Necessary and sufficient conditions for a-contraction

In this paper, we investigate the theory of [Formula: see text]-contraction with shifts with the intention of extending it to intermediate families. The theory of [Formula: see text]-contraction with shifts is used to prove orbital [Formula: see text] stability to shock solutions of conservation laws. In this setting there are strong results for scalar laws and the extremal families of [Formula: see text] systems of conservation laws. The only known results showing contraction of interior families are for the contact family the full Euler system [D. Serre and A. F. Vasseur, The relative entropy method for the stability of intermediate shock waves; the rich case, Discrete Contin. Dyn. Syst. 36(8) (2016) 4569–4577] and the case of rich systems [D. Serre and A. F. Vasseur, About the relative entropy method for hyperbolic systems of conservation laws, in A Panorama of Mathematics: Pure and Applied, Contemporary Mathematics, Vol. 658 (American Mathematical Society, 2014), pp. 237–248]. This investigation culminates in finding necessary and sufficient conditions for which small shocks of general systems are local attractors with respect to the [Formula: see text]-contraction theory.