Schr?dinger Equation Based Stability Analysis Of DNA Functional Model

With the development of bioinformatics,the functions of DNA have been studied extensively and intensively.In this thesis,the discrete model and continuous model are presented,respectively.From the discrete perspective,the structure matrix is given which is based on the analysis of DNA sequence.From the continuous point of view,the models of multi-base network and single-base state are set up and described by the Schr?dinger equation,in addition,the solution expansion property of tree-shaped network and the stability of Schr?dinger equation which is under the influence of time-delay and disturbance are illustrated.1.For discrete case,the analysis of DNA sequence is proposed.From the bases connecting orders,the corresponding structure matrix is obtained.According to the features of the structure matrix and the similarity between different structure matrices,the species that the DNA sequences belong to can be determined.2.The solution expansion of a tree-shaped network described by Schr?dinger equations is studied.According to the spectral analysis of the system operator,the asymptotic expressions of the eigenvalues are deduced.In addition,it can be obtained that the system operator generates a Gevrey class semigroup with ? > 2 from the discussion of the eigenvalues and eigenvectors.Then,the solvability of the system is proved.3.The Schr?dinger equation with boundary input time delay and distributed boundary input time delay are introduced respectively.For the time delay problem,the design method of feedback controller is presented to obtain the stability of the Schr?dinger equation.A ”partial state” predictor is designed for the system and a undelayed system is deduced.Based on the undelayed system,a feedback control strategy is designed to stabilize the time delay system.The exact observability of the undelayed dual system is checked.Then it is shown that the system can be stabilized exponentially under the feedback control.As the distributed time delay system is assumed to be unobservable,the exponential type observer should be designed before predicting the state of the original system.4.The stability of Schr?dinger equations under the external and internal disturbance are investigated respectively.As the adoption of sliding mode control to eliminate the disturbance,the linear system turns into a nonlinear one.Hence,it becomes difficult to analyse the well-posedness and the stability of the system.To obtain the well-posedness of the system,the Lions-LaxMilgram theorem is extended to the semilinear case.The stability of the disturbed systems are discussed according to the positive limit set principle and Lyapunov function.5.The exponential stability of the Schr?dinger equation with a constrained boundary input is considered.Due to the restricted feedback control,the Schr?dinger equation is nonlinear,whose well-posedness is solved by the nonlinear semigroup theory.The asymptotic and exponential stabilities of the system are discussed with the weak topology and Riesz basis method,respectively.