Differential-algebraic systems (descriptor systems) are the essential models for the control and simulation of dynamical processes in all areas of science and engineering that are constrained e.g. by conservation laws, balance conditions or just geometric conditions. When economical, biological or chemical systems are modelled by descriptor systems, in which the state describes concentrations, populations of species, or just numbers of cells, then the solution is a nonnegative vector function. Hence, the numerical methods for the control or simulation should respect this structure. Applications are semi-dicretized advection-diffusion-reaction models that arise in atmospheric chemistry, chemotaxis problems, transport problems or economic models.

• extension of the Perron-Frobenius theory to positive descriptor systems;
• characterization of positivity in the descriptor case;
• stability analysis of positive descriptor systems;
• positivity preserving model reduction;
• doubly nonnegative solutions of projected Lyapunov equations;
• AMG for (singular) M-matrices, convergence results.

• a systematic study of application problems in biological systems including chemotaxis, control of chemical reactions in the human blood system etc.
• extension of controllability, observability, stabilizability to positive descriptor systems;
• parameter identification and model realization methods for positive descriptor systems;
• positivity preserving preconditioners for iterative methods like algebraic multigrid methods and approximate inverse preconditioners;
• positivity preserving optimal control of positive descriptor systems;
• positivity preserving discretization and index reduction methods;