## Diego Napp

## Coding Theory: Convolutional codes

Coding theory - theory of error correcting codes - is one of the most interesting and applied part of mathematics. All real systems that work with digitally represented data, as CD players, TV, fax machines, internet, satellites, mobiles, require to use error correcting codes because all real channels are, to some extent, noisy. Coding theory methods are often elegant applications of very basic concepts and methods of (abstract) algebra. In this talk

we aim to give a general overview of this area and explain the topics our group is particularly interested in.

we aim to give a general overview of this area and explain the topics our group is particularly interested in.

Lígia Abrunheiro

## An optimal control approach to the Herglotz variational problem on the Euclidean sphere

We consider a variational problem that depends on the covariant acceleration on the Euclidean sphere which is an extension of the Herglotz variational problem. The corresponding second-order optimal control problem is described and the generalized Euler-Lagrange equation is deduced from the Hamiltonian equations.

Ricardo Almeida

## A numerical method to solve FDE's and optimization problems with dependence of a fractional operator

We study a new fractional operator, which generalizes the Caputo and the Caputo-Hadamard fractional derivatives. After presenting some important results about the fractional operator, we study calculus of variation problems, where the Lagrangian depends on this fractional operator.

We present sufficient and necessary conditions of first and second order to determine the extremizers of a functional. The cases of integral and holomonic constraints are also considered. An existence and uniqueness theorem for a fractional Caputo type problem, with dependence on the Caputo-Katugampola derivative, is proven. As a numerical procedure to deal with such type of problems, we prove a decomposition formula for the new fractional derivative, which depends on the first-order derivative only. This formula allows us to provide a simple numerical procedure to solve the fractional problem type, by rewriting the fractional problems into an ordinary one.

We present sufficient and necessary conditions of first and second order to determine the extremizers of a functional. The cases of integral and holomonic constraints are also considered. An existence and uniqueness theorem for a fractional Caputo type problem, with dependence on the Caputo-Katugampola derivative, is proven. As a numerical procedure to deal with such type of problems, we prove a decomposition formula for the new fractional derivative, which depends on the first-order derivative only. This formula allows us to provide a simple numerical procedure to solve the fractional problem type, by rewriting the fractional problems into an ordinary one.