Structural Stability and Asymptotic Stability for Linear Multidimensional Systems: a Counterexample * * This work was supported by the ANR MSDOS grant ANR-13-BS03-0005.

摘要

In this paper, we revisit the notions of structural stability and asymptotic stability that are often considered as equivalent in the field of multidimensional systems. We illustrate that the equivalence between asymptotic and structural stability depends on where we define the boundary conditions. More precisely, we show that structural stability implies asymptotic stability when the boundary conditions are imposed on the positive axes. But a carefully designed counterexample shows that the opposite does not hold in this case. This illustrates once again the importance of the boundary conditions when dealing with multidimensional systems.