Block Backstepping for Isotachic Hyperbolic PIDE-ODE Systems

This work extends the theory of backstepping control of
$(m+n)$
hyperbolic PIDEs and
$m$
ODEs to blocks of isotachic states (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, and leads to a block backstepping design. Our motivation is 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. We use a Riemann transformation 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
$L^{2}$
sense. An arbitrarily rapid convergence rate can be obtained by adjusting control parameters. Finally, numerical simulations are presented corroborating the theoretical developments.