Regularity of Linear Systems of Differential Equations on the Axes and Pencils of Quadratic Forms

Abstract

It is considered linear systems of differential equations and investigated questions of regularity of these systems. To explore the regularity it is comfortable to use quadratic form whose derivative with respect to the adjoint system is positive definite. Sometimes it is possible to find such a quadratic form, the derivative of which with respect to the system is non-negative. There are examples showing that in this case we can't say anything about the exponential dichotomy of this system (that is, its regularity). The question arises whether it is possible to combine a certain set of quadratic forms to get such a form, the derivative of which with respect to the system is positive definite. This question is similar to the question that arises in the theory of control: having a set of certain data about an object, can one say something about this object as a whole. It turns out that this is possible, only a set of these quadratic forms should be special, in some sense complete. In the presented article the authors propose to write it with the help of some combination of specific symmetric matrices $S_1, S_2, \dots$ . So we have a quadratic form \[V_{p} =p_{1} \left\langle S_{1} \left(t\right)x,x\right\rangle +p_{2} \left\langle S_{2} \left(t\right)x,x\right\rangle + \dots +p_{k-1} \left\langle S_{k-1} \left(t\right)x,x\right\rangle +\left\langle S_{k} \left(t\right)x,x\right\rangle\] It is proved that the derivative of this quadratic form is positive definite for sufficiently large parameters $p_1, \dots, p_{k-1}$. The results are illustrated by examples.