Admissibility of unbounded control operators.

*(English)*Zbl 0685.93043The 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