Underactuated boundary control of a linearized Saint-Venant equation

Abstract This paper deals with the stabilization of one-dimensional linearized, shock-free Saint-Venant equations, which are widely used in fluid modelling. The approach focuses on four under-actuated hyperbolic partial differential equations, combining Volterra transformations inspired by backstepping techniques. The system is reduced to an integral difference equation, allowing for stability analysis the use of spectral methods. A distributed state feedback controller is designed to eliminate integral terms by solving a Fredholm equation. This result effectively addresses challenges related to under-actuation and lays the foundation for extending stabilization techniques to nonlinear scenarios.