Adaptive Stabilization of an Uncertain Anti-stable Wave PDE in an ODE’s Control Path

This paper presents an adaptive control scheme for boundary stabilization of an anti-stable wave partial differential equation (PDE) coupled with an ordinary differential equation (ODE) in its control path. The study addresses the challenge of unknown coefficients in both the boundary condition and in-domain of the anti-stable wave PDE, specifically the wave propagation speed and anti-damping coefficient. Our adaptive approach substitutes the estimated values of unknown parameters in the non-adaptive control law, and then employs the infinite-dimensional backstepping method in the design. Lyapunov-based update laws are derived for adaptive stabilization using a suitable Lyapunov function. The stability analysis demonstrates that an appropriate functional norm of the closed-loop system remains bounded by an exponential function of its initial value, ensuring the stability of the closed-loop system despite uncertainty. The efficacy of the proposed adaptive scheme is demonstrated through a simulation example.