Input-to-state stability and integral input-to-state stability in various norms for 1-D nonlinear parabolic PDEs

Abstract

For any and any , by using the approximative Lyapunov method, the -integral input-to-state stability (-iISS) in the spatial -norm for a class of 1-D nonlinear parabolic PDEs with distributed in-domain disturbances and under homogeneous Robin boundary conditions is assessed. Furthermore, when , under additional assumptions on the relationship between the coefficients of the equation and the parameter p, the -iISS in the spatial -norm for the consider system is proved. In addition, in the special case of , a refined -iISS estimate in the spatial -norm is established for the considered system.