# On the Controllability of Conformable Fractional Deterministic Control Systems in Finite Dimensional Spaces

Authors: Maher Jneid; Muath Awadalla
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

In this paper, we establish a set of convenient conditions of controllability for semilinear fractional finite dimensional control systems involving conformable fractional derivative. Indeed, sufficient conditions of controllability for a semilinear conformable fractional system are presented, assuming that the corresponding linear systems are controllable. The present method is based on conformable fractional exponential matrix, Gramian matrix, and the iterative technique. Two illustrated examples are carried out to establish the facility and efficiency of this technique.


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## Body

## 1. Introduction

Controllability concepts have played a substantial role in several fields in engineering, control theory, and applied mathematics. In 1960, the controllability was first defined by Kalman [1] as a property of shifting the systems from any initial state value into any state value at a terminal time. This definition was divided into two notions: an exact and an approximate controllability which become more suitable for dealing with control systems in infinite dimensional spaces. The purpose of those notions is the existence of control systems which are approximately controllable, but are not exact (see [2]). In fact, the term exact controllability would refer to as a controllability which is the same as defined by Kalman. However, the definition of approximate controllability is determined by transferring the systems from any initial state value into some small neighbourhood of any point at terminal time in the state space. Later on, many researchers conducted pioneering studies in an attempt to obtain proper controllability conditions (exact and approximate) for the linear and nonlinear control systems (see, for example, [3–8] and the references cited therein).Many problems in the real world can be modelled purely by fractional differential equations (for more details, refer to [9, 10]). This new calculus has pointedly attracted the mathematicians to focus clearly on revealing better results. The concept of controllability was extended to fractional control systems by various investigators. For instance, Sakthivel et al. [11] utilized fixed point approach to prove the controllability of nonlinear fractional systems. Vijayakumar et al. [12] obtained the controllability conditions for fractional integrodifferential neutral control systems with nonlocal conditions. Ma and Liu [13] employed analytic methods and resolvent operator to investigate controllability conditions and continuous dependence of a fractional neutral integrodifferential equation involving state-dependent delay. Jneid [14] derived sufficient conditions of approximate controllability for semilinear integrodifferential systems of fractional order with nonlocal conditions by using compact semigroup operator and Schauder fixed-point theorem. Sakthivel et al. [15] studied the approximate controllability conditions for nonlinear fractional stochastic differential inclusions, providing that the corresponding linear part is approximately controllable. Chokkalingam and Baleanu [16] obtained a set of sufficient conditions for controllability for fractional functional integrodifferential systems involving the Caputo fractional derivative of order α∈0,1 in Banach spaces.Previous works concerning controllability problems for fractional systems have been limited to Riemann, Liouville, and Caputo derivatives, while only one study concerning exact controllability involving conformable fractional derivative (CFD) as a definition of fractional derivative has been done by Jneid [17] till now. In this work, we aim at bringing up this kind of systems to the attention of investigators. Moreover, we derive controllability results for the semilinear conformable fractional system with initial condition ξ:(1)T0qxt=Axt+But+ft,xt,ut,0<t≤τ,0<q≤1,x0=ξ,where T0q is a conformable fractional derivative x∈C0,τ;ℝn, u∈C0,τ;ℝm, A∈ℝn×n and B∈ℝn×m, and f:0,τ×ℝn×ℝm⟶ℝn is an appropriate nonlinear function.The rest of this paper is divided into five sections. In Section1, we provide needed fundamental information related to conformable fractional derivatives and we establish the mild solution of nonlinear systems involving conformable fractional derivative in terms of fractional exponential matrix by using Laplace transform. The controllability conditions for the linear systems are obtained in Section 3. In Section 4, an iterative analysis approach and controllability conditions are exhibited. We give two suitable examples to show the usefulness and effectiveness of this technique in Section 5. Finally, a short conclusion is given in Section 6.


## 2. Preliminaries

Let0<q≤1 and I=0,τ, through the entire article.Definition 1 (see [18]).
The CFD of a given functionf:0,∞⟶ℝ, at t>0 of order q is given as(2)T0qft=limϵ⟶0ft+εt1−q−ftϵ,provided that the right side of this expression exists as a finite number.
Using this new definition of derivative, one can have the following properties which are similar to those of the classical derivative:(a)
For all constantc,T0qc=0.(b)
For alls∈ℝ,T0qts=sts−q.(c)
T0qetq/q=etq/q.Definition 2 (see [18]).
Given a functionf:0,∞⟶ℝ, the conformable fractional Laplace transform of f at t>0 of order q is given as(3)T0qfts=F0qs=∫0∞e−stq/qfttq−1dt.Theorem 1 (see [18]).
Given a differentiable functionf:0,∞⟶ℝ,(4)T0qT0qfts=sF0qs−f0.Moreover,(5)F0qs=Tfqt1/qs,where T is the classical Laplace transform.Consider the conformable fractional system as follows:(6)T0qxt=Axt+ft,t∈I,x0=ξ,where T0q is the conformable fractional derivative operator, x,f∈C0,τ;ℝn, and A is an n×n−matrix. Now, apply conformable fractional Laplace transform on system (6) to obtain(7)X0qs=ξ1sId−A+F0qs1sId−A,where Id is an n×n−identity matrix. Utilizing the relation given in (5) and applying the inverse Laplace transform, one can get the solution of system (6) in this way:(8)xt=eAtq/qξ+∫0teAtq/q−sq/qfssq−1ds,where eAtq/q=∑k=0∞Aktkq/qkk! is called conformable fractional exponential matrix.


## 3. Linear Control Systems

Let us consider a linear conformable fractional control system that is described by(9)T0qxt=Axt+But,t∈I,x0=ξ,where ξ∈ℝn is an initial condition, T0q is the conformable fractional derivative operator, x∈C0,τ;ℝn, u∈C0,τ;ℝm, A∈ℝn×n, and B∈ℝn×m.Lemma 1.
The mild solution of system (9) on I in conformable fractional sense is given by(10)xut=eAtq/qξ+∫0teAtq/q−sq/qBussq−1ds.Proof.
This result follows simply from the forgoing section.Denote the set of admissible controls byUad=C0,τ;ℝm and the reachable set of system (1) by(11)Rξ,τf=xuτ∈ℝn;xu∈C0,τ;ℝn,where there exists u∈Uad such that xu satisfies system (1).Definition 3.
System (9) is said to be controllable on I if Rξ,τ0=ℝn, for any ξ∈ℝn.
In other words, for any givenξ,h∈ℝn, system (9) can be reached to the intended state h at terminal time τ from any initial state ξ. The controllability Gramian matrix for the linear system (9) is defined by(12)Qqt−r=∫rtsq−1eAsq/qBB∗eA∗sq/qds,0≤r≤t≤τ.
For simplicity, we useQqt and Qt−r instead of Qqt−0, and Qqt, respectively.Theorem 2.
System (9) is controllable on I if and only if Qτ is invertible.Proof.
LetQτ be invertible. Then, Qτ−1 exists and we can define a control u as follows:(13)ut=B∗eA∗τq/q−τq/qQτ−1h−eAτq/qξ,0≤t≤τ,with h and ξ arbitrarily chosen from ℝn. Obviously, u∈C0,τ;ℝm.
Now, substituting this control into equation (10) at the terminal time t=τ, we get(14)xuτ=eAtq/qξ+∫0τeAtq/q−sq/qBB∗eA∗tq/q−sq/qQτ−1×h−eAtq/qξsq−1ds=eAtq/qξ+QτQτ−1h−eAtq/qξ=h,which implies the controllability of system (9).
Conversely, let system (9) be controllable on 0,τ. Assume the contrary that Qqτ is not invertible. Therefore, there exists a nonzero vector y∈ℝn so that(15)y∗Qτy=0=∫0τsq−1y∗eAτq/q−sq/qB2ds.
Hence,(16)y∗eAτq/q−sq/qBsq−1=0,∀s∈0,τ.
Leth=0. Since system (9) is controllable, for every initial state value ξ, we can obtain a control u leading the solution xu of (9) into 0 at terminal time τ.
Selectξ=−eAτq/qy. Thus,(17)xuτ=eAτq/qξ+∫0τeAτq/q−sq/qBussq−1ds=0.
This yields(18)y=∫0τeAτq/q−sq/qBussq−1ds.
Multiplying through byy∗ yields(19)y∗y=∫0τy∗eAτq/q−sq/qBussq−1ds=0,which contradicts that y≠0. Hence, the controllability matrix Qqτ is invertible.


## 4. Semilinear Control Systems

Let us consider a semilinear conformable fractional control system that is described by(20)T0qxt=Axt+But+ft,xt,ut,t∈I,x0=ξ,where A,B,ξ,x,anduf are defined as in the previous section.For brevity, for anyτ>0, let X=CI;ℝn. It is clear that the Cartesian product X×Uad is a Banach space equipped with the norm(21)⋅,⋅=⋅X+⋅Uad,where ∀x∈X and ∀u∈Uad, and(22)xX=maxt∈Ixt,uUad=maxt∈Iut.Let us assume the following:(A1)f is bounded and satisfies Lipschitz continuity on X×Uad. That is, for every t∈I,z1,z2∈ℝn, and v1,v2∈ℝm, there exist M>0 and N>0 so that(23)ft,z1,v1≤M,ft,z1,v1−ft,z2,v2≤Nz1−z2+v1−v2.(A2) For everyτ>0,Qτ is invertible.Define the operatorF:X×Uad⟶X×Uad as(24)Fx,ut=Xt,Ut,where(25)Xt=eAtq/qξ+qQteAτq/q−tq/qQτ−1h−eAtq/qξ+∫0teAtq/q−sq/qsq−1fs,xs,usds,−∫0tQt−seA∗τq/q−tq/qQτ−s−1eAτq/q−sq/qsq−1fs,xs,usds,(26)Ut=B∗eA∗τq/q−tq/qQτ−1×h−eAτq/qξ−∫0tQτ−s−1sq−1eAτq/q−sq/qfs,xs,us.Introduce the iterative method as follows:(27)x0ut=eAtq/qξ+QteAτq/q−tq/qQτ−1h−eAτq/qξ,xn+1ut=x0t+∫0teAtq/q−sq/qsq−1fs,xns,unsds−∫0tQt−seA∗τq/q−tq/qQτ−s−1eAtq/q−sq/qsq−1fs,xns,unsds,(28)u0t=B∗eA∗τq/q−tq/qQτ−1×h−eAτq/qξun+1t=u0t−B∗eA∗τq/q−tq/q×∫0tQτ−s−1sq−1eAτq/q−sq/qfs,xns,uns.DenoteΦnt=(xnt,unt for all t∈I and n=0,1,2,….(29)P1=maxt∈Itq−1eAτq/q−tq/q,P2=maxt∈IB∗eA∗τq/q−tq/q,P3=maxt∈IeA∗τq/q−tq/q,γ=maxt∈IQt−1.Lemma 2.
Let the assumptions (A1) and (A2) hold true and letn≥1. Then,(30)Φn+1−Φn≤L1nL2nτn+1n+1!,where(31)L1=MP11+P2γ+QτP3γ,L2=P1N1+QτP3γ+P1P2Mγ.Proof.
Let us start to estimateΦ1−Φ0. By the definition of norm, we have(32)Φ1−Φ0=maxt∈Ix1t−x0t+maxt∈Iu1t−u0t,x1ut−x0ut=∫0teAtq/q−sq/qsq−1fs,x0s,u0sds−∫0tQt−seA∗τq/q−tq/qQτ−s−1eAτq/q−sq/qsq−1fs,x0s,u0sds≤P1M+QτP3P1Mγt.
Similarly,(33)u1t−u0t=t1−qB∗eA∗τq/q−tq/q∫0tQτ−s−1sq−1eAτq/q−sq/qfs,x0s,u0s≤P2P1Mγt.
Combining (32) and (33), we obtain(34)Φ1t−Φ0t≤L1t,where(35)L1=MP11+P2γ+QτP3γ,x2ut−x1ut=∫0teAtq/q−sq/q−Qt−seA∗τq/q−tq/qQτ−s−1eAτq/q−sq/qfs,x2s,u2s−fs,x1s,u1ssq−1ds≤P1N∫0tx2s−x1s+u2s−u1sds+QτP3P1Nγ∫0tx2s−x1s+u2s−u1sds≤P1N1+QτP3γ∫0tx1s−x0s+u1s−u0sds.
In a similar manner, we get(36)u2t−u1t=−B∗eA∗τq/q−tq/q∫0tQτ−s−1sq−1eAτq/q−sq/qfs,x1s,u1s+B∗eA∗τq/q−tq/q∫0tQτ−s−1sq−1eAτq/q−sq/qfs,x0s,u0s≤P1P2Mγ∫0tx1s−x0s+u1s−u0sds.
Combining (35) and (36) yields(37)Φ2−Φ1≤L2∫0tx1s−x0s+u1s−u0sds,where(38)L2=P1N1+QτP3γ+P1P2Mγ.
Substituting (34) into (37), we obtain(39)Φ2−Φ1≤L2∫0tΦ1t−Φ0tds,≤L1L2t22!≤L1L2τ22!.
Applying mathematical induction onn∈ℕ, we get the following estimation:(40)Φn+1−Φn≤L1nL2nτn+1n+1!.Lemma 3.
Let assumptions (A1) and (A2) hold true. Then, the sequence of functionsΦn defined as in (27) and (28) is uniformly convergent.Proof.
IdentifyΦn+1t=Φ0t+Φ1t−Φ0t+Φ2t−⋯+Φn+1t−Φnt as a partial sum of(41)Φ0t+∑k=0∞Φk+1t−Φkt.
Using the relation (40), we have(42)Φn+1t≤Φ0t+∑k=0nΦk+1t−Φkt≤Φ0+L1τ+L1L2τ22!+L12L22τ33!+⋯+L1kL2kτk+1k+1!.
It is easy to see that the sum in the equation (41) is convergent, and hence the sum in the equation (42) also converges as n⟶∞. This implies that the sequence Φn converges since it is a partial sum of a convergent series. According to Weierstrass M-test, this convergence is uniform and hence the limit function, say, Φ, for the sequence Φn is continuous.Theorem 3.
Let the assumptions (A1) and (A2) hold true. Then, the nonlinear mapF admits only one fixed point in X×Uad.Proof.
Thanks to Lemma3 there is a pair x,u∈X×Uad so that xn,un⟶x,u as n⟶∞. Therefore, by taking the limit on both sides in (28) and (42), we see that the pair x,u is a fixed point of F. Suppose that there are two distinct fixed points of F say, x,u and y,v in X×Uad. Then, as in proof of Lemma 2,(43)Fx,ut−Fy,vt≤C∫0tFx,us−Fy,vsds,for some constant C.
LetSt=∫0tFx,us−Fy,vsds. It is clear that S0=0 and ∀t≥0,St≥0. Differentiating with respect to t, we obtain S′t=Fx,ut−Fy,vt and S′t−CSt≤0. Multiplying by exponential quantity e−Ct gives(44)e−CtSt′≤0.
Now, integrating through from zero tot yields(45)e−CtSt≤0.
Since for allt≥0,e−Ct>0, then for all t≥0,St≤0. Therefore, for all t≥0,St=0, and consequently S′t=0. Thus, Fx,ut=Fy,vt for all t≥0. This contradicts the assumption that x,u and y,v are two distinct fixed points. This proof is completed.Theorem 4.
Let assumptions (A1) and (A2) hold true. Then, the semilinear control system (20) is controllable on I.Proof.
Fixξ∈ℝn and h∈ℝn. From Theorem 3, there is a unique mild solution x∈X as defined in (25) which corresponds to a unique control u∈Uad as defined in (26). Hence, xτ=h. Therefore, the semilinear system (20) is controllable on I.


## 5. Examples

Example 1.
Consider the following conformable fractional control system:(46)T0qxt=yt+ut+x2t+5,T0qyt=ut+cosut,where 0≤t≤1,x0,y0∈ℝ2u∈C0,1;ℝ. This system can be expressed in the following general form:(47)T0qzt=Azt+But+ft,zt,ut,where(48)z=xy,A=0100,B=11,ft,zt,ut=x2t+5cosut.
By simple calculations, we obtain(49)eAtq/q=1tqq01.
The controllability Gramian matrix is(50)Q1=∫01sq−1eAsq/qBB∗eA∗sq/qds,=∫01sq−11+sqq21+sqq1+sqq1ds.
After a simple computation, we get(51)Q1=1q+1q2+13q31q+12q21q+12q21q.
Hence, for every0<q≤1,detQ1=−1/q4≠0, which means that Q1 is invertible. In addition, the nonlinear function f is bounded and satisfies Lipschitz condition with respect to z and u with the constant N=1; then, by Theorem 4, the given control system (46) is controllable on 0,1.Example 2.
Consider the following conformable fractional control system:(52)T0qxt=xt+ut+cosxsinx,T0qyt=yt+vt+sinut+vt,where 0≤t≤1,x0,y0∈ℝ2u,v∈C0,1;ℝ. This system can be expressed in the following general form:(53)T0qzt=Azt+But+ft,zt,wt,where(54)z=xy,A=1001,B=1001,ft,zt,wt=cosxsinxsinut+vt.
By easy calculations, we have(55)eAtq/q=etq/q00etq/q.
The controllability Gramian matrix is(56)Q1=∫01sq−1eAsq/qBB∗eA∗sq/qds,=∫01sq−1e2sq/q00e2sq/qds.
After a simple computation, we get(57)Q1=2e2/q1001.
Hence, for every0<q≤1,detQ1=2e2/q≠0, which means that Q1 is invertible. In addition, the nonlinear function f is bounded and satisfies Lipschitz condition with respect to z and w with the constant N=1; then, by Theorem 4, the given control system (52) is controllable on 0,1.


## 6. Conclusion

In this work, the controllability conditions for semilinear conformable fractional deterministic systems are derived under a normal condition, that is, the associated linear system is controllable. The iterative technique is used here to construct a suitable sequence which is under some conditions uniformly convergent to a mild solution of the semilinear system. The present results show that this technique is very effective in finding the mild solution of semilinear control systems involving conformable fractional derivative. Finally, it should be mentioned that the result of this paper can be expanded to diverse kinds of conformable fractional systems in finite and infinite dimensional spaces as well.



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