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Results on exact controllability of secondorder semilinear control system in Hilbert spaces
Advances in Difference Equations volumeÂ 2021, ArticleÂ number:Â 455 (2021)
Abstract
In our manuscript, we extend the controllability outcomes given by Bashirov (Math. Methods Appl. Sci. 44(9):7455â€“7462, 2021) for a family of secondorder semilinear control system by formulating a sequence of piecewise controls. This approach does not involve large estimations which are required to apply fixed point theorems. Therefore, we avoid the use of fixed point theory and the contraction mapping principle. We establish that a secondorder semilinear system drives any starting position to the required final position from the domain of the system. To achieve the required results, we suppose that the linear system is exactly controllable at every noninitial time period, the norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function, and the nonlinear term is bounded. Finally, an example has been presented to validate the results.
1 Introduction
Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time is known or postulated. This is illustrated in classical mechanics where the motion of a body is described by its position and velocity as time varies. For the studies related to the existence of solution for integer and fractionalorder systems, one can refer to [2â€“30]. The concept of controllability is one of the underlying ideas in mathematical control theory. Controllability analysis is used in several reallife problems which include, but are not limited to, rocket launching problems for satellite and aircraft control, missiles and antimissiles problems in defense, regulating inflation rate in the economy, controlling sugar level in the blood, etc. A systematic study of controllability was initiated by Kalman [31] in 1963 when the theory of controllability for timeinvariant and timevarying control systems in statespace form was developed.
Several engineering and scientific problems can be expressed by infinitedimensional differential equations. Therefore, it becomes necessary to discuss the controllability results for infinitedimensional systems. The controllability problems for finitedimensional nonlinear systems have been immensely analyzed by several authors. Many authors have advanced the idea of controllability from finitedimensional systems to infinitedimensional systems and determined appropriate requirements for the controllability of nonlinear systems. Different techniques have been practiced for the examination of controllability including fixed point theorems [32â€“34]. For more problems on controllability and recent progresses on fractional calculus and its applications, please refer to [35â€“47].
Secondorder differential equations represent abstract mathematical interpretations of several partial differential equations which occur in many applications related to the oscillation of fastened bars, the transverse motion of an extensible beam, and several other realworld physical phenomena. Hence it becomes really important to determine the controllability outcomes for this kind of system. Controllability discussions for various secondorder nonlinear systems have been widely studied by several authors [33, 48â€“58]. Fixed point theorems have been utilized greatly in determining the existence and controllability results for different firstorder and secondorder systems which involve large estimations on system constants see [33, 59, 60]. Recently BashirovÂ [1] obtained the exact controllability results for firstorder semilinear systems using a new technique that is based on the piecewise formulation of driving controls and without using fixed point theory. Earlier the same approach was applied in terms of approximate controllability [61]. Motivated by Bashirov [1], we extend the controllability results for secondorder semilinear systems excluding the use of fixed point theory. It is centered on a piecewise formulation of steering controls and does not involve large estimations which are required to apply fixed point theorems. By constructing a piecewise sequence of controls, we determine that a secondorder semilinear system is exactly controllable to the domain of the system operator, that is, it drives the system from any starting position to the required final position from the domain of the system. To achieve the required results, we suppose the subsequent requirements:

(a)
The corresponding linear system is exactly controllable at every noninitial time period.

(b)
The norm of the inverse of the controllability Grammian operator increases as the time approaches zero with the slower rate in comparison to the reciprocal of the square function.

(c)
The nonlinear term is bounded.
Let us consider \(Z=L_{2}[0,b;\mathbb{X}]\) and \(Y=L_{2}[0,b;\mathbb{U}]\) as the function spaces defined on \(J=[0,b]\), \(0\leq b<\infty \), where \(\mathbb{X}\) and \(\mathbb{U}\) are two Hilbert spaces. Consider the following secondorder semilinear control system:
where

1
\(p(t)\) represents the state having values in Hilbert space \(\mathbb{X}\).

2
Control function v is defined from \([0,b]\rightarrow \mathbb{U}\).

3
B is a bounded and linear operator from \(\mathbb{U}\) into \(\mathbb{X}\).

4
The function \(r:[0,b]\times \mathbb{X}\rightarrow \mathbb{X}\) is a purely nonlinear function which produces nonlinearity in the system.

5
\(A:\operatorname{dom}(A)\subseteq \mathbb{X}\rightarrow \mathbb{X}\) is linear, closed where \(\operatorname{dom}(A)\) is a dense subset of X.
The linear system corresponding to (1.1) with state vector \(q(t)\) and control v is defined by
The article is structured in the subsequent manner:

1
Sect.Â 2 presents a few basic results related to control theory and secondorder systems.

2
Sect.Â 3 provides the assumptions which are required to obtain the controllability results.

3
Sect.Â 4 discusses the controllability results using the new technique.

4
Sect.Â 5 presents an example to verify the established outcomes.
2 Auxiliary results
Here, we will review fundamental theories and a few definitions which would be helpful for further discussions.
Definition 2.1
[62] A oneparameter family \(\{C(t),t\in \mathbb{R}\}\) of bounded linear operators mapping the Hilbert space \(\mathbb{X}\) into itself is called a strongly continuous cosine family if and only if

1
\(C(0)=I\);

2
\(C(s+t)+C(st)=2C(s)C(t)\);

3
\(C(t)x\) is continuous in t on \(\mathbb{R}\) for each fixed \(x\in \mathbb{X}\).
If \(\{C(t),t\in \mathbb{R}\}\) is a strongly continuous cosine family in \(\mathbb{X}\), then \(\{S(t),t\in \mathbb{R}\}\) is a oneparameter family of operators in \(\mathbb{X}\) defined by
The infinitesimal generator of a strongly continuous cosine family \(\{C(t), t\in \mathbb{R}\}\) is the operator \(A:\mathbb{X}\rightarrow \mathbb{X}\) defined by
The domain of operator A is defined as
These cosine and sine families defined above and generator A fulfill the following properties.
Lemma 2.2
([28])
Suppose that A is the infinitesimal generator of a cosine family of operators \(\{C(t):t\in \mathbb{R}\}\). Then the following hold:

(1)
There exist \(M' \geq 1\) and \(\omega \geq 0\) such that \(\C(t)\\leq M'e^{\omega t}\), and hence \(\S(t)\\leq M'e^{\omega t}\).

(2)
\(A\int _{s}^{r}S(u)x\,du=[C(r)C(s)]x\) for all \(0\leq s\leq r<\infty \).

(3)
There exists \(N'\geq 1\) such that \(\S(s)S(r)\\leq N' \int _{s}^{r}e^{\omega s}\,ds \) for all \(0\leq s\leq r<\infty \).
The uniform boundedness principle together with \((1)\) implies that both \(\{C(t):t\in J\}\) and \(\{S(t):t\in J\}\) are uniformly bounded and \(M=M'e^{\omega b}\).
Proposition 2.3
([62])
Let \(\{C(t),t\in \mathbb{R}\}\) be a strongly continuous cosine family in \(\mathbb{X}\) with infinitesimal generator A. The following are true:

1
\(S(0)=0\).

2
\(C(t)=C(t)\) and \(S(t)=S(t)\) for all \(t\in \mathbb{R}\).

3
If \(x\in E\), then \(S(t)x,C(t)x\in \operatorname{dom}(A)\) and \(\frac{d}{dt}S(t)x=C(t)x\), and \(\frac{d}{dt}C(t)x=AS(t)x\), where \(E=\{x:C(t)x \textit{ is once continuously differentiable function of }t\}\).

4
If \(x\in \operatorname{dom}(A)\), then \(S(t)x\in \operatorname{dom}(A)\) and \(AS(t)x=S(t)Ax\).

5
If \(x\in \operatorname{dom}(A)\), then \(C(t)x\in \operatorname{dom}(A)\) and \(\frac{d^{2}}{dt^{2}}C(t)x=AC(t)x=C(t)Ax\).

6
If \(x\in E\), then \(\lim_{t\rightarrow 0}AS(t)x=0\).
Proposition 2.4
([62])
Let \(\{C(t), t \in \mathbb{R}\}\) be a strongly continuous cosine family in \(\mathbb{X}\). The operator \(\hat{A}: \mathbb{X}\rightarrow \mathbb{X}\) defined by
with domain \(x\in \mathbb{X}\) for which this limit exists, is the infinitesimal generator of the cosine family \(\{C(t), t\in \mathbb{R}\}\).
Suppose \(U_{ad}=L_{2}[0,b;\mathbb{U}]\), which is the set of admissible controls. We define the mild solution of the given semi linear system (1.1) and its corresponding linear system as follows.
Definition 2.5
The mild solution of system (1.1) is defined by a function \(p(\cdot )\in \mathbb{X}\) which satisfies the following integral equation:
and the mild solution of the corresponding linear system (1.2) is described by the following integral equation:
Definition 2.6
([1])
System (1.1) is said to be approximately controllable in the time interval \([0,b]\) if, for the given starting position \((p_{0}, q_{0})\in \mathbb{X}\) and the required final position \((p_{F}, q_{F})\in \mathbb{X}\) and \(\epsilon >0\), there exists a control function \(v\in U_{ad}\) such that the solution of (1.1) satisfies
where \(p(b)\) is the state value of system (1.1) at time \(t=b\). If \(p(b)=p_{F}\), then the system is said to be exactly controllable. The system is said to be exactly controllable to \(\operatorname{dom}(A)\) on \([0,b]\) if, for the given starting position \((p_{0}, q_{0})\in \mathbb{X}\) and the required final position \((p_{F}, q_{F})\in \operatorname{dom}(A)\), there exists a control function \(v\in U_{ad}\) such that the solution of the system satisfies \(p(b)=p_{F}\), \(p'(b)=q_{F}\).
Remark 2.7
Note that the exact controllability to \(\operatorname{dom}(A)\) lies in between the exact and approximate controllability. Therefore, it is a weaker concept than the exact controllability. In real life applications, sometimes we are more concerned with attaining the points from \(\operatorname{dom}(A)\). If it is possible to reach the points from \(\mathbb{X} \backslash \operatorname{dom}(A)\) as well, then it can be considered as an additional capability of the system.
3 Assumptions
Let us introduce the controllability Grammian operator W associated with linear system (1.2) by
where \(S^{*}(s)\) denotes the adjoint of \(S(s)\).
Theorem 3.1
The corresponding linear system (1.2) is exactly controllable on the interval \([h, b]\) iff \(W(bh)\) is coercive. The control which drives the system from the starting position \((q(h), q'(h))\in X\) to the final position \((p_{F}, q_{F})\in X\) is given by
Proof
The result can be seen in [59]. Moreover, it can be easily verified by substituting the above defined control in the mild solution of the corresponding linear system that it transfers \((q(h), q'(h))\) to \((p_{F}, q_{F})\) on the interval \([h, b]\).â€ƒâ–¡
Remark 3.2
The coercivity of \(W(t)\) indicates that \((W(t))^{1}\) is a bounded linear operator. We say that \(W(t)\) is coercive if there exists \(\gamma >0\) such that \(\langle W(t)x, x\rangle \geq \gamma \x\^{2}\) for all \(x\in X\). Here \(W(0)=0\) and, therefore, it fails to be coercive. But it may be coercive for \(0< t\leq b\). Therefore, the above result holds on \([h, b]\).
To determine the main result, we make the following assumptions on the controllability Grammian operator W and the nonlinear function \(r(t,p)\):

(I)
\(W(t)\) is coercive for all \(0< t\leq b\).

(II)
There exists some \(N\geq 0\) such that
$$ t^{1+\alpha } \bigl\Vert \bigl(W(t)\bigr)^{1} \bigr\Vert \leq N \quad \text{for all } 0< t\leq b, 0\leq \alpha < 1. $$That is, \(\(W(t))^{1}\ \rightarrow \infty \) as t â†’ 0^{+} with the slower rate in comparison to the reciprocal of the square function as
$$ \bigl\Vert \bigl(W(t)\bigr)^{1} \bigr\Vert \leq \frac{N}{t^{1+\alpha }}< \frac{N}{t^{2}}, $$for small values of t.

(III)
The nonlinear function r is Lebesgue measurable in t.

(IV)
r is Lipschitz continuous in p.

(V)
r is bounded in \([0,b]\times \mathbb{X}\), \(i.e\)., there exists \(K>0\) such that
$$ \bigl\Vert r(t,p) \bigr\Vert \leq K \quad \text{for all } (t,p)\in [0,b] \times \mathbb{X}. $$
4 Results on controllability
In this section, we primarily focus on the study of exact controllability of the assumed system.
Theorem 4.1
System (1.1) is exactly controllable to \(\operatorname{dom}(A)\) on the interval \([0,b]\) for every \(b>0\) provided assumptions (I)â€“(V) hold.
Proof: We construct a piecewise sequence of driving controls to formulate the required control function v which drives the given system from the starting position \((p_{0}, q_{0})\in \mathbb{X}\) to the final position \((p_{F}, q_{F})\in \operatorname{dom}(A)\) in the following manner.
For this, consider the sequence \(\{h_{n}\}\) which is defined by \(h_{n}=\frac{b}{2^{n}}\) for \(n=1,2,\ldots \)â€‰.
We have \(\sum_{n=1}^{\infty }h_{n}=b\). For the sake of simplicity, let us take \(h_{0}=0\) and
Then \(\lim_{n\rightarrow \infty }b_{n}=\sum_{k=0}^{\infty }h_{k}=b\).
Using Theorem 3.1, the corresponding linear system (1.2) is exactly controllable on \([b_{0}, b_{1}]\) along with the control
which steers the initial state \(p_{0}\) to \(C(h_{1})p_{F}+S(h_{1})q_{F}\).
That is,
Define v on \([b_{0},b_{1}]\) by letting \(v(\varrho )=v_{1}(\varrho )\). Then, from (2.1), we obtain
For brevity, let \(p(b_{1})=p_{1}\). Next, consider (1.2) on \([b_{1},b_{2}]\). By Theorem 3.1, the control
steers \(p_{1}\) to \(C(h_{2})p_{F}+S(h_{2}) q_{F}\). Writing \(p'(b_{1})=q_{1}\), then the control \(u_{2}(\varrho )\) can be written as
That is,
Define v on \((b_{1},b_{2}]\) by letting \(v(\varrho )=v_{2}(\varrho )\). Then, from (2.1), we obtain
For the sake of convenience, let \(p(b_{2})=p_{2}\). Progressing in this fashion, we acquire a sequence of driving controls
where \(q_{n1}=p'(b_{n1})\).
After combining the above sequence of controls, we get the control function as follows:
and
Now by using the assumption (V), we get
where \(M=\sup_{[0,b]}\C(\varrho )\\) and \(K=\sup_{[0,b]\times X}\r(\varrho ,p)\\).
Since \(C(\varrho )\) is strongly continuous, \(S(0)=0\) and \(\lim_{n\rightarrow \infty }h_{n}=0\).
Therefore, \(\lim_{n\rightarrow \infty }p_{n}=p_{F}\). Also,
Thus, we have
Since \(C(\varrho )\) is strongly continuous, \(S(0)=0\) and \(\lim_{n\rightarrow \infty }h_{n}=0\).
Therefore, \(\lim_{n\rightarrow \infty }q_{n}=q_{F}\).
Next we prove that \(v\in U_{ad}\). Since every \(v_{n}\) of v is continuous on the interval \((b_{n1},b_{n}]\) for \(n=0,1,2,\ldots \)â€‰, hence v is measurable. Also,
Therefore, by (4.3) and (4.4),
Since \(C(h_{0})p_{F}=p_{F}\) and \(KMh_{0}=0\), we obtain
Since \(\lim_{\varrho \rightarrow 0} \frac{C(2t)p_{F}p_{F}}{2t^{2}}=\hat{A}p_{F}\), using proposition (2) results in
which implies
for some \(P, P'>0\).
Then, by using assumption (II),
where \(\q_{F}\\leq C\) and \(\A p_{F} \\leq L\) for some \(C, L>0\).
Since \(0<\alpha <1\), therefore the series
\(\sum_{n=1}^{\infty } \frac{h_{n}}{(h_{n+1})^{\alpha 1}}\) and \(\sum_{n=1}^{\infty } \frac{h_{n}^{2}}{(h_{n+1})^{\alpha 1}}\) are convergent.
This shows that \(v\in U_{ad}\). Therefore, p is continuous and
Thus, \(p(b)=p_{F}\) and \(p'(b)=q_{F}\).
5 Example
Consider the partial differential equation:
Let \(\mathbb{X}=L_{2}[0,\pi ]\) and \(\gamma :[0,b]\times (0,\pi )\rightarrow \mathbb{R}\) be a continuous control function in Ï±. Define the operator \(A:D(A)\rightarrow \mathbb{X}\) by
with \(D(A)=\{\eta \in \mathbb{X}:\eta ,\eta ^{\prime } \text{ are absolutely continuous } \eta ^{\prime \prime }\in \mathbb{X},\eta (0)=\eta (\pi )=0\} \). A is an infinitesimal generator of a strongly continuous cosine family \(C(\varrho )\) on X. Moreover, the spectrum of A consists of eigenvalues \(n^{2}\) for \(n=1,2,3,\ldots \)â€‰, with the associated normalized eigenvectors \(\eta _{n}(s)=(2/\pi )^{1/2}\sin (ns)\). In particular,
The cosine function \(C(\varrho )\) and the sine function \(S(\varrho )\) are defined in the following way:
respectively. Define \(r:J \times \mathbb{X}\rightarrow \mathbb{X}\) by
Let \(v:[0,b]\rightarrow \mathbb{U}\) be defined by
Define the controllability operator in the following way:
Let us choose the system operator A in such a way that \(W(\varrho )\) is coercive for all \(0< \varrho \leq b\) and there exists some \(N\geq 0\) such that
Also, the nonlinear function Ïƒ can be considered satisfying conditions (III)â€“(V).
The considered PDE (5.1) can be converted to (1.1). Therefore, system (5.1) is exactly controllable to \(\operatorname{dom}(A)\).
6 Conclusion
In the present manuscript, the exact controllability to \(\operatorname{dom}(A)\) for a secondorder semilinear system has been discussed using a new technique which avoids fixed point theorems and does not involve large estimations on the system constants. The control function has been formulated by the piecewise construction of steering controls. These results can be further extended for systems with delay or deviated arguments with impulses and fractionalorder systems.
Availability of data and materials
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Arora, U., Vijayakumar, V., Shukla, A. et al. Results on exact controllability of secondorder semilinear control system in Hilbert spaces. Adv Differ Equ 2021, 455 (2021). https://doi.org/10.1186/s13662021036205
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DOI: https://doi.org/10.1186/s13662021036205