Singular estimates and uniform stability of coupled systems of hyperbolic/parabolic PDEs
The present paper researches properties of an accepted acoustic chamber model, subject to unbounded control action of its flexible wall. The model consists of a coupled system of two partial differential equations of different types: a hyperbolic PDE acting whithin the acoustic chamber and a parabolic-like PDE acting on the elastic wall of the chamber. The authors allow the parameter \(\alpha\), which measures the strength of damping in the plate like component, to run over the entire range \(1/2 \leq \alpha \leq 1\), enlarging the scope of the parabolic component from the original Kelvin-Voight damping \(\alpha=1\), considered in prior literature. They seek to achieve two critical control-theoretic properties: (i) a quantitatively precise singular estimate for the operator \(e^{At}B\) as \(t\) tends to \(0\), \(B\) being the unbounded operator, and (ii) a uniform (exponential) stability result of the free dynamics operator \(e^{At}\). As \(\alpha\) decreases form 1 to 1/2, a progressively stronger damping on the boundary of the wave (hyperbolic) component is needed to fulfill goal (i). However, the additional boundary damping introduces instability of the overall free dynamics \(e^{At}\). In order to remove this pathology, an additional static damping is introduced in the boundary conditions of the wave component. This paper gives a precise description of these two intertwined and conflicting aspects for the purpose of achieving the simplest possible model, as a function of \(\alpha\), where both goals (i) and (ii) are attained.
scientific article; zbMATH DE number 1780531