How special are symmetric and nonnegative costs? Approaching  c-concavity through tropical functional analysis, we unveil an analogy with Aronszajn's theorem on the  equivalent characterizations of  Hilbertian kernels, proving that tropical positive semidefinite kernels correspond to a feature map, define monotonous operators, and generate max-plus function spaces endowed with a reproducing property. They furthermore include all the Hilbertian kernels classically studied as well as Monge arrays. We provide tropical  "representer theorems", showing that some infinite dimensional regression and interpolation problems admit solutions lying in finite dimensional spaces. We apply this to optimal control, in which tropical kernels allow one to represent the value function. This talk is based on a joint work with Stéphane Gaubert (INRIA CMAP), to appear in IEOT, https://arxiv.org/abs/2202.11410