Analytic decomposition of differential graded Lie algebras

We prove explit formulas for the decomposition of a differential graded Lie algebra into a minimal and a linear $L_\infty$-algebra. We define a category of metric $L_\infty$-algebras, called Palamodov $L_\infty$ algebras, where the structure maps satisfy a certain convergence condition and deduce a decomposition theorem for differential graded Lie algebras in this category. This theorem serves for instance to prove the convergence of the Kuranishi map assigned to a differential graded Lie algeba.