Boundary Control of Transport-Reaction-Diffusion Cascade with Simultaneous In-Domain and Boundary Couplings
Résumé
In this paper, we consider the control of a transport equation interconnected with a reaction-diffusion equation, resulting in a transport-reaction-diffusion cascade. The couplings between the different subsystems appear in the two boundary conditions of the reaction-diffusion equation and in the domain. These boundary couplings can be interpreted as pointwise delayed inputs acting on the reaction-diffusion equation, while the in-domain coupling is seen as a spatially varying input delay. This combination of inputs presents significant challenges as applying state-of-the-art approaches -such as backstepping transformations or spectral-reduction techniques -are insufficient for addressing the complexities involved. In this paper, we take advantage of the distinct features of these two methods to develop a new effective feedback control strategy for the studied PDE cascade. Additionally, we provide a necessary and sufficient controllability condition for designing a stabilizing state feedback controller. Notably, we illustrate the existence of specific couplings between the two subsystems of the cascade for which stabilization of the system is not achievable.
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