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Admissibility of unbounded control operators. (English) Zbl 0685.93043
The paper’s interest is in the structure of a control operator B, i.e., the representation and admissibility of B for the following linear control system in a Banach space X: $\dot x(t)\quad =\quad Ax(t)+Bu(t).$ Here, A is the generator of a $$C_ 0$$ semigroup $${\mathbb{T}}=({\mathbb{T}}_ t)_{t\geq 0}$$ on X, x(t) the state in X, and u(t) the input in U, U being a Banach space. A family of linear operators $$\Phi =(\Phi_ t)_{t\geq 0}$$ defined by $\Phi_ tu\quad =\quad \int^{t}_{0}{\mathbb{T}}_{t-\sigma}Bu(\sigma)d\sigma,\quad u\in \Omega \quad =\quad L^ p([0,\infty),U)$ enjoys a functional equation, the so-called composition property $\Phi_{\tau +t}(u\diamond_{\tau}v)\quad =\quad {\mathbb{T}}_ t\Phi_{\tau}u+\Phi_ tv,\quad u,v\in \Omega,\quad \tau,t\geq 0,$ where $$1\leq p\leq \infty,$$ and $$u\diamond_{\tau}v\in \Omega$$ means the $$\tau$$-concatenation of u and v; $$=u(t)$$ for $$0\leq t<\tau,$$ $$=v(t-\tau)$$ for $$t\geq \tau.$$ An abstract linear control system on X and $$\Omega$$ is defined as a pair $$\Sigma =({\mathbb{T}},\Phi)$$, where $$\Phi_ t$$ is in $${\mathcal L}(\Omega,X)$$ and enjoys the composition property. The representation theorem is first shown: Let $$1\leq p<\infty$$ and $$\Sigma$$ be an abstract linear control system. Then there is a unique $$B\in {\mathcal L}(U,X_{-1})$$ ensuring the relation (1). Here, $$X_{-1}$$ is a Banach space obtained as the completion of X relative to the norm $$\| (\beta I-A)^{-1}x\|$$, $$\beta\in \rho (A)$$. Thus, B is generally unbounded from U to X. The fact that the converse is not always true leads to the definition of a class of B: An operator $$B\in {\mathcal L}(U,X_{-1})$$ is called p-admissible for $${\mathbb{T}}$$ if $$\Phi_ t\in {\mathcal L}(\Omega,X)$$ for $$t\geq 0$$. Then, if $${\mathbb{T}}$$ is a group and $$p<\infty$$, the necessary and sufficient condition for B to be p-admissible is that $$\{t>0$$; $$\Phi_ tu\in X\}\neq \emptyset$$ for every $$u\in \Omega$$. Some related results are also shown. An application to a one-dimensional wave equation on a bounded interval is given.
Reviewer: T.Nambu

##### MSC:
 93C25 Control/observation systems in abstract spaces 93C20 Control/observation systems governed by partial differential equations 93B15 Realizations from input-output data 93B28 Operator-theoretic methods 93C05 Linear systems in control theory 47D03 Groups and semigroups of linear operators
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