(C,A,B)-pairs in infinite dimensions. (English) Zbl 0553.93037

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 \(S^*(Im E)\subset V^*(Ker D)\).
Reviewer: M.Kono


93C25 Control/observation systems in abstract spaces
47A15 Invariant subspaces of linear operators
47D03 Groups and semigroups of linear operators
93C05 Linear systems in control theory
93C35 Multivariable systems, multidimensional control systems
46C99 Inner product spaces and their generalizations, Hilbert spaces
Full Text: DOI


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