In this paper, we discuss the problem of approximating stability radius appearing in the design procedure of finite-dimensional stabilizing controllers for an infinite-dimensional dynamical system. The calculation of stability radius needs the value of $H_\infty$-norm of a transfer function whose realization is described by infinite-dimensional operators in a Hilbert space. From the computational point of view, we need to prepare a family of approximate finite-dimensional operators and then to calculate the $H_\infty$-norm of their transfer functions. However, it is not assured that they converge to the value of $H_\infty$-norm of the original transfer function. The purpose of this study is to justify the convergence. In a numerical example, we treat parabolic distributed parameter systems with distributed control and distributed/boundary observation.
distributed parameter system, finite-dimensional controller, stability radius, transfer function, semigroup