Inhomogeneous Abstract Differential Equations and Stabilization

Abstract

In the previous five chapters we considered the homogeneous (abstract) differential equation

$$\dot x(t) = Ax(t),\,\,\,\,\,\,\,\,\,\,\,\,x(0) = x_0 .$$

(10.1)

However, for control theoretical questions it is essential to add an input to the differential equation, see e.g. Chapters 3 and 4. Section 10.1 is devoted to infinite-dimensional inhomogeneous differential equations and in Section 10.2 we add an output equation. The obtained formulas will be very similar to those found in Chapter 2. However, C0-semigroups are in general not differentiable on an infinite-dimensional state space and thus the proofs are more involved.