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Destabilizability of linear Hamiltonian systems. (Russian) Zbl 0609.34055
The authors discuss the stability of the zero solution of the so called linear Hamiltonian system (1) $$\dot x=A(t)x$$, $$x\in {\mathbb{R}}^{2n}$$, $$t\in {\mathbb{R}}^+$$; where A(t) is a piecewise continuous, bounded linear operator. Let $$J: {\mathbb{R}}^ n\to {\mathbb{R}}^ n$$ be an operator satisfying $$J^{-1}=J^*=-J$$. The system (1) is called linear Hamiltonian if JA(t) is a selfadjoint operator for any $$t\in {\mathbb{R}}^+.$$
Let $${\mathcal H}$$ denote the set of such systems with the sup norm $$\| A\| =\sup_{t>0}\| A(t)\| =\sup_{t>0}\{\sup_{| x| =1}(| A(t)\cdot x|)\}$$ and $$\dot {\mathcal H}^ a$$periodic subset of $${\mathcal H}$$. The authors recall the following result of V. I. Arnol’d. A system $$A\in \dot {\mathcal H}$$ is linear Hamiltonian if and only if its Cauchy operator X(t,s), $$t,s\in {\mathbb{R}}^+$$ is symplectic, that is $$J^*X^*JX'=I$$ (the identity operator). The system A is said to be symplecticly transformable into system B if there exists a symplectic Lyapunov transformation $$y=L(t)x$$ transforming $$\dot x=A(t)x$$ into $$\dot y=B(t)y.$$
The authors prove a number of interesting results, such as: A system $$A\in {\mathcal H}$$ is stable if and only if it can be symplectically transformed into the system $$\dot y\equiv 0$$. In the space $${\mathcal H}$$ the stable systems form a nowhere dense set. If $${\mathcal H}_ 0(A)$$ denotes the subset of systems $$B\in {\mathcal H}$$ such that $$\lim_{t\to \infty}\| B(t)-A(t)\| =0,$$ then the following theorem is true: For any stable system $$A\in {\mathcal H}$$ there exists an unstable system $$B\in {\mathcal H}_ 0(A)$$.
Reviewer: V.Komkov

MSC:
 34D20 Stability of solutions to ordinary differential equations 34A30 Linear ordinary differential equations and systems 70H05 Hamilton’s equations