Stabilization in the supremum norm of wave PDE/nonlinear ODE cascades

In a recent result we solved the problem of stabilization of the cascade of a wave PDE with a general nonlinear ODE in the H₂ × H₁ norm of the wave PDE state. In this article we present stability results in the lower C¹ × C⁰ norm for general nonlinear ODEs. In our stability analysis we use arguments based on both Lyapunov functionals and explicit solutions. We specialize our general design for wave PDE-ODE cascades to the case of a wave PDE whose uncontrolled end does not drive an ODE but is instead governed by a nonlinear Robin boundary condition (a “nonlinear spring”). This is the first global stabilization result for wave equations that incorporate non-collocated destabilizing nonlinearities of superlinear growth.