# On Partial Complete Controllability of Semilinear Systems

Authors: Agamirza E. Bashirov; Maher Jneid
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Many control systems can be written as a first-order differential equation
if the state space enlarged. Therefore, general conditions on controllability, stated for
the first-order differential equations, are too strong for these systems. For such systems
partial controllability concepts, which assume the original state space, are more
suitable. In this paper, a sufficient condition for the partial complete controllability of
semilinear control system is proved. The result is demonstrated through examples.


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## Body

## 1. Introduction

A concept of controllability, defined by Kalman [1] in 1960 for finite dimensional control systems, is a property of attaining every point in the state space from every initial state point for a finite time. Further studies on this concept in infinite dimensional spaces demonstrated that it is suitable to consider its two versions: a stronger version of complete controllability and a weaker version of approximate controllability. The reason for these versions was the fact that many infinite dimensional control systems are not completely controllable while they are approximately controllable (see Fattorini [2] and  Russell [3]). The necessary and sufficient conditions for complete and approximate controllability concepts are almost completely studied and presented in, for example, Curtain and Zwart [4], Bensoussan [5], Bensoussan et al. [6], Zabczyk [7], Bashirov [8], Klamka [9], and so forth for linear systems; Balachandran and Dauer [10, 11], Klamka [12], Mahmudov [13], Li and Yong [14], and so forth for nonlinear systems; Sakthivel et al. [15–17], Yan [18], and so forth for fractional differential systems; and Ren et al. [19] for differential inclusions.Recently, in Bashirov et al. [20, 21] the partial controllability concepts were initiated. The idea of these concepts is that some control systems, including higher order differential equations, wave equations, and delay equations, can be written as a first-order differential equation only by enlarging the dimension of the state space. Therefore, the theorems on controllability, which are formulated for control systems in the form of first-order differential equation, are too strong for them because they involve the enlarged state space. In such cases the partial controllability concepts became preferable, which assume the original state space. The basic controllability conditions for linear systems, including resolvent conditions from Bashirov and Mahmudov [22] and Bashirov and Kerimov [23] (see also [24–26]), are extended to partial controllability concepts by just a replacement of the controllability operator by its partial version.In this paper our aim is to study the partial complete controllability of semilinear systems. The controllability concepts for semilinear systems are intensively discussed in the literature (see Balachandran and Dauer [10, 11], Klamka [12], Mahmudov [13], Sakthivel et al. [15, 17], and references therein). A basic tool of study in these works is fixed point theorems. In this paper, we also use one of the fixed point theorems, a contraction mapping theorem, and find a sufficient condition for the partial complete controllability of a semilinear control system.The rest of this paper is organised in the following way. In Section2 we set the problem, give basic definitions, and motivate the partial controllability concepts by considering a higher order differential equation, a wave equation, and a delay equation. Section 3 contains the proof of the main result. In Section 4, we demonstrate the main result in the examples. Finally, Section 5 contains directions of further research regarding partial controllability concepts.


## 2. Setting the Problem and Motivation

Consider the basic semilinear control system(1)xt′=Axt+But+f(t,xt,ut),0<t≤T,x0∈X,
on the interval [0,T] with T>0, where x and u are state and control processes. We assume that the following conditions hold. (A)
Xand U are separable Hilbert spaces, H is a closed subspace of X, and L is a projection operator from X to H;(B)
A is a densely defined closed linear operator on X, generating a strongly continuous semigroup eAt, t≥0;(C)
B is a bounded linear operator from U to X;(D)
f is a nonlinear function from [0,T]×X×U to X, satisfying that
(i)
f is continuous on [0,T]×X×U;(ii)
f is Lipschitz continuous with respect to x and u that is, for all t∈[0,T], u,v∈U and x,y∈X,
(2)∥f(t,x,u)-f(t,y,v)∥≤K(∥x-y∥+∥u-v∥)
for some K≥0;(E)
Uad=C(0,T;U) is the space of all continuous functions from [0,T] to U. Define the controllability and L-partial controllability operators Qt and Q~t by
(3)Qt=∫0t‍eAsBB*eA*sds,Q~t=LQtL*,
where L* is the adjoint of L. The L-partial controllability operator becomes the controllability operator if L=I (the identity operator). We will also assume the following condition;(F)
Q~T is coercive; that is, there is γ>0 such that 〈Q~Th,h〉≥γ∥h∥2 for all h∈H.Note that this condition implies the existence ofQ~T-1 as a bounded linear operator and ∥Q~T-1∥≤1/γ. Respectively, the linear system associated with (1) (the case when f=0) is L-partially complete controllable on the interval [0,T] (see, Bashirov et al. [20, 21]).The above conditions imply the existence of a unique continuous solution of (1) in the mild sense for every u∈Uad and x0∈X (see Li and Yong [14] and Byszewski [27]); that is, there is a unique continuous function x from [0,T] to X such that
(4)xt=eAtx0+∫0t‍eA(t-s)(Bus+f(s,xs,us))ds.Let(5)DTx0={h∈H:∃u∈Uadsuchthath=LxT}.
Following Bashirov et al. [21], the semilinear control system (1) is said to be L-partially complete controllable on Uad if DTx0=H for all x0∈X. Similarly, the semilinear system in (1) is said to be L-partially approximate controllable on Uad if DTx0¯=H for all x0∈X, where DTx0¯ is the closure of DTx0. If H=X, these are just well-known complete and approximate controllability concepts, respectively. In this paper, we study the concept of L-partial complete controllability.The reason for studyingL-partial controllability concepts is that many systems can be written in the form of (1) if the original state space is enlarged. Therefore, suitable controllability concepts for such systems are the L-partial controllability concepts with the operator L projecting the enlarged state space to the original one. Here are some examples of such systems, which are discussed in Bashirov [8], Section  3.1.1, in more details.Example 1.
Consider the system(6)xt(n)=f(t,xt,xt′,…,xt(n-1),ut),
assuming that its state space is the one-dimensional space ℝ. The ordinary controllability concepts for this system are the equality to or denseness in ℝ of the respective attainable set. We can write this system as the first-order differential equation
(7)yt′=Ayt+F(t,yt,ut)
if
(8)yt=[xtxt′⋮xt(n-2)xt(n-1)],A=[01⋯0000⋯00⋮⋮⋱⋮⋮00⋯0100⋯00],F(t,y,u)=[00⋮0f(t,x,x′,…,x(n-1),u)].
The state space of this system is the n-dimensional Euclidean space ℝn and, respectively, its attainable set is a subset of ℝn. Therefore, the controllability concepts of the system for y are stronger than those of the system for x. But if we define the projection operator L by
(9)L=[10⋯00]:ℝn⟶ℝ,
then the L-partial controllability concepts of the system for y become the same as the ordinary controllability concepts of the system for x.Example 2.
Consider the nonlinear wave equation(10)∂2xt,θ∂t2=∂2xt,θ∂θ2+f(t,xt,θ,∂xt,θ∂t,ut),
where x is a real-valued function of two variables t≥0 and 0≤θ≤1. The state space of this system is L2(0,1) (the space of square integrable functions on [0,1]). This system can be written as the first-order abstract differential equation
(11)yt′=Ayt+F(t,yt,ut)
if
(12)yt=[xt,θ∂xt,θ∂t],A=[0Id2dθ20],F(t,y,u)=[0f(t,y1,y2,u)],
where y∈L2(0,1)×L2(0,1). The state space L2(0,1)×L2(0,1) of the system for y is the enlargement of the state space L2(0,1) of the system for x. This is a cost that is paid to bring the wave equation to the form of first-order differential equation.  The ordinary controllability concepts for the system (11) are too strong for the system (10). If
(13)L=[I0]:L2(0,1)×L2(0,1)⟶L2(0,1),
then L-partially controllability concepts of the system for y become ordinary controllability concepts of the system for x.Example 3.
Consider the system(14)xt′=f(t,xt,∫-ε0‍xt+θdθ,ut),
which contains a simple distributed delay in the nonlinear term, assuming that x is a real-valued function. Then the state space is ℝ. To bring this system to a system without delay, enlarge ℝ to ℝ×L2(-ε,0) and define L2(-ε,0)-valued function
(15)[x-t]θ=xt+θ,t≥0,-ε≤θ≤0.
Then for
(16)yt=[xtx-t],A=[000ddθ],F(t,y,u)=[f(t,x,x-,u)0],
the above system can be written as the abstract system
(17)yt′=Ayt+f(t,yt,ut).
Similar to the previous examples, one can easily observe that the ordinary controllability concepts for the system (17) are too strong for the system (14), but the L-partial controllability concepts of the system for y with
(18)L=[I0]:ℝ×L2(0,1)⟶ℝ
are exactly the ordinary controllability concepts of the system for x.These examples motivate a study of the partial controllability concepts. In this paper it is proved that under the conditions (A)–(F), the system in (1) is L-partially complete controllable.


## 3. Main Result

DenoteX~=C(0,T;X). Then X~×Uad is a Banach space with the norm
(19)∥(·,·)∥X~×Uad=∥·∥X~+∥·∥Uad.Lemma 4.
Under the conditions (A), (B), and (C),(20)∥Qt∥≤∥QT∥,∥Q~t∥≤∥Q~T∥,0≤t≤T.Proof.
It is easy to see thatQt=Qt* and 〈Qtx,x〉≥0 for all x∈X. Hence,
(21)∥Qt∥=sup∥x∥=1〈Qtx,x〉.
Then
(22)〈QTx,x〉=∫0T‍〈eAsBB*eA*sx,x〉ds=〈Qtx,x〉+∫tT‍〈eAsBB*eA*sx,x〉ds=〈Qtx,x〉+∫tT‍∥B*eA*sx∥2ds≥〈Qtx,x〉.
This implies ∥Qt∥≤∥QT∥. The conclusion of the lemma regarding Q~T follows from 〈Q~tx,x〉=〈QtL*x,L*x〉.The proof of the following lemma appears in different forms in several papers, for example, Mahmudov [13]. Our proof is a minor modification of them.Lemma 5.
Assume that the conditions (A)–(F) hold and take arbitraryh∈H. Then for the operator G:X~×Uad→X~×Uad, defined by
(23)G(y,v)(t)=(Y(t),V(t)),0≤t≤T,
where
(24)Y(t)=-QteA*(T-t)L*Q~T-1L∫0T‍eA(T-s)f(s,ys,vs)ds+∫0t‍eA(t-s)f(s,ys,vs)ds,V(t)=B*eA*(T-t)L*Q~T-1×(h-LeATx0-L∫0T‍eA(T-s)f(s,ys,vs)ds),
the following inequality holds:
(25)∥G(y,v)-G(z,w)∥≤(1+∥QT∥Mγ+∥B∥Mγ)×MKT(∥y-z∥+∥v-w∥),
where
(26)M=sup0≤t≤T∥eAt∥.Proof.
Let(y,v) and (z,w) be two functions in X~×Uad such that G(y,v)=(Y,V) and G(z,w)=(Z,W). Then,
(27)∥G(y,v)-G(z,w)∥X~×Uad=∥Y-Z∥X~+∥V-W∥Uad.
Here, ∥Y-Z∥X~ can be estimated as follows:
(28)∥Y-Z∥=maxt∈[0,T]∥∫0teA(t-s)(f(s,ys,vs)-f(s,zs,ws))ds-QteA*(T-t)L*Q~T-1L×∫0TeA(T-s)(f(s,ys,vs)-f(s,zs,ws))ds∥≤maxt∈[0,T](M+M2∥Qt∥∥Q~T-1∥)×∫0T‍∥f(s,ys,vs)-f(s,zs,ws)∥ds≤M(1+M∥QT∥∥Q~T-1∥)×∫0T‍∥f(s,ys,vs)-f(s,zs,ws)∥ds≤(1+∥QT∥Mγ)MK∫0T(∥ys-zs∥+∥vs-ws∥)ds≤(1+∥QT∥Mγ)MKT(∥y-z∥+∥v-w∥).
Similarly, for ∥V-W∥Uad, we have
(29)∥V-W∥=maxt∈[0,T]∥∫0TeA(T-s)B*eA*(T-t)L*Q~T-1L×∫0TeA(T-s)(f(s,ys,vs)-f(s,zs,ws))ds∥≤M2∥B∥∥Q~T-1∥∫0T‍∥f(s,ys,vs)-f(s,zs,ws)∥ds≤M2∥B∥1γK∫0T‍(∥ys-zs∥+∥vs-ws∥)ds≤∥B∥MγMKT(∥y-z∥+∥v-w∥).
Combining (28) and (29), we obtain the demanded inequality.Lemma 6.
Under the conditions (A)–(F), if(30)(1+∥QT∥Mγ+∥B∥Mγ)MKT<1,
then the operator G, mapping X~×Uad into X~×Uad, has a unique fixed point (x,u)∈X~×Uad.Proof.
By Lemma5, G is a contraction mapping. Also, the space X~×Uad is complete. Hence, G has a fixed point.Theorem 7.
Under the conditions (A)–(F) and (30), the semilinear system (1) is L-partially complete controllable on [0,T].Proof.
Take anyx0∈X and h∈H. Show that there is u∈Uad such that h=LxT. To this end, consider u, defined as follows:
(31)ut=B*eA*(T-t)L*Q~T-1×(h-LeATx0-L∫0T‍eA(T-s)f(s,xs,us)ds).
Substituting (31) in (4) and applying Fubini’s theorem (see Bashirov [8], p. 45), we obtain
(32)xt=eAtx0+∫0t‍eA(t-s)BB*eA*(t-s)eA*(T-t)L*Q~T-1(h-LeATx0)ds-∫0t‍∫0T‍eA(t-s)BB*eA*(t-s)eA*(T-t)L*Q~T-1LeA(T-r)×f(r,xr,ur)drds+∫0t‍eA(t-s)f(s,xs,us)ds=eAtx0+QteA*(T-t)L*Q~T-1(h-LeATx0)+∫0t‍eA(t-s)f(s,xs,us)ds-∫0T‍∫0t‍eA(t-s)BB*eA*(t-s)eA*(T-t)L*Q~T-1LeA(T-r)×f(r,xr,ur)dsdr=eAtx0+QteA*(T-t)L*Q~T-1(h-LeATx0)+∫0t‍eA(t-s)f(s,xs,us)ds-∫0T‍QteA*(T-t)L*Q~T-1LeA(T-r)f(r,xr,ur)dr=eAtx0+QteA*(T-t)L*Q~T-1(h-LeATx0)+∫0t‍eA(t-s)f(s,xs,us)ds-QteA*(T-t)L*Q~T-1L∫0T‍eA(T-s)f(s,xs,us)ds.
According to Lemma 6, there is a unique pair (x,u)∈X~×Uad, satisfying (31) and (32). So, u∈Uad. Furthermore, we have
(33)LxT=L(∫0TeA(T-s)eATx0+QTL*Q~T-1(h-LeATx0)+∫0T‍eA(T-s)f(s,xs,us)ds-QTL*Q~T-1L∫0TeA(T-s)f(s,xs,us)ds)=LeATx0+LQTL*Q~T-1(h-LeATx0)+L∫0T‍eA(T-s)f(s,xs,us)ds-LQTL*Q~T-1L∫0T‍eA(T-s)f(s,xs,us)ds=h.
Thus, there is u∈Uad which steers x0 to xT with LxT=h. This means that the semilinear system (1) is L-partially complete controllable on [0,T] as desired.Remark 8.
DecomposingQT in the form
(34)QT=[Q~TRTRR*PT],
where RT:H⊥→H and PT:H⊥→H⊥ are other components of QT besides Q~T and H⊥ is an orthogonal complement of H in X, one can calculate
(35)〈QTh,h〉=〈Q~Th1,h1〉+2〈RTh2,h1〉+〈PTh2,h2〉,
where h1=Lh∈H and h2=h-Lh∈H⊥. Therefore, the coercivity of QT implies the same ofQ~T. But, the converse is not true. Theorem 7 is powerful in the cases when Q~T is coercive, but PT is not.Example 9.
Theorem7 establishes just sufficient condition of L-partial complete controllability. In this example we will demonstrate that this is not a necessary condition. We will consider a simple case of L=I when L-partial complete controllability reduces to complete controllability. Consider the one-dimensional control system
(36)xt′=2xt+2ut,x0∈ℝ.
This is a linear system, and the controllability operator of this system is equal to
(37)∫0T‍4e4tdt=e4T-1>0foreveryT>0.
According to the theory of controllability for linear systems, this system is controllable (completely) for every T>0.
Have another look at this system by writing it as(38)xt′=xt+ut+f(xt,ut),x0∈ℝ,
where
(39)f(x,u)=x+u.
Here, f satisfies the Lipschitz condition with K=1. Also, A=B=1, implying ∥B∥=1 and M=sup[0,T]∥eAt∥=eT. Furthermore,
(40)QT=∫0T‍e2tdt=e2T-12.
So, ∥QT∥=γ=(e2T-1)/2. Then the inequality (30) becomes
(41)(1+eT+2eTeT-1)eTT<1.
The limit of the left-hand side in this inequality when T→∞ is equal to ∞. This means that there is a sufficiently large T such that the conditions of Theorem 7 do not hold for this T, although the system under consideration is completely controllable. Thus, Theorem 7 states a sufficient condition which is not a necessary condition.


## 4. Examples

We demonstrate the features ofL-partial complete controllability in the following examples of control systems.Example 1.
Consider the system of differential equations(42)xt′=yt+but,x0∈ℝ,yt′=f(t,xt,yt,ut),y0∈ℝ
on [0,T], where u∈Uad=C(0,T;ℝ). Besides the complete controllability property, that is,
(43){(x,y)∈ℝ2:∃u∈Uadsuchthat(xT,yT)=(x,y)}=ℝ2,
we can investigate the partial complete controllability property, that is,
(44){x∈ℝ:∃u∈UadsuchthatxT=x}=ℝ.
We can write this system in ℝ2 as the following semilinear system:
(45)zt′=Azt+F(t,zt,ut)+But,
where
(46)zt=[xtyt],A=[0100]B=[b0],F(t,z,u)=[0f(t,x,y,u)],
assuming that
(47)z=[xy].
It can be calculated that
(48)eAt=[1t01]=[1001]+[0t00].
Hence,
(49)∥eAt∥≤1+t≤1+T,0≤t≤T.
The controllability operator is
(50)QT=∫0T‍eAtBB*eA*tdt=b2T[1000].
Hence, QT is not coercive, and the conditions for complete controllability, based on coercivity of QT, fail for this example. Although system (42) can still be complete controllable for properly selected functions f, we can investigate the partial complete controllability for this system being interested in just the first component xt of zt.
LetL=[10]. Then
(51)Q~T=LQTL*=b2T>0.
This means that the linear system associated with the semilinear system (45) is L-partially complete controllable. Furthermore, the inequality (30) becomes
(52)(1+b2T(1+T)b2T+b(1+T)b2T)(1+T)TK<1,
or, simplifying,
(53)K<b(1+T)(1+T+2bT+bT2).
This establishes a relation between Lipschitz coefficient K and terminal time moment T. Depending on K, T must be taken sufficiently large to satisfy (53). So, the system (42) is L-partially complete controllable for the time T if the Lipschitz coefficient K, related to f, satisfies (53).Example 2.
Delay equations are typical for application of partial controllability concepts. Consider a nonlinear delay equation(54)xt′=axt+but+f(t,xt,∫-ε0‍xt+θdθ,ut),x0=ξ,xθ=ηθ,-ε≤θ≤0,
on [0,T], where 0<ε<T, η∈L2(-ε,0;ℝ), and u∈Uad=C(0,T;ℝ).
Similar to Example3, introduce the function x-:[0,T]→L2(-ε,0;ℝ) by
(55)[x-t]θ=xt+θ,0≤t≤T,-ε≤θ≤0.
This function satisfies
(56)x-t′=(ddθ)x-t,x-0=η,0<t≤T.
Denote by 𝒯t, t≥0, the semigroup generated by the differential operator d/dθ and let Γ be the integral operator from L2(-ε,0;ℝ) to ℝ, defined by
(57)Γh=∫-ε0‍hθdθ,h∈L2(-ε,0;ℝ),
noticing that ∥Γ∥≤ε. Then for
(58)yt=[xtx-t],ζ=[ξη]∈ℝ×L2(-ε,0;ℝ),
we can write system (54) as
(59)yt′=Ayt+F(t,yt,ut)+But,y0=ζ,
where
(60)A=[a00ddθ],F(t,y,u)=[f(t,x,Γx-,u)0],B=[b0],
assuming that
(61)y=[xx-]∈ℝ×L2(-ε,0;ℝ).
The semigroup, generated by A, has the form
(62)eAt=[eat00𝒯t],t≥0.
Therefore, the controllability operator for system (54) can be calculated as
(63)QT=∫0T‍eAtB*BeA*tdt=∫0T‍[b2e2at000]dt=[b2(e2aT-1)2a000].
This is definitely not a coercive operator.Taking into account that the original system is given by (54) and (59) is just representation of (54) in the standard form, which enlarges the original state space ℝ to ℝ×L2(-ε,0;ℝ), we observe that the complete controllability for system (54) is in fact L-partial complete controllability for system (59) if
(64)L=[10]:ℝ×L2(-ε,0;ℝ)⟶ℝ.
Calculating partial controllability operator, we obtain
(65)Q~T=LQTL*=b2(e2aT-1)2a>0,
which is coercive.Furthermore, using(66)∥eAt∥≤1+eaT,∥QT∥=γ=b2(e2aT-1)2a,
we write the inequality (30) in the form
(67)(1+1+eaT+2ab(eaT-1))(1+eaT)TK<1.
If the Lipschitz coefficient K of the function F and terminal time moment T satisfy this inequality, then system (54) is completely controllable, which in turn means that system (59) is L-partially complete controllable.


## 5. Conclusion

In this paper a sufficient condition for partial complete controllability of a semilinear control system is proved. This is a continuation of the pioneering research that has been done by Bashirov et al. [20, 21] about partial controllability concepts. A research in this way, concerning partial complete and approximate controllability for semilinear deterministic and stochastic systems, has already been done and awaiting for publication. There are other kinds of systems which besides semilinearity include other features, for example, impulsiveness, fractional derivative issue, and so forth. The result of this paper can be extended to these systems as well.



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