## Section: New Results

### Optimization tools and applications

#### Optimal structure of gas transmission trunklines

Participants : A. André, F. Bonnans.

We have finalized our result on the optimal structure of gas transmission trunklines. Suppose a gas pipeline is to be designed to transport a specified flowrate from the entry point to the gas demand point. Physical and contractual requirements at supply and delivery nodes are known as well as the costs to buy and lay a pipeline or build a compressor station. In order to minimize the overall cost of creation of this mainline, the following design variables need to be determined: the number of compressor stations, the lengths of pipeline segments between compressor stations, the diameters of the pipeline segments, the suction and discharge pressures at each compressor station. To facilitate the calculation of the design of a pipeline, gas engineers proposed, in several handbooks, to base their cost-assessments on some optimal properties from previous experiences and usual engineering practices: the distance between compressors is constant, all diameters are equal, and all inlet (resp. outlet) pressures are equal. The goals of this paper are (1) to state on which assumptions we can consider that the optimal properties are valid and (2) to propose a rigorous proof of the optimal properties (based on nonlinear programming optimality conditions) within a more general framework than before. The paper will appear in [11] .

#### Time-homogenization result for the dynamics of dislocation densities

Participants : A. Briani, R. Monneau.

In [24] we are interested by the dynamics of dislocations densities in a metal submitted by a periodic stress. Our main aim is to try to give a description of the densities of dislocations after “a long time”. By an homogenization procedure in the framework of viscosity solution we obtain that at the limit the dislocation density fulfills a diffusion equation.