The Properties Of Resolvent Operators And Their Applications In Fractional Evolution Systems

Since differential equations with fractional derivatives have broad applicability in describing many physical problems with memory features and genetic properties,these equations have been the active research issues in recent years.When the frac-tional derivatives are utilized to describe some practical applications in viscoelastic materials and used in the definition of state space description,including stabili-ty,observability and controllability,Riemann-Liouville type fractional derivatives operators or Hilfer type(general Riemann-Liouville type)are more suitable than Caputo type.Therefore,we lay special emphasis on investigating Riemann-Liouville fractional evolution systems and Hilfer fractional evolution equations in the present thesis.On the other hand,since the resolvent approach,a generalization of the semigroup method,is convenient and efficient in studying abstract Volterra equa-tions and fractional evolution systems,the goal of the present thesis is to explore the properties of resolvents(including stability,subordination principle,approximation and compactness)and the properties of the operators related to resolvent(includ-ing continuity and compactness),and cope with the approximate controllability of Riemann-Liouville fractional evolution systems and the time optimal control prob-lem of Hilfer fractional evolution equations.The structure of the present thesis is as follows:Chapter 1 contains the research background and our main work of the current thesis.Chapter 2 is intended to collect some preliminaries,including the definitions and important results of fractional calculus and Mittag-Leffler functions,the con-cepts and important properties of semigroups and resolvents,and some notions and conclusions from multi-valued analysis.Chapter 3 discusses the uniform stability of a resolvent family {Rh(t)}t?0,de-pending on a parameter h.We first present a GGP type theorem on {Rh(t)}t?0 and give some sufficien-t conditions to guarantee the uniform stability of {Rh(t)}t?0 by utilizing Fourier analysis method.Then,under some suitable conditions,we derive that the weak LP-stability of {Rh(t)}t?0 can imply its uniform stability by employing the GGP type theorem and the adjoint theory of resolvent.Our work generalizes some pre-vious results on the uniform stability of a Co-semigroup family {Th(t)}t?0 and a resolvent family {R(t)}t?0,without depending on the parameter h.Chapter 4 deals with the subordination principle and approximation of a/3 order and ? type resolvent {T?,?(s)}s>0.We first introduce the definition of exponential boundedness for s ?s0(s0>0)of the resolvent and propose some sufficient conditions to guarantee exponen-tial boundedness for s ? s0.We then establish the subordination principle of{T?,?(s)}s>0 by using probability density functions and the conditions.Then,when the resolvent generated by A is exponentially bounded for s ? s0,we verify that kA(k ? 0)also generates a resolvent {T?,?k(s)}s>0.Furthermore,we investigate the approximation of{T?,?k(s)}s>0.With the help of the subordination principle,it is convenience to give applications about Hilfer fractional evolution systems and Riemann-Liouville fractional evolution equations.Additionally,the approximation theorem can enable us to treat the time optimal control problem by Meyer approx-imation.In Chapter 5,by employing the technique combining integral contractor,re-solvent and space decomposition,we analyze the approximate controllability of the following system in a Hilbert space H:where A:D(A)(?)H ?H is a generator of an ? order resolvent {T?(t)}t>0 and f:J×H ? H is a nonlinear function with a regular integral contractor assumption.Furthermore,u? L2(J;U)and B?L(L2(J;U);L2(J;H)),where J?[0,b]and U is a Hilbert space.Under suitable assumptions of the operator t1-?T?(t),we first introduce a no-tion of mild solutions to the system by the resolvent properties and exhibit the existence and uniqueness of solutions by exploiting an integral contractor condi-tion.Then,we deal with the approximate controllability problems by utilizing an approach of space decomposition.We end up exploring an example to illuminate that the hypotheses on t1-?T?(t)are suitable.Emphasis here is that the technique combining integral contractor,resolvent and space decomposition can enable us to treat all previous work on approximate controllability problems of integer-order evo-lution equations and fractional evolution systems with Caputo type derivatives or Riemann-Liouville type in Hilbert spaces.Chapter 6 is devoted to exploring the following Riemann-Liouville fractional delay control system in a Banach space V:where 0<?<1,A generates a ? order resolvent {R?(t)}t>0,? is continuous on[-r,0],y(t)=?(?)t1-?y(t)for t? J:=[0,b],y(0)=lim t?0+y(t),yt(?)=y(t +?)for t ? J and ??[-r,0],f is a nonlinear function without Lipschitz assumption.Furthermore,u?Lp(J;U)and B ?L(Lp(J;U);LP(J;V))(p>1/?),where U is a Banach space.We first propose a suitable concept of mild solutions to the system by uti-lizing the resolvent method and the convolution technique.Then,we display the topological characteristics(compactness and R? properties)of the corresponding solution set by employing the resolvent approach.In addition,with the aid of the topological characteristics of solution set and the idea of resolvent,we analyze the approximate controllability for the system without the Lipschitz condition of the nonlinear terms.We end up addressing a fractional diffusion control system by utilizing our theoretical findings.Our results generalize and extend some recent results on this topic.Moreover,it is worth pointing out that our technique can be applied to the approximate controllability problems of integer-order evolution equations and fractional evolution systems with Riemann-Liouville type derivatives or Caputo type in Banach spaces.In Chapter 7,with the help of the approximation theory of the ?-order and ?-type resolvent,we handle the time optimal control problem of the following system in a Banach space V by employing Meyer approximation:where 0<?<1,0???1,the notation Jt?(1-?)means the ?(1-?)order fractional integral operator,A generates a ? order and ? type resolvent ?T?,?(t)}t>0 which is exponentially bounded for s ? s0(s0>0),f:[0,T]xV ? V is a nonlinear function without Lipschitz assumption.Furthermore,Vad is an appropriate admissible set and B ? L?([0,T],L(Y,V)),where Y is a separable and reflexive Banach space.We first introduce a suitable notion of mild solutions to this system by using convolution technique and resolvent theory.Then,we display the existence result by employing Schauder theorem and the properties of resolvent which are similar to the the semigroup property.However,the uniqueness of solutions cannot be guar-anteed.Furthermore,with the help of the approximation theory of the ? order and? type resolvent established in Chapter 4,by utilizing the approach of formulating minimizing sequences twice(First,for the fixed control w,we set up the minimiz-ing sequences of the state.Then,we construct the minimizing sequences of the control),we find the optimal state-control pair of the Meyer problem P? without the Lipschitz continuity of the nonlinear term f.Moreover,with the aid of the approximation theory of the resolvent,we seek the optimal state-control pair of the system by Meyer approximation.We end up a simple application to demonstrate the validity of our theoretical findings.This chapter provides a new technique to study the time optimal problem.Furthermore,our work generalizes some previous results on the time optimal controls.Chapter 8 provides our conclusions and our future work.