Stabilization of Underactuated Linear Coupled Reaction–Diffusion PDEs via Distributed or Boundary Actuation

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.