  In this paper, we study the stability of a one-dimensional Bresse system with
infinite memory type control and/or with heat conduction given by Cattaneo's
law acting in the shear angle displacement. When the thermal effect vanishes,
the system becomes elastic with memory term acting on one equation. Unlike [6],
[10], and [22], we consider the interesting case of fully Dirichlet boundary
conditions. Indeed, under equal speed of propagation condition, we establish
the exponential stability of the system. However, in the natural physical case
when the speeds of propagation are different, using a spectrum method, we show
that the Bresse system is not uniformly stable. In this case, we establish a
polynomial energy decay rate. Our study is valid for all other mixed boundary
conditions and generalizes that of [6], [10], and [22].
