Optimal placement of actuators and sensors for control of nonequilibrium dynamics

In this paper, we provide a systematic convex programming-based approach for the optimal locations of static actuators and sensors for the control of nonequilibrium dynamics. The problem is motivated with regard to its application for control of nonequilibrium dynamics in the form of temperature in building systems and control of oil spill in oceanographic flow. The controlled evolution of a passive scalar field, modeling the temperature distribution or the density of oil dispersant, is governed by the linear advection partial differential equation (PDE) with spatially located actuators and sensors. Spatial locations of actuators and sensors are optimized to maximize the controllability and observability of the linear advection PDE. Linear transfer Perron-Frobenius and Koopman operators, associated with the advective velocity field, are used to provide analytical characterization for the controllable and observable spaces of the advection PDE. Set-oriented numerical methods are used for the finite dimensional approximation of the transfer operators and in the formulation of the optimization problem. Application of the framework is demonstrated for the optimal placement of actuators for the release of dispersant for oil spill control.