TY - INPR A1 - Geshkovski, B A1 - Zuazua, E T1 - Optimal actuator design via Brunovsky’s normal form N2 - In this paper, by using the Brunovsky normal form, we provide a reformulation of the problem consisting in finding the actuator design which minimizes the controllability cost for finite-dimensional linear systems with scalar controls. Such systems may be seen as spatially discretized linear partial differential equations with lumped controls. The change of coordinates induced by Brunovsky’s normal form allows us to remove the restriction of having to work with diagonalizable system dynamics, and does not entail a randomization procedure as done in past literature on diffusion equations or waves. Instead, the optimization problem reduces to a minimization of the norm of the inverse of a change of basis matrix, and allows for an easy deduction of existence of solutions, and for a clearer picture of some of the problem’s intrinsic symmetries. Numerical experiments help to visualize these artifacts, indicate further open problems, and also show a possible obstruction of using gradient-based algorithms – this is alleviated by using an evolutionary algorithm. Y1 - 2021 ER - TY - JOUR A1 - Geshkovski, B A1 - Zuazua, E T1 - Controllability of one-dimensional viscous free boundary flows N2 - In this work, we address the local controllability of a one-dimensional free boundary problem for a fluid governed by the viscous Burgers equation. The free boundary manifests itself as one moving end of the interval, and its evolution is given by the value of the fluid velocity at this endpoint. We prove that, by means of a control actuating along the fixed boundary, we may steer the fluid to constant velocity in addition to prescribing the free boundary’s position, provided the initial velocities and interface positions are close enough. Y1 - 2021 U6 - https://doi.org/https://doi.org/10.1137/19M1285354 VL - 59 IS - 3 SP - 1830 EP - 1850 ER - TY - INPR A1 - Geshkovski, B A1 - Zuazua, E T1 - Control and Deep Learning: Some connections N2 - This note is an extended abstract for a talk given by the second author during the workshop ”Challenges in Optimization with Complex PDE-Systems”, at Oberwolfach, in February 2021. It is superfluous to state the impact that deep learning has had on modern technology, as it powers many tools of modern society, ranging from web search to content filtering on social networks. A key paradigm of deep learning is that of supervised learning, which may be seen as a compound and high-dimensional simultaneous control problem. This is the viewpoint adopted by our group. And here we present some of our main findings. Y1 - 2021 ER -