Input-to-state stabilisation of 1-D time-varying parabolic PDEs involving Dirichlet boundary disturbances by static backstepping control

This paper addresses the problem of input-to-state stabilisation for a class of time-varying parabolic PDEs with Dirichlet and Robin boundary disturbances, as well as in-domain disturbances. A static backstepping boundary feedback control employing a time-invariant kernel function is developed, which allows significantly reducing the computational burden in controller design and implementation. The so-called generalised Lyapunov method is applied in the assessment of the input-to-state stability (ISS) of parabolic PDEs, which, compared to the non-Lyapunov methods, considerably eases the establishment of the ISS with respect to the Dirichlet and Robin boundary disturbances in the spatial $ L<^>p $ Lp-norm for the closed-loop system whenever $ p\in [2,\infty ) $ p is an element of[2,infinity). Numerical simulations are conducted to illustrate the validity of the controller and the obtained theoretical results.