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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.

MSC:
 93C25 Control/observation systems in abstract spaces 93C15 Control/observation systems governed by ordinary differential equations