Coupled intervals for discrete symplectic systems
Consider discrete symplectic systems of the form \(Z_{k+1}=s_{k}z_{k} , k\in[0, N]\) where \(z_{k}=(x_{k}, u_{k})\) is a real \(2n\)-vector, \(s_{k}=(A_{k}, B_{k}, C_{k}, D_{k})\) is a real symplectic \(2n\times 2n\)-matrix and definiteness of the corresponding quadratic functional \(F_{0}(z)=\sum_{k=0}^{N} Z_{k}^{T}{S_{k}^{T}HS_{k}-H}z_{k}\), where \(H=(0, 0, I, 0)\) is a \(2n\times 2n\)-matrix and \([0, N]\) is a discrete interval with indicated endpoints. It was shown that the oscillation theory of discrete Hamiltonian systems \(\Delta x_{k}=A_{k}X_{k+1}+B_{k}u_{k}, \Delta u_{k}=C_{k}X_{k+1}-A_{k}^{T}u_{k}\), \(k\in[0, N]\) where \(B_{k}\) and \(C_{k}\) are symmetric and \(I-A_{k}\) is invertible, extends naturally to discrete sympletic system. Recently, the authors derived new characterizations of the definiteness of the Hamiltonian quadratic functional associated with the discrete Hamiltonian systems via the motion of a (strict) coupled interval. In this paper the authors extend the above mentioned Hamiltonion coupled interval notion to the discrete symplectic system and relate it to the definiteness of the corresponding quadratic functional. The main difficulty in this generalization lies in the fact that the evolution matrix \(\Phi_{k+1}=A_{k}A_{k-1}\dots A_{j}\) may be singular in the general symplectic case, as apposed to the Hamiltonian case where the matrices \(A_{k}=(I-A_{k})^{-1}\) are invertible.
scientific article; zbMATH DE number 5126042
scientific article