Classification of systems under state and output transformations (Q1269949)

Apparently, the title of this paper has not been translated correctly; it should rather be ``Classification of systems under state and input transformations''. This paper studies linear control systems of the form \[ \dot x= Ax+ Bu,\tag{\(*\)} \] \(x\in K^n\), \(u\in K^m\), \(K\) is a field of characteristic \(0\), \(m< n\), acted on by a subgroup of \(G= GL_n(K)\times GL_m(K)\) of the feedback group \[ (T, S)(A, B)\mapsto (TAT^{-1}, TBS^{-1}),\tag{\(**\)} \] \(T\in GL_n(K)\), \(S\in GL_m(K)\), consisting of the state and input coordinate changes. By assumption, the acting group has dimension lower that the dimension of the ambient system space. The paper focuses on characterizing the set of so-called prestable systems under the action \((**)\), i.e., an open invariant set of control systems, composed of closed orbits. Using methods of algebraic geometry and invariant theory, the following main results have been established. \(\bullet\) If the so-called extended power of \((A,B)\) is reachable then \((A,B)\) is prestable, and not vice versa (Corollary 1). \(\bullet\) If \(n\) is not divisible by \(m\), a semiinvariant \(g\) has been found such that, if \(g(A, B)\neq 0\) then \((A, B)\) is prestable (Theorem 4). \(\bullet\) For \(m=n-1\) a semiinvariant \(P\) has been constructed such that, if \(P(A,B)\neq 0\) then \((A,B)\) is prestable (Corollary 2), prestability has been proved to be equivalent to regularity (Proposition 2, Theorem 6), and a canonical form of \((A,B)\) is derived. \(\bullet\) A further condition for prestability has been obtained asserting that \((A,B)\) is prestable if it is stable (i.e., the orbit of \((A,B)\) is closed and its dimension equals the dimension of the acting group) under the action \((**)\) of the product of special linear groups \(SL_m(K)\times SL_m(K)\) (Theorem 7).