The Study Of Continuous-time Linear Systems’ Reachable Sets Under The Bounded Control Function

In the20th century, the linear control theory and practical application have a significantimpact on humans’ productions and social activities. The linear systems are considered to bethe basic research objects in the field of systems and controllabilities. For the control theory,controllability and observability are the classical theories. Controllability is the ability thatthe control function influences on the systemic behavior. Researching on reachable set has asignificant impact on the controllability. For the classical control theory, control function isjust an observable function. In this paper, we will discuss the linear continuous-time systemsunder the bounded control function.Strictly speaking, most of the problems in real life are the non-linear systems. Forsome not serious non-linear systems which only work in the small areas near the workingpoint, usually we study the nature of them by using approximate linear models. Thereforestudying the theories of linear systems play a pivotal role.First we will introduce the length and the perimeter of the boundary of the two-dimension linear discrete-time systems’ reachable sets under the bounded control function.For the linear continuous-time systems, the research of this aspect is very scanty. In thispaper we will give some conclusions of the linear continuous-time systems under thebounded control function.This paper contains three parts. First we will prove that throughthe continuous-time linear systems satisfy the rank of Kalman,it may be not control, thengive the necessary condition to the linear continuous-time systems under the boundedcontrol function. Second when the control function is the convex set, we will prove thereachable set is the convex set too. Last we will discuss the range of the reachable setsof the two-dimension linear continuous-time systems and give the relationship among thereachable set and the system matrix and the control matrix.