Disconjugacy, transformations and quadratic functionals for symplectic dynamic systems on time scales
| Title in English | Disconjugacy,transformations and quadratic functionals for symplectic dynamic systems on time scales |
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| Authors | |
| Year of publication | 2001 |
| Type | Article in Periodical |
| Magazine / Source | J. Difference Equations Appl. |
| MU Faculty or unit | |
| Citation | |
| Field | General mathematics |
| Keywords | Symplectic dynamic system; time scale; quadratic functional; Roundabout theorem |
| Description | In this paper we study qualitative properties of the so-called symplectic dynamic system (S) z^\Delta=S(t)z on an arbitrary time scale T, providing a unified theory for discrete symplectic systems (T=Z) and differential linear Hamiltonian systems (T=R). We define disconjugacy (no focal points) for conjoined bases of (S) and prove, under a certain minimal normality assumption, that disconjugacy of (S) on the interval under consideration is equivalent to the positivity of the associated quadratic functional. Such statement is commonly called Jacobi condition. We discuss also solvability of the corresponding Riccati matrix equation and transformations. This work may be regarded as a generalization of the results recently obtained by the second author for linear Hamiltonian systems on time scales. |
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