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Algebraic systems theory towards stabilization under parametrical and degree changes in the polynomial matrices of linear mathematical models. (English) Zbl 0698.93060
Summary: This paper deals with the stabilization of the linear time-invariant finite-dimensional control problem specified by the following linear spaces and subspaces on \({\mathbb{C}}:\) \(\chi\) (state \(space)=\chi_*\oplus \chi_ d\), U (input \(space)=U_ 1\oplus U_ 2\), Y (output \(space)=Y_ 1\oplus Y_ 2\), together with the linear mappings: \(Q_ s=\chi \times U\times [0,t]\to \chi\) associated with the evolution equation of the \(C_ 0\)-semigroup S(t) generated by the matrices, of real or complex entries \(A\in {\mathcal L}(\chi,\chi)\) and \(B\in {\mathcal L}(U,\chi)\) of the differential system \[ \dot x(t)=Ax(t)+Bu(t),\quad x(0)=x_ 0; \] \(Q_ 0:\) \(\chi\) \(\times U\times [0,t]\to Y\) (output equation) and \(Q_ f:\) \(Y_ 2\times [0,t]\to U_ 2\) (feedback law). The stabilization for variations in the values of the parameters and structures of the above matrices with respect to a nominal system (of state space \(\chi_*)\) is investigated. The study is made in the context of algebraic systems theory and it includes the variation of the degrees, but not of the orders, of the associated polynomial matrices with respect to the nominal ones.
MSC:
93D15 Stabilization of systems by feedback
93B25 Algebraic methods
93C25 Control/observation systems in abstract spaces
93C05 Linear systems in control theory
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