Spatial controllability and multiple actuator placement for dissipative partial differential equation systems

This manuscript considers the problem of actuator placement for transport-reaction processes, which are mathematically modelled by linear parabolic partial differential equations. The novelty of the proposed optimization scheme is that it considers the controllability of specific modes while at the same time minimizes the spillover effects due to intermediate range modes. Using modal decomposition for space discretization, along with the notions of spatial and modal controllabilities, the semi-infinite optimization problem is formulated as a nonlinear optimization problem in appropriate L? spaces. To address the problem of actuator clusterization, a modification to earlier efforts is introduced, which takes the form of an additional constraint in the location optimization. The formulated optimization problem is subsequently solved using standard search algorithms. The proposed method is successfully applied to a representative process, modelled by a one-dimensional parabolic PDE, where the optimal location of multiple actuators is computed.