Exponential stability of semigroups associated with some operator models in mechanics.
The authors consider equations in a Hilbert space \(H\) of the form \[ \ddot x + B\dot x + Ax = 0\quad t\geq 0, \] where \(B\) is an unbounded symmetric operator in \(H\) (it corresponds to damping), and \(A\) is a uniformly positive and selfadjoint operator in \(H\) (it corresponds to the potential energy of the system), and \(x(t)\) is a function with values in \(H\). Such equations arise as consequences of linearizations in various problems of elasticity theory and hydromechanics. It is proved that the linear solution operator associated with this equation is a generator of a \(C_0\)-semigroup in the energy space. In addition, the mentioned semigroup is exponentially stable if the operator \(B\) is uniformly positive and the operator \(A\) dominates \(B\) in the sense of quadratic forms.
scientific article; zbMATH DE number 2120972