As is known, bounds of the resolvent of a matrix in the right complex half-plane yield bounds of solutions of homogeneous and inhomogeneous linear differential equations with this matrix. We ask two basic
questions:

- Up to which size of structured perturbations are the resolvent norms of the perturbed matrices within a given bound in the right complex half-plane?

- For a given size of structured perturbations, what is the smallest common bound for the resolvent norms of the perturbed matrices in the right complex half-plane?

This is considered for general linear structures such as complex or real matrices with a given sparsity pattern or with restricted range and corange, or special classes such as Toeplitz or Hankel matrices.
Conceptually, we combine unstructured and structured pseudospectra in a joint pseudospectrum, allowing for the use of resolvent bounds as with unstructured pseudospectra and for structured perturbations as with structured pseudospectra. The above questions are addressed by an algorithm which solves eigenvalue optimization problems via suitably discretized rank-1 matrix differential equations. The talk is based on joint work with Nicola Guglielmi.