relations

If

Let

We now specify the initial function and admissible control function on

Since

If

The proof of the theorem is similar to that of Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

The present research was partially supported by the National Fundamental Research Grant Scheme FRGS of Malaysia (no. 01-01-13-1228FR).