Stabilization of a class of linear control systems generating C 0-semigroups

We have so far studied stabilization of linear parabolic systems. These systems are characterized by the sectorial operator L such that the resolvent (λ − L)⁻¹ exists in a sector with angle greater than π. As a result, the semigroup e^−tL generated by −L is analytic in t > 0 (see Section 4, Chapter 2). We study in this chapter a somewhat more general class of linear systems, and show that the stabilization scheme developed in Chapter 4 effectively works for these systems with some technical changes in the setting. In engineering applications, linear systems other than parabolic systems appear, such that they generate not analytic semigroups but a class of C ₀-semigroups, e.g., those appearing in delay-differential equations (see, e.g., [16, 61]). The properties of C ₀-semigroups are less nicer than those of analytic semigroups. In addition to non-analyticity of semigroups, the infinitesimal generators are not sectorial, that is, the resolvents do not exist in a sector with angle greater than π.