Necessary conditions for incremental stability and the second order variations

The main goal of this paper is to investigate the links between two incremental stability tests. First conditions, based on the dissipativity framework, leads to test the existence of a suitable available storage function satisfying Hamilton-Jacobi-Bellman type equations. Second conditions, deduced of the mean value theorem in norm, leads to test the finite gain stability of all the linearizations (Gateaux derivatives) of the nonlinear system. The main contribution of this paper is to point out how the Jacobi like necessary conditions, i.e. the second order variations of the dissipativity criteria, allows one to connect the test based on the mean value theorem in norm to the one based on the dissipativity framework.