The Control Of Two Classes Of Non-autonomous Linear Systems Based On Floquet Theory And KAM Theory

The study of non-autonomous linear systems occupies an important position in mathemat-ics,and has a wide range of applications in quantum mechanics,material physics and other fields.In particular,the problem of stabilization of non-autonomous systems has attracted more and more attention in the fields of physical and chemical processes,biology,economics,and en-gineering.This paper studies the stabilization of two special classes of non-autonomous linear systems using the reducibility of dynamical systems.The first class are the the n-dimensional systems with periodic coefficient.This type of sys-tem is derived from the attitude control of the satellite,the vibration attenuation of the helicopter,and the research of crystals.Firstly,this paper uses Floquet theory to construct a transformation with the same period and the original periodic system is conjugated into an autonomous linear system by the transformation.Then we use this transformation to design a time-varying linear feedback controller,and combined with the Lyapunov stability theory we prove the asymptot-ically stability of the periodic system under the suitable conditions.This result holds for the n dimensional systems.Finally,we enumerate the numerical simulations in the three cases in the group SL(2,R),which are hyperbolic system,elliptic system,and one dimensional linear stationary Schr?dinger equation with periodic potential.These results verify the effectiveness of the method.The second class are 2-dimensional finitely differentiable systems with quasi-periodic co-efficient.From the perspective of the number of frequencies,the quasi-periodic system is a high-dimensional extension of the periodic system.Such systems play a fundamental role in the fields of quantum Hall effect,spectrum theory of quasi-crystal,and control of cold atom.Firstly,we design a quasi-periodic time-varying linear feedback controller under the assump-tion that the system is reducible,and combined with the Lyapunov stability theory to prove that the system can achieve asymptotically stability under appropriate conditions.Next we prove the full measure reducibility of a class of SL(2,R)system.That is,for almost all rotation numbers,the class of the systems can be conjugated into an autonomous system.Since the re-ducibility of quasi-periodic system depends not only on the structure of the system itself,but also on the number-theoretic properties of frequencies,we only need those frequencies with a slower Diophantine approximation speed.The idea of the proof is to treat the quasi-periodic coefficients as small perturbation of the constant coefficients,and expand the quasi-periodic coefficients into Fourier series.By using KAM iteration,one can construct iterative sequences,and gradually eliminate the time-dependent coefficients of each order while retaining the con-stant coefficients.Hence we can obtain an autonomous system in the final.This paper further extends this method,where we limit the number of iterations,increase the perturbation,and promote the high-dimensional system.Finally,a quasi-periodic system with two frequencies is constructed for numerical simulation,and the method is verified.