We present necessary and sufficient conditions for the lower semicontinuity and exactness of the infinimal convolution $f^*\nabla g^*$, where $f^*$ and $g^*$ denote respectively the conjugate of two convex functions $f$ and $g$. Our goal is to characterize the stability of a minimization problem: $\inf_x\varphi(x,0)$ where $\varphi$ is given by $\varphi(x,u)=f(x)+g(x-u)$.