Global Controllability And Approximate Controllability Of Fractional Order Differential Dynamical Systems

Controllability of differential dynamical systems is a basic property which is very helpful in many real world applications.With the development of science and technolo-gy,the controllability of differential dynamical systems plays an increasingly important role.So,the research of controllability has interested more and more mathematicians and scientists.The controllability of fractional order differential dynamical systems comes from the integer order ones and develops very rapidly in recent years.In this dissertation,we study the global controllability and almost controllability of fractional order differential dynamical systems.In the first chapter,the background and the development of fractional order con-trollability are introduced.In the seeond chapter,we give some relevant basic theory which is the foundation of our main results in the later chapters.The main conclusions of this thesis are established in the next three chapters.The Schaefer's fixed point theorem combining with Mittag-Leffler matrix func-tion is one of the efficient methods dealing with the global controllability of nonlinear dynamical system and integro-differential dynamical system.In the third chapter,we consider the global controllability of the following nonlinear fractional dynamical system:where 1<??2,-Aisan n×n matrix and B is an n,× m matrix,x(t)? R~n,u ? L~?(J,Rm),t?[0,b,]? J and f:J×R~n×Rm ?R~n n2 is continuous,~CD~? is Caputo differential operator.To analysis the nonlinear differential dynamical system(0.0.9),we consider the relevant linear system:Theorem 0.0.1 The linear system(0.0.10)is global controllable on J if and only if Grammian matrix is positive definite,where*denotes the matrix transpose.Theorem 0.0.2 If the linear system(0.0.10)is global controllable on J and the continuous function f satisfy the following condition:then the nonlinear system(0.0.9)is global controllable on J.Similarly,we study the fractional order integro-differential systems using the same method:where 1<??2,A is an n×n matrix and B is an n × m matrix,x(t)? R~n,u ? L~?(J,Rm,),t J,and f:J× R~n × Rm × Rm ?R~n,q:J× J × R~n × Rm ? R~n are continuous.Theorem 0.0.3 If the linear system(0.0.10)is global controllable on J,and exist a constant M>0,for all,s ? J,z,v ?&Cn(J)satisfying then integro-differential system(0.0.11)is global controllable on J.Based on the above results,we establish the global controllability of arbitrary higher order damping dynamic system in the fourth chapter:where p-1<?? p<q-1<?? q,q?p-1,p,q ? N,A is an n × n matrix,B is an n × m matrix,x(t)?R~n,u ?L?(J,Rm),t ?[0,b,],and.is a nonlinear continuous function.To analysis the higher order damping dynamic system(0.0.12),we consider the relevant linear system:where p-1<??p,q-1<??q and q?p-1,p,q?N,A is an n×n matrix,B is an n × m matrix,x(t)? R~n,u ? L~2(J,Rm).Theorem 0.0.4 The linear system(0.0.13)is global controllable on J iff the Grammian matrix is invertible.Theorem 0.0.5 If the following assumptions are satisfying:(1)For each t E J,the function f(t,·,·,·,··):J × R~n x Rm × R~n × R~n? R~n is continuous,the function f(·,x(·),u(·),y(·),z(·)):J ? R~n is strongly measurable for each x,y,z ? R~n,u ? Rm;(2)||f(t,x(t),u(t),CDax(t),cD?x(t))|| ? M,where t ? J,x E R~n,u ? Rm M?R.And the linear system(0.0.13)is global controllable on J,then,the nonlin.ear system(0.0.12)is global controllable on J.In the fifth chapter,we consider the approximate controllability of the following fractional differential dynamical system:where 1/2<a ? 1,t ? J =[0,b],A:D(A)(?)H ? H is the infinitesimal gen-erator of a uniformly bounded Co-semigroup T(t)on a separable Hilbert space H,f(t,x(t),~CD~?_x(t)))?(?)F(t,x(t),CD?x(t)).The notation(?)F stands for the generalized Clarke subdifferential of a locally Lipschitz function,·,·):H?R,see[118].The control function u ? L~2(L,U),U is a Hilbert space,B:U ? H is a bounded linear operator.Theorem 0.0.6 For each u ?L~2(J,U),we assume that the following hypotheses hold:(1)The Co-semigroup T(t)is compact and supt(0,?))||T(t)||?M;(2)F:J × H × H ? R is a function such that:(a)the function t ?(t,x,~CD~?_x)is measurable for all x ? H;(b)the function x ? F(t,x,~CD~?_x)is locally Lipschitz a.e.t ? J.Then the system(0.0.15)has a mild solution on J.Theorem 0.0.7 If for each h(·)? L~2(J,H),the,re exists a function g(·)? R(B)such that ?h = ?g,where R(B)denotes the range of operator B and R(B)is the closure of R(B)andT(t)is a differentiable semigroup,then the linear system is approximately controllable on J.Theorem 0.0.8 Assume that the f(t,x(t),~CD~?_x(t?satisfies exist A:L~2(J,H)?L~2(J,H),?(x)= {?? L~2(J,H):? ?(?)F,t ? J}.And the linear system(0.0.16)is approximately controllable,then,the non,lirnear system(0.0.15)is approximately con-trollable on J.We give an example to illustrate our main results at the end of each chapter.