| In recent years,fractional differential equations are widely used in many fields such as engineering,physics,economics,etc.The theory of operator semigroups and resolvent in Banach spaces is an important tool of handling fractional differential equations in infinite spaces.The concepts of controllability and optimal controls play crucial roles in control theory.Therefore,it is theoretical and practical demand to study the controllability and optimal controls for fractional differential equations by using the theory of semigroup and resolvent under certain conditions.This dissertation deals with the necessary and sufficient conditions of control-lability for linear and nonlinear fractional differential systems,and the existence of Lagrange optimal controls and time optimal controls governed by fractional differ-ential systems.The arrangement of the dissertation is as follows:In Chapter 1,we introduce research background,research states home and abroad as well as the main work of this dissertation.In Chapter 2,we recall some preliminaries of this dissertation,including the definitions and related properties of fractional integral and derivatives,semigroups and C-semigroups,resolvent and multivalued maps.The generation theorem of semigroups,C-semigroups and resolvent.In Chapter 3,we deal with the controllability of the following fractional linear differential system:where 0<??1.A generates an exponentially bounded C-semigroup {S(t)}t?0,x(t)?X.u?Lp(J,Y)(p>1/?).X,Y are Banach spaces.The definitions of mild solutions for fractional linear differential systems are given by Laplace transformation,probability density function as well as the defi-nition and properties of C-semigroup.Moreover,the definitions of controllability for the linear system are established.On this basis,on one side,we first investi-gate the necessary and sufficient conditions of exact controllability and exact null controllability in reflexive Banach spaces X and Y for the system in a operator form.Moreover,we remove the reflexivity of X,and the necessary and sufficient conditions of exact controllability and exact null controllability in the operator form are also hold by using different proof methods.Then,the necessary and sufficient conditions of exact controllability and exact null controllability in a resolvent form are researched under the conditions that X,Y are Hilbert spaces and p = 2.On the other side,we first prove the necessary and sufficient conditions of approximate controllability and approximate null controllability for linear systems in a operator form.Then,we assume that X,X*are strictly convex,and the necessary and suffi-cient conditions of approximate controllability and approximate null controllability for linear systems in a resolvent form are obtained in reflexive Banach space X and Hilbert space Y using duality mapping.At last,the approximate controllability of nonautonomous fractional differential system and semilinear fractional differential system with C being a regular operator are discussed under the assumption that the corresponding linear system is approximately controllable.The results in this chapter improve and generalize the corresponding conclusions of integer-order linear system,and the fractional linear system with A being the generator of a strongly continuous semigroup.In Chapter 4,we study the approximate controllability of the following frac-tional differential system with nonlocal conditions:where 1<q<2.A generates a resolvent {Sq(t)}t?0.x(·)?X.u(·)? L2(J,U).X,U are Hilbert spaces.The definitions of mild solutions of the system are obtained by using the tool of convolution,resolvent as well as the relevant operators generated by resolven-t.On this basis,we firstly prove that the relevant operators have properties of compactness and continuity in the uniform operator topology with the aid of the compactness and continuity in the uniform operator topology assumptions on the resolvent.Secondly,the expression for the control function is obtained by the corre-sponding linear regulator problem.Thirdly,the Lipschitz continuity of f is removed,and the existence results of fractional semilinear system are obtained by making full use of the properties of resolvent and the relevant operators generated by resolvent as well as Schauder's fixed point theorem.In addition,the compactness of nonlocal item g is weakened by using approximation techniques.Finally,the approximately controllable result of the above semilinear control system is acquired under the as-sumption that the corresponding linear system is approximately controllable,which improves and extends some results on this topic.In Chapter 5,we study the following Lagrange optimal control problem(P):where the cost function J(x,u)=?0bL(t,x(t),u(t))at,and(x,u)satisfies the fol-lowing composite fractional semilinear relaxation systems:where 0<?<1.A generates a resolvent {S1-?(t)}t?0.x(-)?X and u(·)?LP(J,Y).X is a Banach space,and Y is a reflexive Banach space.U:J ?2Y\{(?)}is a admissible control set.f:J x X?X.The definitions of mild solutions of the relaxation systems are obtained by Laplace transformation and the definition of resolvent.Based on this,on one hand,we assume that the nonlinear function is locally Lipschitz continuous,and the exis-tence and uniqueness of mild solution is acquired using generalized Banach contrac-tion principle.On this basis,the Lagrange optimal feasible solutions are obtained by establishing minimizing sequence and the Gronwall inequality.On the other hand,we derive the existence results of fractional relaxation systems by Schauder's fixed point theorem under the compactness and continuity in the uniform operator topology assumptions on the resolvent.Furthermore,applying the idea of con-structing minimizing sequences twice,the Lagrange optimal feasible solutions are also obtained,which implies that the uniqueness of mild solution is not the sufficient condition of the existence of Lagrange optimal feasible solutions.The results in this chapter improve and generalize the related conclusions in literatures.In Chapter 6,we study the following time optimal control problem(Q):where AdWT and U0 represent the sets of feasible solutions and control functions under certain conditions,respectively.The feasible solutions(y,u)fulfils the following fractional differential system with Riemann-Liouville derivatives:where 0<7<1.y(t)?X and u(t)?Y.X is a Banach space,and Y is a reflexive Banach space.A generates a C0 semigroup {T(t)}t?0.Uad is an admissible set for control functions.The definitions of mild solutions for the system with Riemann-Liouville frac-tional derivatives are given by using Laplace transformation,probability density function as well as the definition of semigroup in the space C1-?([0,d],X).On this basis,we firstly prove the compactness and continuity in the uniform opera-tor topology as well as similar semigroup properties of the the relevant operator S?(t)(t>0)generated by the semigroup by using the compactness of semigroup.Secondly,the existence results of the system are obtained using those properties and Schauder's fixed point theorem.Thirdly,the approach of establishing time minimizing sequences twice is applied to acquire the time optimal feasible solutions without the Lipschitz continuity of nonlinear function.In addition,the reflexivity of state space is removed by making full use of compact method.Finally,an exam-ple is given to illustrate the main conclusions.Our work essentially improve and generalize the corresponding results in literatures. |
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