Abstract : In this minisyposium we discuss the recent developments in control and optimization.
We analyze solutions to PDE in a general framework and develop a new control and optimization methods and theoretic analysis.
The control PDE analysis includes the constrained stochastic Nash game,
optimal control problems in metric spaces and
optimal control of poroelastic systems and frequency dependent Hautus tests for controllability.
We apply our methods to specific examples in biomedicine, population dynamics and Economics
Also, we develop a new theoretic framework for optimal control and optimization
based on Rockafellar's perturbation theory
to analyze and solve general nonsmooth convex minimization and
monotone inclusion problems.
[02838] PDE Constrained Optimization with Non-smooth Learning Informed Structures
Author(s) :
Michael Hintermueller (Weierstrass Institute Berlin)
Abstract : A class of optimization problems subject to PDEs with components resulting from ReLU-based neural network models is studied analytically and numerically. Concerning the analysis it is shown that direct classical smoothing of the non-smooth structures leads to issues concerning the existence of solutions. Further, for the non-smooth setting stationarity conditions are derived and used numerically. With respect to the numerical solution, a bundle-free minimization algorithm relying on possible smoothing on the level of directional derivatives is introduced and analyzed, and a brief report on computational tests is provided.
[03725] Some optimal control problems in metric spaces
Author(s) :
Hasnaa Zidani (INSA Rouen Normandie)
Othmane Jerhaoui (INSA Rouen Normandie)
Averil Prost (INSA Rouen Normandie)
Abstract : In this talk, we will discuss some optimal control problems in metric spaces (e.g. stratified systems, centralized control problems in Wasserstein space). We are mainly interested in the characterisation of the value function as viscosity solution of an adequate Hamilton-Jacobi (HJ) equation. For this, we introduce a notion of viscosity solutions for HJ equations in some metric spaces. This notion is based on test functions that are directionally differentiable and can be represented as a difference of two semi-convex functions. Under mild assumptions on the Hamiltonian and on the metric space, we can derive the main properties of viscosity theory: the comparison principle and Perron's method.
[03787] Exact controllability for systems describing plate vibrations. A perturbation approach.
Author(s) :
Marius Tucsnak (University of Bordeaux)
Abstract : The aim of this talk is to describe new exact controllability properties of systems described by perturbations of the classical Kirchhoff plate equation. We first consider systems described by an abstract plate equation with a bounded control operator. The generator of these systems is perturbed by bounded operators which are not necessarily compact, thus not falling in the range of application of compactness-uniqueness arguments. Our first main result is abstract and can be informally stated as follows: if the system described by the corresponding unperturbed abstract wave equation, with the same control operator, is exactly controllable (in some time), then the considered perturbed plate system is exactly controllable in arbitrarily small time. The employed methodology is based, in particular, on frequency-dependent Hautus type tests for systems with skew-adjoint operators.
When applied to systems described by the classical Kirchhoff equations, our abstract results, combined with some elliptic Carleman-type estimates, yield exact controllability in arbitrarily small time, provided that the system described by the wave equation in the same spatial domain and with the same control operator is exactly controllable. The same abstract results can be used to prove the exact controllability of the system obtained by linearizing the von K\'arm\'an plate equation around a real analytic stationary state. This leads, via a fixed-point method, to our second main result: the nonlinear system described by the von K\'arm\'an plate equations is locally exactly controllable around any stationary state defined by a real analytic function.
[03789] A Perturbation Framework for Convex Minimization with Nonlinear Compositions
Author(s) :
Luis Briceño-Arias (Universidad Técnica Federico Santa María)
Patrick L. Combettes (North Carolina State University)
Abstract : We introduce a framework based on Rockafellar's perturbation
theory to analyze and solve general nonsmooth convex minimization
and monotone inclusion problems involving nonlinearly composed
functions as well as linear compositions.
[03864] Analysis and Control in Poroelastic Systems
Author(s) :
Lorena Bociu (NC State University)
Abstract : We answer questions related to tissue biomechanics via wellposedness, sensitivity analysis, and optimal control problems for fluid flows through deformable porous media. These results are relevant for many applications in biology, medicine and bio-engineering. We focus on the local description of the problem, which involves implicit, degenerate, nonlinear poroelastic systems, as well as scenarios where the global features of the problem are accounted for through a multi-scale coupling with a lumped hydraulic circuit.
