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A note on generalized invariant subspaces for infinite-dimensional systems. (English) Zbl 1060.93031

Consider the infinitesimal generator \(A(\alpha)\) which is defined as \(A_0 + \alpha \underline{A}\), where \(\alpha\) is a real vector, and \(\underline{A}\) is a vector consisting of bounded operators. Similarly, one defines the operator-valued function \(B(\beta)\). The authors define the subspace \(V\) to be generalized controlled \(S(A,B)\) invariant if there exists a bounded feedback \(F\) such that for all \(\alpha\) and \(\beta\) the subspace \(V\) is invariant under the semigroup generated by \(A(\alpha) + B(\beta) F\). The main result states that \(V\) is generalized controlled \(S(A,B)\) invariant if and only if there exists a feedback \(F\), vectors \(\alpha, \tilde{\alpha}, \beta, \tilde{\beta}\) with \(\alpha < \tilde{\alpha}\) and \( \beta < \tilde{\beta}\) (pointwise) such that \(V\) is invariant under the semigroups generated by \(A(\alpha) + B(\beta) F\) and by \(A(\tilde{\alpha}) + B(\tilde{\beta}) F\). They obtain similar results for generalized conditional \(S(C,A)\) invariance and for generalized \(S(C,A,B)\) invariance.

MSC:

93B27 Geometric methods
93C25 Control/observation systems in abstract spaces
93B52 Feedback control
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