Two conjectures on the admissibility of control operators.

*(English)*Zbl 0763.93041
Estimation and control of distributed parameter systems, Proc. Int. Conf., Vorau/Austria 1990, ISNM 100, 367-378 (1991).

[For the entire collection see Zbl 0732.00029.]

The paper considers linear control systems described by the differential equation \(x'(t)=Ax(t)+Bu(t)\). Here, \(A\) generates a \(C_ 0\) semigroup on a Hilbert space \(X\). \(X\) is dense in a larger Hilbert space \(X_{-1}\). The problem is to determine which operator are admissible, meaning that for any input function \(u\in L^ 2[0,\infty)\) (with value in \(V\)), the differential equation has an \(X\)-valued solution. It is conjectured that a necessary and sufficient condition for admissibility is \((sI-A)^{-1} B\leq K(\text{Re }s)\), for all \(s\) in some right half-plane. If true, this would be a generalization of the Carleson measure theorem. Several results are proved which tend to support the conjecture: (1) the condition is necessary, (2) if \(U\) is finite-dimensional and \(A\) is normal then the condition is sufficient, (3) if \(e^{At}\) is left invertible then the condition is sufficient, (4) if \(A\) is normal and \(e^{At}\) is analytic then the condition is sufficient.

The paper considers linear control systems described by the differential equation \(x'(t)=Ax(t)+Bu(t)\). Here, \(A\) generates a \(C_ 0\) semigroup on a Hilbert space \(X\). \(X\) is dense in a larger Hilbert space \(X_{-1}\). The problem is to determine which operator are admissible, meaning that for any input function \(u\in L^ 2[0,\infty)\) (with value in \(V\)), the differential equation has an \(X\)-valued solution. It is conjectured that a necessary and sufficient condition for admissibility is \((sI-A)^{-1} B\leq K(\text{Re }s)\), for all \(s\) in some right half-plane. If true, this would be a generalization of the Carleson measure theorem. Several results are proved which tend to support the conjecture: (1) the condition is necessary, (2) if \(U\) is finite-dimensional and \(A\) is normal then the condition is sufficient, (3) if \(e^{At}\) is left invertible then the condition is sufficient, (4) if \(A\) is normal and \(e^{At}\) is analytic then the condition is sufficient.

Reviewer: A.Feintuch (Beersheva)