Backstepping Stabilization of 2×2 Linear Hyperbolic PDEs Coupled with Potentially Unstable Actuator and Load Dynamics
Résumé
We consider the problem of full-state feedback exponential stabilization of the zero equilibrium of a (possibly unstable) coupled 2 × 2 system of linear first-order hyperbolic partial differential equations (PDEs). Unlike previous works, boundary couplings to linear ordinary differential equations (ODEs) at both boundaries are considered (under some condi- tions) and actuation is only available through one of these ODE dynamics. This structure can arise when considering linear (or linearized) systems of balance laws with finite-dimensional actuator and load dynamics. The feedback law proposed in this paper is constructed using an invertible transform based on the (infinite-dimensional) backstepping method and a suitable target system that together guarantee the exponential stability of the origin of the closed-loop system in an adequate norm.
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