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In this paper, we study the $a$-contraction property of small extremal shocksfor 1-d systems of hyperbolic conservation laws endowed with a single convexentropy, when subjected to large perturbations. We show that the weightcoefficient $a$ can be chosen with amplitude proportional to the size of theshock. The main result of this paper is a key building block in the companionpaper, [{arXiv:2010.04761}, 2020], in which uniqueness and BV-weak stabilityresults for $2\times 2$ systems of hyperbolic conservation laws are proved.

Sharp a-contraction estimates for small extremal shocks