  Recently, the concept of k-contraction has been introduced as a promising
generalization of contraction for dynamical systems. However, the study of
k-contraction properties has faced significant challenges due to the reliance
on complex mathematical objects called matrix compounds. As a result, related
control design methodologies have yet to appear in the literature. In this
paper, we overcome existing limitations and propose new sufficient conditions
for k-contraction which do not rely on matrix compounds. Our design-oriented
conditions stem from a strong geometrical interpretation and establish a
connection between kcontraction and p-dominance. Notably, these conditions are
also necessary in the linear time-invariant framework. Leveraging on these
findings, we propose a feedback design methodology for both the linear and the
nonlinear scenarios.
