DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2021164 DYNAMICAL SYSTEMS IDENTIFYING THE HEAT SINK J. D. Audu a , A. Boumenir ∗,b , K. M. Furati a and I. O. Sarumi a (a) Department of Mathematics, King Fahd University of Petroleum and Minerals Dhahran, Saudi Arabia (b) Department of Mathematics, University of West Georgia, GA 30118, USA Abstract. In this paper we examine the identification problem of the heat sink for a one dimensional heat equation through observations of the solution at the boundary or through a desired temperature profile to be attained at a certain given time. We make use of pseudo-spectral methods to recast the direct as well as the inverse problem in terms of linear systems in matrix form. The resulting evolution equations in finite dimensional spaces leads to fast real time algorithms which are crucial to applied control theory. 1. Introduction. The control and monitoring of heat transfer plays an important role in combustion theory, conservation of energy, steel and iron casting processes, and chemical reactors to name a few. This task can be described by the simple inverse problem of reconstructing the heat sink coefficient q appearing in    u t (x, t)= u xx (x, t)+ q(x)u(x, t), (x, t) ∈ (0,π) × (0, ∞), u(0,t)=0, u(π,t)=0, t> 0, u(x, 0) = f (x), x ∈ (0,π), (1) from observations of the solution. We shall be interested in the following two problems: A. Controllabillity: Given an initial condition u(x, 0), find q so the solution reaches a given target profile within time T, i.e. u(x, T ) for 0 <x<π, B. Identification: Determine the coefficient q in (1) from reading u(π,t) for T<t<T + ǫ. The controllability problem (A) is important in manufacturing, engine perfor- mance, and in the design of microchips. The identification problem (B) is in fact an inverse problem that arises in many engineering applications and fabrication processes, where q is an unknown to be determined, [4, 5]. Once q known, one can predict the heat distribution at later times and control it. It is well known that the heat dissipation and cooling are important factors in steel forging, welding and many fabrication processes. In reverse engineering, a snapshot of a rod would reveal its heat profile u(x, T ), and from this thermal imaging one can reconstruct q and find hidden defects. This shows the close connection and interaction between inverse problems and control theory, [6, 9]. 2020 Mathematics Subject Classification. 35R30, 65M32. Key words and phrases. Inverse problems, heat equation, parameter identification, reconstruc- tion algorithms. This work is supported by KFUPM grant SB191022. ∗ Corresponding author: A. Boumenir. 1