In this lecture we shall present some recent results in collaboration with B. Geshkovski (MIT) on the design of optimal sensors and actuators for control systems. We shall mainly focus in the finite-dimensional case, using the Brunovsky normal form. This allows to reformulate the problem in a purely matricial context, which permits rewriting the problem as a minimization problem of the norm of the inverse of a change of basis matrix, and allows us to stipulate the existence of minimizers, as well as non-uniqueness, due to an invariance of the cost with respect to orthogonal transformations. We will present several numerical experiments to both visualize these artifacts and also point out towards further directions and open problems, in particular in the context of PDE infinite-dimensional models.[-]

We model, simulate and control the guiding problem for a herd of evaders under the action of repulsive drivers. The problem is formulated in an optimal control framework, where the drivers (controls) aim to guide the evaders (states) to a desired region of the Euclidean space.
Classical control methods allow to build coordinated strategies so that the drivers successfully drive the evaders to the desired final destination.
But the computational cost quickly becomes unfeasible when the number of interacting agents is large.
We present a method that combines the Random Batch Method (RBM) and Model Predictive Control (MPC) to significantly reduce the computational cost without compromising the efficiency of the control strategy.
This talk is based on joint work with Dongnam Ko, from the Catholic University of Korea.[-]