Exact boundary controllability in thermoelasticity with memory.
The authors contribute another paper to a fast growing field of controlling elastic or thermoelastic systems. There are almost too many high quality papers to follow all results in this field. The names of Lasiecka, Triggiani, Lagnese, Zuazua, Lebeau, Krabs, Sklyar, Avalos, Shubov, Barbu, Liu, Yan, Yao, Tikhomirov, Lyashko, and many others published in recent years works which come to mind as one studies the recent publications related to these topics. Of course, there are various possibilities of choosing the mathematical model for very similar processes. In studies of beam controllability one can choose the Euler-Bernoulli model of an elastic medium, or of a thermoelastic medium, without memory, or with memory or the Timoshenko model with or without memory, or with some specified type of memory, which cannot be described by the usual convolution product. Considering the Kirchhoff plate or Timoshenko plate, or similar operators, usually with some limitations on the time variable, opens new vistas for people working in this field. The authors point out correctly the physical limitations of the Fourier law of heat conduction as it is applied to thermoelasticity. Looking for different constitutive assumptions they have adapted the Gurtin-Pipkin model for propagation of temperature. In this model the thermoelastic system is fully hyperbolic, and the propagation speed is finite. \textit{G. Avalos} and \textit{I. Lasiecka} proved in their paper in [SIAM J. Control Optimization 38, No. 2, 337--383 (2000; Zbl 0948.35015)], that in the Kirchhoff model the thermoelastic system is partially-exactly controllable. The word ``partially'' implies that only the displacement is exactly controllable. Reading their paper it seems that it would be difficult to improve their results. The present authors show that in the Gurtin-Pipkin model the system is exactly boundary controllable (i.e. exactly boundary controllable in both displacements and temperature).
scientific article; zbMATH DE number 2002538