  In this paper, we investigate the rapid stabilization of N-layer Timoshenko
composite beams with anti-damping and anti-stiffness at the uncontrolled
boundaries. The problem of stabilization for a two-layer composite beam has
been previously studied by transforming the model into a 1-D hyperbolic
PIDE-ODE form and then applying backstepping to this new system. In principle
this approach is generalizable to any number of layers. However, when some of
the layers have the same physical properties (as e.g. in lamination of repeated
layers), the approach leads to isotachic hyperbolic PDEs (i.e. where some
states have the same transport speed). This particular yet physical and
interesting case has not received much attention beyond a few remarks in the
early hyperbolic design. Thus, this work starts by extending the theory of
backstepping control of (m + n) hyperbolic PIDEs and m ODEs to blocks of
isotachic states, leading to a block backstepping design. Then, returning to
multilayer Timoshenko beams, the Riemann transformation is used to transform
the states of N-layer Timoshenko beams into a 1-D hyperbolic PIDE-ODE system.
The block backstepping method is then applied to this model, obtaining
closed-loop stability of the origin in the L2 sense. An arbitrarily rapid
convergence rate can be obtained by adjusting control parameters. Finally,
numerical simulations are presented corroborating the theoretical developments.
