Aspects of controllability and observability for time-varying PDE systems

There are many industrial and biological reaction-diffusion systems which exhibit time-varying features where certain parameters of the system change during the process. The underlying transport-phenomena are often modelled using parabolic partial differential equations (PDEs) with time-varying coefficients which describe the dynamics of the process. Often it is of interest to control this dynamical behaviour such as the regulation of temperature or concentration, and one approach is the use of infinite-dimensional systems theory to represent the PDE models, with time-varying process parameters, as abstract nonautonomous evolution equations on appropriately defined function spaces. In contrast to timeinvariant control problems, the theory for controllability and observability for time-varying systems is less well established. In this work, we consider some pertinent aspects regarding the controllability and observability of nonautonomous infinite-dimensional systems. An example is considered for which the conditions for exact, null, and approximate controllability and observability are verified, and some observations regarding the influence of time-varying input and measurement operators are provided.