In recent years there has been a growing interest in studying
differential equations possessing a wide range of qualitative
and structural characteristics.
Special examples are some problems in optimization and control
theory which can be reviewed as gradient flows on Riemannian manifolds.
Here, we consider ordinary differential equations for computing
Euclidean norm diagonal balanced realizations arising in linear
systems theory.
Particularly, since no algebraic method exists for finding
such a realizations, we propose differential approaches based
on the evaluation of the limiting solution of different gradient
flows with respect to appropriate cost functions.