Observer design for an IPDE with time-dependent coefficients

This paper develops an exponentially convergent observer for a reaction-advection-diffusion integro-partial differential equation (IPDE) with time-dependent coefficients, via the PDE backstepping method. For the (I)PDEs with timedependent coefficients, the backstepping transform gain kernel system is an (integral) evolution equation, and its coefficients also depend on time, which makes the derivation of the well-posedness of its solution to this (I)PDE nontrivial. The majorant argument is powerful in dealing with this difficulty, which is utilized in some existing literatures and is also employed in this study. To the best of the authors' knowledge, there are no existing results of observer design for the class of IPDEs with time-dependent, possibly unbounded, coefficients (which have possibly unbounded derivatives) on infinite time interval. Indeed, all the previous references for stabilization or observer design problem of an PDE with time-dependent coefficients consider a finite time interval, or the coefficients (and their derivatives) are required to be bounded with respect to the time variable. This paper could thus serve as a starting point for the study of these IPDEs.