Necessary and Sufficient Stability Conditions via the Eigenvalues-Eigenvectors Method: an Application to the Magnetic Bénard Problem

We study the magnetic Bénard problem (MBP) with the Lyapunov direct method and obtain necessary and sufficient conditions of (conditional) nonlinear stability. An operative method (which rests upon the classical eigenvalues–eigenvectors theory) is introduced to define an optimal Lyapunov function for the linearized problem. In the case of stress-free boundary conditions, the linearized system of MBP is associated to a linear ordinary differential system. The study of stability of zero solution of this system (with the eigenvalues–eigenvectors method and Lyapunov theory) gives in a canonical way the Lyapunov function for the control of stability.