Necessary and sufficient conditions for $a$-contraction

In this paper we investigate the theory of $a$-contraction with shifts with the intention of extending it to intermediate families. The theory of $a$-contraction with shifts is used to prove orbital $L^2$ stability to shock solutions of conservation laws. In this setting there are strong results for scalar laws and the extremal families of $n\times n$ systems of conservation laws. The only known results showing contraction of interior families are for the contact family the full Euler system and the case of rich systems due to Serre and Vasseur '16. This investigation culminates in finding necessary and sufficient conditions for which small shocks of general systems are local attractors with respect to the $a$-contraction theory.