Structural stability, asymptotic stability and exponential stability for linear multidimensional systems: the good, the bad and the ugly

In this paper, we investigate three concepts of stability for linear two-dimensional systems: the ‘good’ structural stability (an algebraic property linked to the location of the roots of a certain characteristic polynomial), the ‘bad’ asymptotic stability (roughly the trajectory converges to the equilibrium point) and the ‘ugly’ exponential stability (the rate of convergence is at least exponential). More precisely, we show that for a usual set of boundary conditions taken along the positive semi-axes, structural stability and exponential stability are equivalent notions. For this particular set of boundary conditions, we further prove that structural stability implies asymptotic stability but a counterexample shows that asymptotic stability does not imply structural stability which is a major difference compared to the one-dimensional case. This also highlights the importance of the boundary conditions when one works with multidimensional systems.