In this chapter we give conditions under which the operators B and C can be interpreted as the control operator, respectively, observation operator of a Lp∣Reg-well-posed linear system with a given semigroup generator A. In this case we call B and C admissible for A. If B and C can be interpreted as the control and observation operators of the same system, then they are jointly admissible. We are also interested in whether or not the system is stable (this is often refered to as infinite time admissibility). We furthermore discuss admissibility questions specifically related to the L2-well-posed Hilbert space case (i.e, Y, X, and U are Hilbert spaces). This leads us to a study of H2-spaces.

Introduction to admissibility

As we have shown in Chapter 4, every well-posed linear system Σ = has a set of generators (which determine the system node). In the general case these consist of the semigroup generator A, the control operator B, and the combined observation/feedthrough operator C & D. In the compatible and regular cases the operator C & D can be replaced by the extended observation operator C∣W and the corresponding feedthrough operator D. We have given some sufficient conditions on A, B, C & D, C, C∣W, and D for these operators to be the generators of a (possibly compatible) Lp∣Reg-well-posed linear system.