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Center for Nonlinear Studies |
The endeavor to establish proof-based stabilizing boundary feedback control laws for PDE systems has yielded several pivotal discoveries, primarily centered around the concept of PDE backstepping. However, it is unfortunate that the inherent complexity of the control design approach has resulted in controller gain functions that are often not readily computable. This talk discusses new progress in harnessing the computational capabilities provided by Machine Learning techniques to improve the feasibility PDE control laws by speeding up the computation of gain kernel functions that emanate from PDE backstepping design. Deep neural networks that approximate nonlinear function-to-function mappings, i.e., operators, which are called DeepONet, is proven to be capable of encoding entire PDE control methodologies, such as backstepping, so that, for each new functional coefficient of a PDE plant, the backstepping gains are obtained through a simple function evaluation. I will present the approximation of multiple (coupled) nonlinear operators resulting from the control of a simple counter-convecting hyperbolic system, namely, a Goursat form kernel PDE. Such a coupled kernel PDE problem arises in several canonical hyperbolic PDE problems: oil drilling, Saint-Venant model of shallow water waves, and Aw-Rascle model of stop-and-go instability in congested traffic flow or networks of gas-electric network systems. Furthermore, the approximation of multiple (cascaded) nonlinear operators arising in the control of PDE systems from distinct PDE classes, a reaction-diffusion plant, which is a parabolic PDE, with input delay, which is a hyperbolic PDE, leading to a composition of operators defined by a single hyperbolic PDE in Goursat form and one parabolic PDE on a rectangular domain, will be discussed. In both cases, substituting the DeepONet-approximated gains, guarantees mathematically provable exponential stability of the closed-loop plant. This is joint work with Shanshan Wang at University of Shanghai for Science and Technology and Miroslav Krstic at UC San Diego.
Mamadou Diagne holds the position of Assistant Professor with the Department of Mechanical and Aerospace Engineering at UC San Diego. He completed his Ph.D. in 2013 at the Laboratory of Automation, Process Engineering and Pharmaceutical Engineering, University Claude Bernard Lyon I (France). Between 2017 and 2022, he held the position of Assistant Professor with the Department of Mechanical Aerospace and Nuclear Engineering at Rensselaer Polytechnic Institute. From 2013 to 2016, he did postdoctoral research at both UC San Diego and University of Michigan. His research focuses notably on the control of partial differential equations (PDEs) as well as coupled PDE-ODE systems. He has made significant contributions in the domains of delay systems control, adaptive control, and event-based control. His expertise extends to controlling fluid dynamics and flow systems, with practical applications in water systems, traffic systems, supply chains, and production systems. He received the NSF CAREER Award in 2020.
Host: Anatoly Zlotnik (T-5)