Towards optimal actuator placement for dissipative PDE systems in the presence of uncertainty

We consider the issue of actuator placement for transport-reaction processes when there is significant time-varying disturbance present. Such processes are commonly mathematically modeled by perturbed linear dissipative partial differential equations (PDEs). The proposed method is based on previous work by the authors on actuator placement for PDEs, however the presence of noise and/or model uncertainty precludes their direct application. By Using modal decomposition for space discretization and employing the concept of spatial and modal norms, an optimization problem is formulated that considers the controllability of specific modes, minimizes the spillover effects to the fast modes and takes explicitly into consideration the spatial distribution of noise or model uncertainty. The proposed method is successfully applied to a representative one-dimensional parabolic PDE, where the optimal location of multiple actuators is computed.