Prossimi seminari del Dipartimento di Matematica

We address the control of Partial Differential equations (PDEs) with unknown parameters. Our objective is to devise an efficient algorithm capable of both identifying and controlling the unknown system. We assume that the desired PDE is observable provided a control input and an initial condition. The method works as follows, given an estimated parameter configuration, we compute the corresponding control using the State-Dependent Riccati Equation (SDRE) approach. Subsequently, after computing the control, we observe the trajectory and estimate a new parameter configuration using Bayesian Linear Regression method. This process iterates until reaching the final time, incorporating a defined stopping criterion for updating the parameter configuration. We also focus on the computational cost of the algorithm, since we deal with high dimensional systems. To enhance the efficiency of the method, indeed, we employ model order reduction through the Proper Orthogonal Decomposition (POD) method. The considered problem's dimension is notably large, and POD provides impressive speedups. Further, a detailed description on the coupling between POD and SDRE is also provided. Finally, numerical examples will show the accurateness of our method across.

Si crede spesso che matematica e fisica siano due discipline separate da un chiaro confine rimasto stabile nel corso della storia: è questo il tacito presupposto assunto dagli autori di storie della matematica e della fisica. Il significato dei due termini è invece cambiato profondamente nel tempo insieme allo status epistemologico delle due scienze. Una riflessione sulla storia dei due concetti può essere utile per individuare le caratteristiche essenziali delle scienze esatte, mettere in discussione l’esistenza di un confine invalicabile tra fisica e matematica e immaginare possibili scenari di evoluzione futura.

The aim of the talk is to analyze an evolutionary Φ- Laplacian problem, with singular and convective reactions: u_t -A u= f+g .The differential operator A considered is an elliptic operator, driven by a Young function Φ, while the reaction terms are Carathéodory functions obeying suitable growth conditions. The problem possesses three features of interest: • the operator A can be non-homogeneous, and with unbalanced growth; • the reaction term f is singular (i.e., it behaves like u^{−γ} with γ ∈ (0, 1)) and f(x, ·) can be non-monotone); • the reaction term g is convective (i.e., it depends on ∇u). We will first introduce the functional setting of the problem, by recalling the most relevant function spaces involved and their basic properties. Secondly, the main issues concerning both singular and convective terms are highlighted, together with the sub-solution and freezing techniques. Finally, we will briefly sketch the proof of our existence result, based on a priori estimates and a semi-discretization (in time) procedure. The seminar will have an introductory fashion, under the trend proposed by the cycle ASK (Analysis Student Kernel), for young analysis researchers, at the University of Bologna (https://sites.google.com/view/askbologna/home?authuser=1).