The Estimates Of Eigenvalues For The Solution Of The Lyapunov Matrix Diferential Equations In Linear Systems

The matrix theory is widely used in the areas of control theory, dynamic pro-gramming, statistics, ladder networks, transport theory and stochastic fltering. Inlinear control systems, the discussion of many important properties such as control-lability, stability and observability can often be changed to solving related nonlinearmatrix equations.This paper, by using control inequalities, we provide lower bounds on sum-mation of eigenvalues (including the trace) for the solution of Lyapunov matrixdiferential equation in stationary-constant linear systems as well as upper and low-er bounds on summation of eigenvalues (including the trace) for the solution ofLyapunov matrix diferential equation in time-varying linear systems, which im-prove and generalize some existing results.In chapter one, we present some background knowledge of two kinds of Lya-punov matrix diferential equations, and introduce some symbols and defnitionused in this paper.In chapter two, by using control inequalities, integral inequalities and eigenval-ue inequalities of special matrix product, combined with Schur triangulation theo-rem and properties of exponential matrix, we obtain lower bounds on summationof eigenvalues (including the trace) for the solution of Lyapunov matrix diferentialequation in stationary-constant linear systems, which improve some existing resultsin some cases. The superiority will be illustrated by numerical examples.In chapter three, by using optimization inequalities and properties of convexfunction, combined with spectral decomposition theorem, the relationship on theeigenvalues for the derivative of matrix and the derivative for the eigenvalues ofmatrix as well as inequality techniques, upper and lower bounds on summation ofeigenvalues (including the trace) for the solution of Lyapunov matrix diferentialequation in time-varying linear systems are derived, which improve and generalizesome existing results. Numerical examples show the efectiveness.