Floquet theory for first-order delay equations and an application to height stabilization of a drone's flight

In this paper, we proposed a version of the Floquet theory for delay differential equations. We demonstrated that very natural assumptions for control in technical applications can lead us to a one-dimensional fundamental system. This approach allowed researchers to work with classical methods used in the case of ordinary differential equations. On this basis, new original unexpected results on the exponential stability were proposed. For example, in the equation $ x'(t)+a(t)x(t-\tau(t)) = 0 $, $ t\in[0, \infty) $, we avoided the assumption on the smallness of the product $ \sup_{t\in[0, \infty)}a(t)\sup_{t\in[0, \infty)}\tau(t) < 3/2 $ for asymptotic stability. We obtained that in the case of $ \omega $-periodic coefficient and delay, the fact that the period $ \omega $ was situated in a corresponding interval can lead to exponential stability. We then applied our new tests of stability to the stabilization of a drone's flight, where smallness of the noted above product could not be achieved from a technical point of view. For an equation with periodic coefficient and delay, we got a formula of the solution's representation on the semiaxis.