Backward error analysis of generalized eigenvalue problems preserving block structures of matrices

This paper considers the backward error analysis of an approximate eigenpair of blockwise structured matrix pencils that becomes an exact eigenpair of an appropriately minimal perturbed block matrix pencil. The obtained perturbed pencil preserves the structures of different blocks for the Frobenius norm. In application, we discuss the different pencils arising in continuous-time linear quadratic optimal control problems, discrete-time linear quadratic optimal control, and port-Hamiltonian descriptor systems in optimal control. We also present several numerical examples to illustrate our framework.