Sufficient conditions on input-to-stabilisation of time-varying semi-linear systems with disturbances in Hilbert spaces

Abstract

In this paper, we investigate the problem of input-to-state stabilisation of time-varying semi-linear systems with disturbance in Hilbert spaces. An original adaptive input-to-state stabilisation strategy to derive this exceptional problem under the nonlinearities modelling is proposed using the Lyapunov method. Since all the states are not measurable, a Luenberger observer is designed to estimate unmeasurable states for input-to-state stabilisation. Thus, in the design procedure, first, a Luenberger observer is designed, and then the adaptive output feedback controller is constructed via the estimated states and adaptation law. Moreover, we give a sufficient condition, in terms of a perturbation term, to guarantee the input-to-state stability of the time-varying semi-linear systems under a stabilising nonlinear feedback law. An example of a controlled reaction-diffusion equation is given.