Mini-workshop: Analysis, Numerics and Control

Date: Fri. May 26, 2023:

Event: FAU DCN-AvH Mini-workshop: Analysis, Numerics and Control
Organized by: FAU DCN-AvH, Chair for Dynamics, Control, Machine Learning and Numerics – Alexander von Humboldt Professorship at FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg (Germany)

14:00H
Title: Convex functions and its applications
Speaker: Prof. Dr. Çetin Yildiz
Affiliation: Visiting Scientist at FAU DCN-AvH from Atatürk University, Turkey

Abstract. The theory of inequalities has been the focus of researchers in the last decade. One of the main reasons for this interest is the definition of convexity. This definition firstly has appeared in Jensen’s (the celebrated Danish mathematician) papers in 1905. The convex functions has experienced a rapid development because the convex functions are closely related to the theory of inequalities and many important inequalities (such as Hermite-Hadamard inequality, Minkowski inequality, Jensen inequality) are consequences of the applications of convex functions.
The definition of the convex function can be represented as follows:
Definition: A function $f: \mathcal{I} \subset \mathbb{R} \rightarrow \mathbb{R}$ is said to be convex if

$f (ta + (1-t)b) \leq tf(a) + (1-t)f(b)$

holds for all $a,b \in \mathcal{I}$ and $t \in [0,1].$

The Hermite-Hadamard inequality, which is the main result of convex functions’ widespread application and excellent geometrical interpretation, has received a lot of attention in fundamental mathematics. Recent years have seen a rapid development in the theory of inequality. Important inequalities, such as the Hermite-Hadamard inequality, are one of the most important reasons for this development. It is worth reflecting on the fact that the theories of inequality and convexity are closely related to one another. In recent years, several new extensions, generalizations, and definitions of novel convexity have been given, and parallel developments in the theory of convexity inequality, particularly integral inequalities theory, have been emphasized.

Inequalities involving fractional integrals are a special focus of the calculus of non-integer order, widely known as Fractional Calculus. This subject deals with the generalization of integrals and derivative operators. In the literature, there are several definitions for fractional integral operators, such as Hadamard integral, Riemann-Liouville fractional integral, Caputo-Fabrizio fractional integral, Riemann-Liouville fractional integral, and conformable fractional integral.
In this presentation, we aim to explain convex functions and important inequalities.

14:30H
Title: Optimal control problems for multi-species diffusion-reaction equations in a porous medium
Speaker: Prof. Dr. Hari Mahato
Affiliation: Visiting Scientist at FAU DCN-AvH from IIT Kharagpur.
Abstract. We present an optimal control problem associated to a chemical transportation phenomena in a periodic porous medium. Our control problem consists of a $L^2$ -cost (objective) functional which is a function of the control (input) and the state variables and subject to a set of constraints (diffusion-reaction-precipitation model). Posing controls on the porous part of the medium, we set up a convex minimization problem to characterize an arbitrary control to be an optimal control. We first obtain the existence of solution of both state-equations (diffusion-reaction model) and optimal control. Then, we resolve a relation between optimal control and solution of adjoint state of state-equations. Later, we do homogenization (upscaling) of the optimal control problem via. rigorous two-scale convergence and periodic unfolding method.

WHERE?

On-site / Online

On-site:
Room Übung 4. Department Mathematik.
FAU, Friedrich-Alexander-Universität Erlangen-Nürnberg.
Cauerstraße 11, 91058 Erlangen.

Online:
Zoom meeting link
Meeting ID: 614 4658 1599 | PIN code: 914397

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