Optimal Stability and Eigenvalue Multiplicity

We consider the problem of minimizing over an affine set of square matrices the maximum of the real parts of the eigenvalues. Such problems are prototypical in robust control and stability analysis. Under nondegeneracy conditions, we show that the multiplicities of the active eigenvalues at a critical matrix remain unchanged under small perturbations of the problem. Furthermore, each distinct active eigenvalue corresponds to a single Jordan block. This behavior is crucial for optimality conditions and numerical methods. Our techniques blend nonsmooth optimization and matrix analysis.