**Alessia Mandini (PUC, Rio de Janeiro - Brazil) Symmetries and reduction in symplectic geometry**

**Abstract**

In these lectures I will describe how symmetries in a symplectic space can be used to reduce the complexity of the system via symplectic reduction. I will treat in detail the regular and the singular cases, giving examples to illustrate the phenomena. In particular we will discuss moduli spaces of polygons and other examples.

**Antonio Ponno (University of Padova, Italy) Introduction to Hamiltonian PDEs: an overview of models and methods**

**Abstract**

Hamiltonian Partial Differential Equations (PDEs) are introduced starting from a single class of models that includes most of the PDEs of interest to physics, ranging from gas dynamics to quantum mechanics.

The main topics of the Hamiltonian formalism, such as (canonical and noncanonical) transformations, symmetries, first integrals, and perturbation theory are presented and discussed through explicit examples.

The theory of Lax-integrable systems is also shortly discussed.

**Olivier Cots (IRIT, Toulose, France) ****Introduction to geometric and numerical methods in optimal control with application in medical imaging**

**Abstract**

An optimal control problem (OCP) is an infinite dimensional optimization problem with algebraic and differential constraints. A simple example of OCP in Lagrange form is the following. Find a command law which steers a dynamical control system from an initial configuration to a target point while minimizing an objective function representing the cost of the trajectory followed by the system during the transfer. The optimal solution can be found as an extremal, solution of the Maximum Principle and analyzed with the techniques of geometric control. As an example, the contrast and saturation problems in Magnetic Resonance Imaging (MRI) are considered. These two control problems are modeled as Mayer problems in optimal control, with a single input affine control system, with respectively a state in dimension two (four) for the saturation (contrast) problem. An analysis with the techniques of geometric control is used first to obtain an optimal synthesis in the case of the saturation problem while for the contrast problem (of higher dimension), this analysis is used first to reduce the set of candidates as minimizers and then to construct the numerical methods. This leads to a numerical investigation combining direct, indirect (multiple shooting) and homotopy methods.