## Contributed Session Fri.1.H 0111

#### Friday, 10:30 - 12:00 h, Room: H 0111

**Cluster 19: PDE-constrained optimization & multi-level/multi-grid methods** [...]

### Optimal control of PDEs with advection terms

**Chair: Mohamed Abdul majeed Al Lawati**

**Friday, 10:30 - 10:55 h, Room: H 0111, Talk 1**

**Anis Younes**

The Navier-Stokes problem in velocity-pressure formulation: Convergence and optimal control

**Coauthors: Mohamed Bouchiba, Abdenaceur Jarray**

**Abstract:**

We study the nonlinear Navier-Stokes problem in

velocity-pressure formulation. We construct a sequence of a Newton-linearized

problems and we show that the sequence of weak solutions converges towards

the solution of the nonlinear one in a quadratic way. A control problem on the

homogeneous problem is considered .

**Friday, 11:00 - 11:25 h, Room: H 0111, Talk 2**

**Sebastian Pfaff**

Optimal boundary control for nonlinear hyperbolic conservation laws with source terms

**Coauthor: Stefan Ulbrich**

**Abstract:**

Hyperbolic conservation laws arise in many different applications such as traffic modelling or fluid mechanics. The difficulty in the optimal control of hyperbolic conservation laws stems from the occurrence of moving discontinuities (shocks) in the entropy solution. This leads to the fact that the control-to-state mapping is not differentiable in the usual sense.

In this talk we consider the optimal control of a scalar balance law on a bounded spatial domain with controls in source term, initial data and the boundary condition. We show that the state depends shift-differentiably on the control by extending previous results for the control of Cauchy problems. Furthermore we present an adjoint-based gradient representation for cost functionals. The adjoint equation is a linear transport equation with discontinuous coefficients on a bounded domain which requires a proper extension of the notion of a reversible solution. The presented results form the basis for the consideration of optimal control problems for switched networks of nonlinear conservation laws.

**Friday, 11:30 - 11:55 h, Room: H 0111, Talk 3**

**Mohamed Abdul majeed Al Lawati**

A rational characteristic method for advection diffusion equations

**Abstract:**

We present a characteristic method for the solution of the two-dimensional advection diffusion equations which uses Wachspress-type rational basis functions over polygonal discretizations of the spatial domain within the framework of the Eulerian-Lagrangian localized adjoint methods (ELLAM). The derived scheme maintains the advantages of previous ELLAM schemes and generates accurate numerical solutions even when large time steps are used in the simulation. Numerical experiments are presented to illustrate the performance of the method and to investigate its convergence numerically.