  Due to their relevance in controller design, we consider the problem of
determining the $\mathcal{L}^2$-gain, passivity properties and conic relations
of an input-output system. While, in practice, the input-output relation is
often undisclosed, input-output data tuples can be sampled by performing
(numerical) experiments. Hence, we present sampling strategies for discrete
time and continuous time linear time-invariant systems to iteratively determine
the $\mathcal{L}^2$-gain, the shortage of passivity and the cone with minimal
radius that the input-output relation is confined to. These sampling strategies
are based on gradient dynamical systems and saddle point flows to solve the
reformulated optimization problems, where the gradients can be evaluated from
only input-output data samples. This leads us to evolution equations, whose
convergence properties are then discussed in continuous time and discrete time.
