Optimal H2-design with the smoothed spectral abscissa

is a smooth function and thus relatively easy to optimize. For the H2-norm to exist however, the system matrix A must be stable, i.e. all its eigenvalues must reside in the left half complex plane. This is equivalent to demanding that the real part of the rightmost eigenvalue of A, the so-called spectral abscissa α(A), be negative. Objective functions derived from α(A) unfortunately are in general not smooth, so one typically ends up with a nonsmooth optimization problem for the stabilization.