  This work concerns the exponential stabilization of underactuated linear
homogeneous systems of m parabolic partial differential equations (PDEs) in
cascade (reaction-diffusion systems), where only the first state is controlled
either internally or from the right boundary and in which the diffusion
coefficients are distinct. For the distributed control case, a
proportional-type stabilizing control is given explicitly. After applying modal
decomposition, the stabilizing law is based on a transformation for the
ordinary differential equations (ODE) system corresponding to the comparatively
unstable modes into a target one, where the calculation of the stabilization
law is independent of the arbitrarily large number of these modes. This is
achieved by solving generalized Sylvester equations recursively. For the
boundary control case, under appropriate sufficient conditions on the coupling
matrix (reaction term), the proposed controller is dynamic. A dynamic extension
technique via trigonometric change of variables that places the control
internally is first performed. Then, modal decomposition is applied followed by
a state transformation of the ODE system, which must be stabilized in order to
be written in a form where a dynamic law can be established. For both
distributed and boundary control systems, a constructive and scalable
stabilization algorithm is proposed, as the choice of the controller gains is
independent of the number of unstable modes and only relies on the
stabilization of the reaction term. The present approach solves the problem of
stabilization of underactuated systems when in the presence of distinct
diffusion coefficients, the problem is not directly solvable, similarly to the
scalar PDE case. Keywords: Linear parabolic PDE systems, underactuated systems,
stabilization, modal decomposition
