(C,A,B)-pairs in infinite dimensions
This paper considers infinite-dimensional systems on a real separable Hilbert space X of the form \(\dot x=Ax+Bu\), \(y=Cx\) and introduces a (C,A)-invariant, a T(C,A)-invariant, an (A,B)-invariant and a T(A,b)- invariant subspace where A is the infinitesimal generator of a strongly continuous semigroup T(t) on X and \(B\in L[R^ m,X]\), \(C\in L[X,R^ k]\). Consider the feedback processor on the Hilbert space W of the form \(\dot w=Nw+My\), \(u=Lw+Ky\), where M, L, K are bounded linear operators and N is the infinitesimal generator of a strongly continuous semigroup on W. Defining \(x^ e=(x,w)\), we obtain the closed-loop operator \[ A^ e=\left[ \begin{matrix} A+BKC\\ M\end{matrix} \begin{matrix} BL\\ N\end{matrix} \right]. \] \(A^ e\) is also an infinitesimal generator of a semigroup \(T^ e(t)\). If (S,V) is a trajectory-invariant (C,A,B)-pair and \(S\subset D(A)\), then there exist an extension space \(W\cong V/S\), bounded operators M, L, K, a linear operator N on W and a closed subspace H of \(X^ e=X\oplus W\) which is \(A^ e\)-invariant. Further, N is an infinitesimal generator, the feedback processor is well defined and H is invariant under the semigroup generated by \(A^ e\). Consider the system on X \(\dot x=Ax+Bu+Eq\), \(y=Cx\), \(z=Dx\), where \(E\in L[R^ r,X]\), \(D\in L[X,R^ l]\), q is the disturbance and z is the output to be decoupled. DDPM is to design a feedback processor such that the output z is decoupled from the disturbance q, i.e. \(D^ e\int^{t}_{0}T^ e(t-s)E^ eq(s)ds=0\), \(t\geq 0\), where \(T^ e(t)\) is the semigroup generated by the closed-loop system operator \(A^ e\). Let \(S^*(L)\) and \(V^*(K)\) denote the infimal T(C,A)-invariant subspace containing a closed subspace L and the supremal T(C,A)-invariant subspace contained in a closed subspace K respectively. Assume that \(S^*(Im E)\) and \(V^*(Ker D)\) exist and satisfy \(S^*(Im E)\subset D(A)\). Then, DDPM is solvable if \(S^*(Im E)\subset V^*(Ker D)\).
scientific article; zbMATH DE number 3882296