Xiuwen LIZhenhai LIU

1.School of Science,Nanjing University of Sciences and Technology,Nanjing 210094,China

2.Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing,Yulin Normal University,Yulin 537000,China

E-mail:641542785@qq.com;zhhliu@hotmail.com

Jing LI

Hunan Province Key Laboratory of Mathematical Modelling and Analysis in Engineering;Department of Mathematics and Statistics,Changsha University of Science and Technology,Changsha 410114,China

E-mail:lijingnew@126.com

Chris TISDELL

Faculty of Science,The University of New South Wales,UNSW,Sydney 2052,Australia

E-mail:cct@unsw.edu.au

Abstract In this paper,we are concerned with the existence of mild solution and controllability for a class of nonlinear fractional control systems with damping in Hilbert spaces.Our fi rst step is to give the representation of mild solution for this control system by utilizing the general method of Laplace transform and the theory of(α,γ)-regularized families of operators.Next,we study the solvability and controllability of nonlinear fractional control systems with damping under some suitable sufficient conditions.Finally,two examples are given to illustrate the theory.

Key words existence;controllability;fractional control systems;damping;regularized resolvent family

1 Introduction

The purpose of this paper is to study the existence of mild solution and controllability of the following nonlinear fractional control systems with damping

Fractional calculus,as an extension of ordinary calculus,allows for the representation of the long-memory and non-local dependence of many processes and provides an excellent instrument for the description of memory and hereditery properties in a model.Consequently,both the ordinary and the partial di ff erential equations of fractional order have drawn great applications in the mathematical models and processes in the fi elds of physics,aerodynamics,electrodynamics,electricity mechanics,etc.See e.g.[3,6–11,13,15,18,25]and references therein.As stated in[10,22],fractional partial di ff erential equations became especially important for the modeling of the so called anomalous phenomena in nature and in the theory of the complex systems.In this connection,the so-called time-fractional di ff usion equation which is obtained from the classical di ff usion equation by replacing the fi rst-order time derivative by a fractional derivative of order α with 0< α <1 was explicitly introduced in physics to describe di ff usion in media with fractal geometry(special types of porous media)and became important and useful for di ff erent applications.

However,in the real material,the existence of damping is inevitable.Therefore,anomalous di ff usion equations with damping became an active area of investigation in the fi eld of applications.It is well known that tuned mass dampers o ff er a relatively simple and e ff ective way of reducing excessive vibrations of high rise buildings,towers and chimneys.Such as the following classical integer order di ff erential equation with damping associated with a simple linear oscillator system excited by white noise,

where x(t)denotes the displacement of the structure and f(t)is the external force,which is assumed to be a white noise.The constant β is the damping ratio and the number δ is the natural frequency for the structure.However,many commercially available dampers or damping materials indicate that the derivative with respect to time on a linear viscous damper model usually has the fractional characteristics and is thus appropriate to introduce the α-th fractional derivative to the displacement.In this connection,the damping equation(?)thus converts into the below form

As we know,for α =1 the equation is a linear viscous damping model and for α =0 the equation is a linear restoring model.It is assumed that α∈[0,1]and in the interval 0<α<1 the device contributes with both sti ff ness and damping.In recent years,the dynamics and vibration analysis of fractional order damped systems were of great interest to researchers(see e.g.[2,6,14,19,20,23,24]and the references therein).

Our motivation for studying system(1.1)comes from recent investigations where a related class appears in connection with partial di ff erential and Cauchy time processes.We also mention that the concepts of controllability,when they were fi rst introduced by Kalman in 1960,were set at the center of control theorem and soon they were generalized to the in fi nite dimensional context.Currently,controllability problems for fractional control systems were examined in a number of publications(see e.g.,[2,4,5,12,16,17]).However,to the best of our knowledge,there is still little information known for the controllability of the nonlinear fractional control systems with damping and this fact is the motivation of the present work.Our fi rst main result,stated as Theorem 3.1 and 3.3,provide the global and local existence and uniqueness of solutions in a mild sense of problem(1.1).Next,in Theorem 4.3,we present our main controllability result of the control system(1.1).

This paper has fi ve sections.In the next section,we include some basic de fi nitions,notations and results needed in this paper.Section 3 presents existence results for mild solutions of the fractional control systems(1.1).Section 4 is devoted to our controllability result.In the last section,two concrete applications of our main results are provided.

2 Preliminaries

For the Hilbert space H with the norm k ·kH,H?denotes its dual and h·,·iHstands for the duality pairing between H?and H.By C(J,H)we denote the Banach space of continuous functions from J=[0,b]into H equipped with the norm k xkC(J,H)=sup kx(t)kH.AC(J,H)is the space of functions which are absolutely continuous on J andand f(m?1)(·) ∈ AC(J,H)}.For the Hilbert spaces H and U,L(H,U)means the space of bounded linear operators from H to U and we write L(H)when U=H.The notation R(λ,A)=(λI ? A)?1represents the resolvent of A with a complex number λ being in the resolvent set ρ(A)and the real part of λ is denoted by Reλ.The Laplace transform of a function f∈L1(R+,H)is de fi ned by

for suitable λ such that the integralf(t)dt is absolutely convergent on H.We denote the convolution of two functions supported only on[0,∞)by

and the Laplace transform of the convolution(f?g)(t)is given by

Now,for the convenience of the readers,we fi rst present some useful de fi nitions and fundamental facts of fractional calculus theory,which can be found in[10,22].

(a)Sα(t)is strongly continuous on R+and Sα(0)=I;

(b)Sα(t)D(A)? D(A)and ASα(t)x=Sα(t)Ax for all x ∈ D(A),t≥ 0;

(c)for all x∈D(A),the following equation holds:

In this case,Sα(t)is called the(α,γ)-regularized family generated by A.

Related to the family Sα,we also introduce the following two families Pα,Qα:R+→ L(H)de fi ned by

The following result is analogous to the C0-semigroup and the cosine family,which is a direct consequence of[19,Remark 2.2].

Lemma 2.6Let A be a closed linear densely de fi ned operator on the Hilbert space H.Then A is the generator of an(α,γ)-regularized family if and only if there exist w ≥ 0 and a strong continuous operator Sα:R+→ L(H)such that{λα(λ+γ):Reλ >w}? ρ(A)and

Combining(2.2)and(2.4),we have

By(2.3)and(2.4),we obtain

Remark 2.7Because of the uniqueness the Laplace transform,we know that

(1)for the case α =0, γ =0,Sα(t)becomes a strongly continuous semigroup generated by A(for more details about it,one can see[1,Theorem 3.1.7]or[21,page 8]);

(2) as in the situation α =1, γ =0,Sα(t)turns out to be a strongly continuous cosine function generated by A and Pα(t)is called sine function associated with Sα(t)(see[1,Proposition 3.14.4]for more details).

Based on Lemmas 2.4 and 2.6,we also have the following lemma.

Lemma 2.8Let 0 ≤ α ≤ 1 and h ∈ L1(J,H),if x(·) ∈ C1(J,H)such that Ax(·) ∈L1(J,H)is a solution of the problem

then x satis fi es the following equation

ProofSuppose that x satis fi es(2.7),then we have

It follows from Lemma 2.4 that

From the above equation and taking the initial conditions into account,we get

Let λ >0 such that λα(λ + γ) ? ρ(A)and take the Laplace transform on both sides of(2.9),then we have

Therefore,we get

Finally,taking the inverse Laplace transform to(2.10)and invoking the Laplace transforms of the operators Sα(t),Pα(t),Qα(t)and combination of formula(2.1)gives

Remark 2.9We say that a function x∈C1(J,H)is a solution of(2.7)if x(t)∈D(A)for all t∈J and the(2.7)holds(cf.[19,page 187]).By Lemma 2.8,we know that x∈C1(J,H)is a solution of problem(2.7),then x is a solution of the Volterra integral(2.8).In this connection,it is therefore nature to de fi ne the following concept of mild solution for problem(2.7).

De fi nition 2.10For h ∈ L1(J,H),a function x(·)∈ C(J,H)is said to be a mild solution of system(2.7),if it satis fi es

3 Existence of Solutions

In this section,we present the existence and uniqueness of mild solutions for problem(1.1)under some appropriate sufficient conditions by a well known fi xed point theorem.

To obtain the global existence of mild solution for system(1.1),we suppose that

H(A):A is the in fi nitesimal generator of an exponentially bounded(α,γ)-regularized family Sα(t)on Hilbert space H,i.e.,there are constants M ≥1 and w≥0 such that

H(f):The nonlinear function f(t,x)is continuous in t and uniformly Lipschitz continuous with respect to x,i.e.,there exists a function l∈L1(J,R+),such that

Now,we are in the position to present our fi rst main result in this section.

Theorem 3.1Assume that conditions H(A)and H(f)hold.Then for each given control function u(·)∈ L1(J,U),problem(1.1)has a unique mild solution on C(J,H).

ProofFor any given control function u(·) ∈ L1(J,U),we de fi ne an operator as follows

Combining(3.1)–(3.3)and induction on n,it is easy to get that

Remark 3.2Hypothesis H(f)is suitably and widely used in guaranteeing the existence and uniqueness of solutions for nonlinear systems.For example,Pazy considered the existence and uniqueness of mild solution for semilinear initial problem by assuming that the function f satis fi es H(f)with l(t)≡l(constant)(cf.[21,Theorem 6.1.2]).Kumar and Sukavanam[12,Theorem 3.1.]proved the existence and uniqueness of the mild solution for the fractional order semilinear systems with bounded delay under the assumption that the nonlinear function satis fi es the Lipschitz condition H(f)with l(t)≡l(constant).Recently,Liu and Li[17,Theorem 3.1]obtained the existence and uniqueness of mild solutions for a class of fractional evolution di ff erential equations involving Riemann-Liouville fractional derivatives under the condition H(f)with l(t)≡l(constant).

The uniform Lipschitz condition of the function f in Theorem 3.1 assures the global(i.e.,de fi ned on J=[0,b])existence of mild solution for system(1.1).If we suppose that f only satis fi es a local Lipschitz condition,that is

H’(f):The nonlinear function f(t,x)is continuous in t for t≥ 0 and locally Lipschitz with respect to x,uniformly in t on bounded interval,that is,for every τ≥ 0 and constant r≥ 0,there exists a function L(t,r)∈L1(J,R+),such that

holds for all x,y∈H with kxkH≤r,kykH≤r and t∈[0,τ],then we have the following local version of Theorem 3.1.

Theorem 3.3For every given control u(·)∈ L1(J,U),if conditions H(A)and H’(f)hold,then there is a tmax≤∞such that problem(1.1)has a unique mild solution on[0,tmax].Moreover,if tmax<∞,then

ProofFor every given control u(·) ∈ L1(J,U),we start by showing that problem(1.1)has a unique mild solution x on an interval[0,t1]and t1is chosen below by

Hence F maps the ball Brof radius r into itself.

Next,we know that the nonlinear function f satis fi es the uniform Lipschitz condition on the interval[0,t1]and thus as in the proof of Theorem 3.1,F possesses a unique fi xed point on BR,which is the desired solution of system(1.1)on the interval[0,t1].

From what we have just proved,we can show that if x is a mild solution of system(1.1)on the interval[0,t1],it can be extended to the interval[0,t1+ ε]with ε>0 by de fi ning on[t1,t1+ε],x(t)=y(t),where y(t)is the solution of the following integral equation

4 Controllability Results

In the remainder of this section,we study the controllability result for the nonlinear fractional control systems with damping in Hilbert spaces.For Hilbert spaces H and U be,we identify H with H?and U with U?.If T ∈ L(H,U),let T?be its adjoint,then of course T?∈ L(U,H)and it can be checked that kT?k=kTk=.

To begin our study on the controllability,we fi rst state the following concept.

De fi nition 4.1The fractional control system(1.1)is said to be controllable on the interval J=[0,b],if for every initial values x0,x1∈H and any given fi nal state xb∈H,there exists a control u∈L2(J,U)such that a mild solution x of system(1.1)satis fi es x(b)=xb.

It is convenient at this point to introduce the controllability operatorassociated with system(1.1)as follows

It is readily from condition(4.1)that Γ is a contraction operator on C(J,H).Hence,according to Banach’s fi xed point theorem,we obtain problem(1.1)has a unique solution which implies that system(1.1)is controllable on J.The proof is completed. ?

5 Applications

In this section,we give two examples to demonstrate how to utilize our results.

Example 5.1Consider the following 3-dimensional nonlinear fractional control system,

where x(t) ∈ R3,u(t) ∈ R2,the matrices A ∈ R3×3,B ∈ R3×2and the nonlinear function f:J×R3→R3are given by

For a vector v=(v1,v2,v3)?∈R3and?is the transpose of(v1,v2,v3),we consider the maximum norm kvkR3=k(v1,v2,v3)?k=max{|v1|,|v2|,|v3|}on R3and it is easy to get that

Hence,we can obtain that the rank of the matrix[B,AB,A2B]is equal to 3.Next,from the Cayley-Hamilton theorem,we have

Hence,the nonlinear function f satis fi es condition H(f)and so system(5.4)has a unique mild solution on C([0,b],H)by Theorem 3.1.Moreover,if condition H(W)is satis fi ed,then system(5.4)is controllable on[0,b].