A remark on strong stability of linear Hamiltonian systems
Consider the linear Hamiltonian system \[ (*)\quad \dot x=Ax,\quad x\in R^{2n},\quad A\in sp(2n,R), \] so that \[ A=JH,\quad H^ T=H,\text{ and } J=\left[ \begin{matrix} 0\\ -I\end{matrix} \begin{matrix} I\\ 0\end{matrix} \right]. \] The system (*) (or \(A\in sp(2n,R))\) is called stable if the equilibrium solution \(x\equiv 0\) is Lyapunov stable. It is called strongly stable if all linear Hamiltonian systems with \(A'\in sp(2n,R)\) sufficiently close to A are stable. Define the n quadratic first integrals by \(h_ k(x)=(-1)^ k(1/2)<JA^{2k-1}x,x>,\) \(k=1,2,...,n\), where \(<\cdot,\cdot >\) is the standard scalar product in \(R^{2n}\). The following characterization of strong stability is obtained. Proposition 1. A linear Hamiltonian system is strongly stable if and only if some linear combination of the n quadratic first integrals \(h_ k\), \(k=1,2,...,n\), is a nondegenerate definite quadratic form. A similar characterization of strong stability for elements of the symplectic group Sp(2n,R). \(S\in Sp(2n,R)\) is called stable if \(\| S^ k\|\) is bounded for all \(k\in Z\). It is called strongly stable if all \(S'\in Sp(2n,R)\) sufficiently close to S are stable. Define \(f_ k(x)=<JS^ kx,x>\), \(k=1,2,...,n.\) Proposition 2. \(S\in Sp(2n,R)\) is strongly stable if and only if some linear combination of the n quadratic first integrals \(f_ k\), \(k=1,2,...,n\), is a nondegenerate definite quadratic form.
scientific article; zbMATH DE number 4156985