Disconjugacy and transformations for symplectic systems
Let \(I\) be the \(n\times n\) identity matrix, and let \(J=\left (\begin{smallmatrix} 0 & I \\ -I & 0 \end{smallmatrix} \right)\). The \(2n\times 2n\) matrix \(S=\left( \begin{smallmatrix} A & B \\ C & D \end{smallmatrix} \right)\) is called symplectic if \(S^TJS=J\) holds, and the system \(z_{k+1} =S_kz_k\), \(0\leq k\leq N\), is symplectic if the matrix \(S\) is symplectic. A solution \(z= {x\choose u}\) to the system has a generalized zero in \([k,k+1]\) if \(x_k\neq 0\), \(x_{k+1} \in\operatorname{Im} B_k\) and \(x^T_k B^+_kx_{k+1} =0\), where \(B^+_k\) denotes the Moore-Penrose inverse of the matrix \(B_k\). The system is called disconjugate on \({\mathcal J}=[0,N]\cap\mathbb{Z}\) if no solution of the system has more than one generalized zero (if \(x_0=0\) no zero). Theorem 1 shows a lot of equivalent facts about disconjugacy, possession of generalized zeros and other similar properties of the solution to the symplectic system. Similar results are obtained relative to the reciprocal symplectic system \(z_{k+1}= S^{-1}_k z_k\). The problems of eventually disconjugate solutions and disconjugacy preserving transformations are considered as well.
scientific article; zbMATH DE number 1143464
scientific article