Region of Attraction of a Backstepping Observer for the Quasilinear Heat Equation
It has long been of interest to understand when control and observer-design methodologies developed for linear dynamical systems can provide local guarantees when applied to nonlinear systems. However, beyond establishing local stability, it is also important to characterize the region of attraction and performance in terms of system parameters and design gains--something that linearization-based approaches cannot achieve. For instance, although increasing an observer gain improves the convergence rate in the linear setting (as in backstepping design), the relationship between this gain and the convergence rate in the nonlinear setting need not be monotonic; understanding these dependencies is therefore crucial for reliable operation. In light of this discussion, we consider the one-dimensional quasilinear heat equation with state-dependent diffusivity, and design a boundary-output observer based on the backstepping design for a linear heat equation with constant diffusivity. Viewing the quasilinear system as a perturbation of the linear one, we establish exponential stability of the origin for the estimation error dynamics in $H^1$, with an estimate of the region of attraction depending on the system parameters, observer gains, and the mismatch between the nonlinear diffusivity and the constant design diffusivity. Importantly, the estimation error converges to zero rather than merely to a neighborhood scaling with this mismatch, even though, in contrast to backstepping-based stabilization of nonlinear PDEs, the mismatch need not decay along trajectories and may remain bounded away from zero, acting as a persistent state-dependent multiplicative perturbation. A technical challenge was to perform a sufficiently-fine Lyapunov analysis that does not yield overly conservative results such as mere boundedness of the estimation error.