Control theory for PDEs has been quite exhaustively developed for model problems (heat and wave equations). But other important models in applications, of hybrid nature, remain poorly understood. This is particularly the case for models involving memory terms in viscoelasticity.
Our recent contributions in this area are inspired in the interpretation of memory models as the coupling of PDEs with infinite-dimensional ODEs. The presence of ODE components in the system explains the failure of controllability if the control is confined on a space-support which is time-independent. This motivates the use of our moving control strategy, making the control move covering the whole domain, introducing the transport effects that the ODE is lacking.
This theory applies to some model examples and an in-depth effort is needed to cope with key issues such as nonlinearity, general relaxation kernels, memory terms in the principal part of the PDE, etc. which are particularly complex for wave models, because of the lack of regularizing effects.
Based on our past experience in developing numerical solvers, the numerical counterparts for the control of models with memory terms will also be developed.
This topic also opens up the perspective of applications to other areas where ODE-PDE models arise, such as population dynamics.