Optimal placement of sensors and actuators

On May 7, 2024 Prof. Enrique Zuazua will talk on “Optimal placement of sensors and actuators” at the Matemáticas y memoria, an online seminar on Analysis and Partial Differential Equations organized by Uptc – Pedagogical and Technological University of Colombia, UNAM – University of Atlántico and GAPC – Ghent Analysis PDE Center.

Abstract. In this lecture, we will discuss the optimal placement of sensors and actuators for control problems, with a focus on the partial differential equation (PDE) setting, particularly in wave and heat-like processes. We will introduce relaxed versions of the problems, enabling spectral characterization and analysis. To better understand the intrinsic complexity of the original time-dependent problem, we will concentrate on the finite-dimensional case, utilizing the Brunovsky normal form. This approach will allow us to reformulate the problem within a purely matrix context, facilitating the rewriting of the problem as a minimization problem involving the norm of the inverse of a change of basis matrix. This formulation enables us to establish the existence of minimizers, as well as non-uniqueness due to the invariance of the cost with respect to orthogonal transformations.

Finally, we will address the problem from a purely geometric perspective, developing approaches to determine optimal shapes for sensor and actuator placement that are universally applicable, independent of the specific PDE model under consideration. We will also present several numerical experiments to visualize these artifacts and also to indicate further directions and open problems, particularly in the context of PDE infinite-dimensional models.

Tue. May 7, 2024 at 21:00H (Berlin time)

Online on YouTube