Weak disconjugacy, weak controllability, and genera of conjoined bases for linear Hamiltonian systems
| Authors | |
|---|---|
| Year of publication | 2022 |
| Type | Article in Periodical |
| Magazine / Source | Annali di Matematica Pura ed Applicata |
| MU Faculty or unit | |
| Citation | |
| Web | https://link.springer.com/article/10.1007/s10231-022-01194-x |
| Doi | http://dx.doi.org/10.1007/s10231-022-01194-x |
| Keywords | Linear Hamiltonian system; Weak disconjugacy; Weak controllability; Genus of conjoined bases; Nonoscillation; Maximal order of abnormality; Principal solution at infinity |
| Description | In this paper, we discuss mutual interrelations between the notions of weak disconjugacy and weak controllability for linear Hamiltonian differential systems. These notions have been used in connection with the study of exponential dichotomy, nonoscillation, and dissipative control processes for these systems [e.g. (Johnson et al., in: Nonautonomous linear Hamiltonian systems: oscillation, spectral theory and control developments in mathematics, Springer, Cham, 2016)]. As our main results, we derive characterizations of the weak controllability and weak disconjugacy in terms of properties of certain subspaces arising in the recently introduced theory of genera of conjoined bases for linear Hamiltonian systems (Sepitka in J Dyn Differ Equ 32(3):1139-1155, 2020). We also present new results regarding the zero value of the maximal order of abnormality of the system in terms of a weak controllability condition, or in terms of a weak disconjugacy condition when the system is nonoscillatory and satisfies the Legendre condition. In our accompanying comments, we highlight the connections of the theory of genera of conjoined bases with the existence of principal solutions at infinity, which arise in the study of weakly disconjugate linear Hamiltonian systems. The results in this paper may be regarded as a completion and clarification of the previous considerations in the literature about the weak disconjugacy and weak controllability conditions for linear Hamiltonian systems [e.g. (Fabbri et al. in: J Math Anal Appl 380(2):853-864, 2011), (Johnson et al., in Nonautonomous linear Hamiltonian systems: oscillation, spectral theory and control developments in mathematics, Springer, Cham, 2016)]. |
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