Exact-approximate boundary controllability of thermoelastic systems under free boundary conditions (Q2712199)

Let me start the review with the same introduction as the recent review of a paper by the same authors: This paper is a continuation of the works of the first author in cooperation with Lasiecka, and those of Lasiecka and Triggiani. If we equate most of the physical coefficients to one, modelling the free vibration of a thin plate coupled with heat transfer and we use the Kirchoff plate model and Fourier's law of heat transfer, we may derive the author's system: NEWLINE\[NEWLINEw_{tt}- \gamma\Delta w_{tt}+ \Delta^2w+ \alpha\Delta\theta= 0\quad \text{and}\quad \beta\theta_t- \eta\Delta\theta+ \sigma\theta- \alpha\Delta w_t= 0NEWLINE\]NEWLINE with given initial data, and with applied boundary control but somewhat different boundary conditions, including the Robin temperature condition.NEWLINENEWLINENEWLINEUnlike the assumptions made by Lasiecka and Lagnese concerning the smallness of the coupling parameter \(\alpha\), no such restrictions are offered in this paper. Geometric restrictions on the star-shaped property of the domain also have been omitted.NEWLINENEWLINENEWLINEThe authors state that it is known (since 1998) that for free boundary conditions, thermoelastic semigroup decomposes, into a damped Kirchoff plate semigroup, and a compact perturbation, which can be ignored in controllability estimates. Thus the control of the \(w\)-variable becomes, the well studied control problem for the Kirchoff plate. Then the multiplier technique allows direct estimation of energy terms. The authors state that in their case these estimates are polluted by boundary terms that are not easily taken care of in the earlier energy estimates. However, some of the previous estimates proved by Triggiani and Lasiecka come to the rescue.NEWLINENEWLINENEWLINEThe main theorem states that given an \(\varepsilon> 0\) there exists a time \(T^*>0\), such that for all \(T> T^*\) there exists an admissible control such that assigned terminal data for the mechanical variables is exactly satisfied, while temperature \(\theta\) is within \(\varepsilon\) of the assigned value. The thermal control is permitted to be infinitely smooth.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].