Simultaneous Controllability And Observability Of Infinite-Dimensional Double Linear Systems

The control theory of infinite-dimensional linear systems has many applications inastronvigation,navigation,industrial process,sociometric fields and so on. Fromthe year 1970s, people have started researching the model of elastic vibrating systemsand vibrating control, and have many important achiements in spectral-analysis ofvibrating systems,controllability,feedback-stabilitation and so on. In addition,people also have some profound results in the fields of pole assignment of single-inputand single output linear systems,frequency-domain judgement of the stability oflinear systems,optimal control time of linear systems,max-value principle ofgeneral infinite-dimensional linear systems,control of people systems,the sumfertility-rate critical value,the population forecasting and so on. But there are manyproblems which need to be solved for distribution parameter systems, and especiallythe boundary control problems which is one of the most important problems, havemany essential problems to be solved. In recent years, people have comprehensiveresearch in boundary control of elastic systems. Especially, people allways have beeninterested in researching exact controllability and exact observability for distributionparameter systems and obtain some profound results. In addition, people start look-ing for the judging conditions for simultaneous exact controllability and simultaneousexact observability of two or more systems, that is to say, whether we can find thejudging conditions for simultaneous exact controllability and simultaneous exact ob-servability of two or more systems with the same inputs and outputs. Sometimes itis difficult to obtain the exact controllability and exact observability for simple lin-ear system in finite time, so it is meaningful to research simultaneous approximatecontrollability and simultaneous approximate observability for some linear systems ininfinite time. In this paper, except the preliminaries in chapter one, we consider the si-multaneous exact (approximate) controllability and observability of infinite-dimensionlinear systems under some conditions from chapter two to chapter four.In chapter two, firstly we obtain the sufficient and necessary conditions for si-multaneous approximate controllability and simultaneous approximate observabilityin infinite time for systems (A₁,B₁, C₁) and (A₂, B₂, C₂) with finite-rank inputs andoutputs by means of matrix rank conditions. where the generators of systems A_i are the following defined self-adjoint operators: A_iz=sum from n=1 to∞λ_nⁱ sum from j=1 to r_nⁱ＜z,φ_njⁱ＞φ_njⁱ, z∈X_i i=1, 2Secondly, we get the sufficient and necessary conditions for simultaneous approximatecontrollability and simultaneous approximate observability in infinite time of doubleRiesz-spectral systems though the properties of single Riesz-spectral system.In chapter three, we use the spectral sets of the generators of linear systems to getthe sufficient conditions for simulateous approximate controllability of the followingtwo systems in appropriate situation:Here, we extend the conclusions about simultaneous approximate controllability ofdouble linear systems of Weiss in [14].In chapter four, we use the Hautus conditions of exact observability of the systemgenerated by skew-adjoint operators to get the sufficient conditions for the simultane-ous exact observability of the following two systems:The originality innovation in my paper is that, we mainly study from the spectralsets of generators of double systems and use the knowledge of semigroup theory andfunctional analysis to get the sufficient and necessary conditions of simultaneous ap-proximate controllability and simultaneous approximate observability in infinite timeof the double infinite-dimension linear systems with finite-rank inputs and outputs. Atthe same time, we also obtain some sufficient conditions to judge simulateous approx-imate controllability,simulateous approximate observability and simulateous exactobservability of the two systems.