In this work, two enhanced versions of Wirtinger’s inequality are developed. These improvements arise when considering a weighted sum of multiple Wirtinger’s inequalities. Depending on the context, one of the proposed refinements may be applicable than the other. Finally, a simple application of such refinements is presented.
Suppose that \(u(x)\) is a real function with a period of \(2\pi\). if \(u(x) \in H^{1}(0,2\pi)\), the classical form of the Wirtinger’s inequality is written as follows [1, Chapter 2]:
\[\int_{0}^{2\pi} u(x)^{2}\,dx \leq \int_{0}^{2\pi} u'(x)^{2}\,dx \quad \text{if } \left\{ \int_{0}^{2\pi} u(x)\,dx = 0 \right\}, \tag{1}\label{eq:1}\]
Two important generalizations of Inequality (1) existing in the literature are as follows [1, Chapter 2]: \[\int_{a}^{b} u(x)^{2}\,dx \leq \left( \frac{b-a}{2\pi} \right)^{2} \int_{a}^{b} u'(x)^{2}\,dx \quad \text{if } \left\{ u(a)=u(b) \land \int_{a}^{b} u(x)\,dx = 0 \right\}, \tag{2}\label{eq:2}\] \[\int_{a}^{b} u(x)^{2}\,dx \leq \left( \frac{b-a}{\pi} \right)^{2} \int_{a}^{b} u'(x)^{2}\,dx \quad \text{if } \left\{ u(a)=u(b)=0 \vee \int_{a}^{b} u(x)\,dx = 0 \right\}. \tag{3}\label{eq:3}\]
It is possible that the function \(y(x)\) does not satisfy any of the conditions of Inequality (3). By substituting \(u(x)=y(x)-\overline{y}\), Inequality (3) can be written in the following generalized form: \[\int_{a}^{b} \left( y(x)-\overline{y} \right)^{2}\,dx \leq c^{2} \int_{a}^{b} y'(x)^{2}\,dx, \tag{4}\label{eq:4}\] where \(\overline{y}=\frac{1}{b-a}\int_{a}^{b} y(x)\,dx\) and \(c=\left( \frac{b-a}{\pi} \right)\). Inequality (4) holds for every function \(y(x)\) if only \(y \in H^{1}(a,b)\). The refinement presented in this paper is based on Inequality (4).
The Wirtinger inequality has widespread applications in the stability analysis of delay differential equations. References [2– 10] provide various applications of the Wirtinger’s inequality. References [11– 15] also present several generalizations and refinements of the Wirtinger inequality. It is important to note that a weighted sum of multiple Wirtinger inequalities can have various applications, ranging from stability analysis of systems to estimating a lower bound for the period of an orbit. The Lyapunov–Krasovskii functional, used in the stability analysis of systems, leads to the weighted sum of multiple Wirtinger inequalities [2]. In reference [16] (p. 33), it is proven that by summing multiple Wirtinger inequalities, one can estimate a lower bound for the period of the system.
If we consider Inequality (4) as a generalized form of the Wirtinger’s inequality, then it is obvious that the weighted sum of \(n\) Wirtinger’s inequalities for a set of \(n\) functions \(\left\{ y_{i}(x) \right\}_{i=1}^{n}\) can be written as follows:
\[\sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx \leq c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx, \tag{5}\label{eq:5}\] where \(\left\{ m_{i} \right\}_{i=1}^{n}>0\) and \(M=\sum\limits_{i=1}^{n} m_{i}\). In this paper, it is proved that two refinements of Inequality (5) exist, as follows: \[\sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx + \left\{ \begin{array}{r} q_{1}^{2}\\ q_{2}^{2} \end{array} \right\} \leq c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx, \tag{6}\label{eq:6}\] where \(q_{1}^{2}\) and \(q_{2}^{2}\) depend on \(\left\{ y_{i}(x) \right\}_{i=1}^{n}\), \(\left\{ y_{i}'(x) \right\}_{i=1}^{n}\), and \(\left\{ m_{i} \right\}_{i=1}^{n}\). Finally, in §3, A simple example of applying these refinements is presented.
The primary outcome of this paper is presented in the next theorem.
Assumption: Assume \(y_{1},\cdots,y_{n} \in H^{1}(a,b)\) are real-valued functions on \((a,b)\).
Theorem 1. Consider a set of \(n\) Wirtinger inequalities applied to functions \(\left\{ y_{i}(x) \right\}_{i=1}^{n}\) as follows: \[\int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx \leq c^{2} \int_{a}^{b} y_{i}'(x)^{2}\,dx, \qquad i=1,\ldots,n, \tag{7}\label{eq:7}\] Where \(\overline{y}_{i}=\frac{1}{b-a}\int_{a}^{b} y_{i}(x)\,dx\) and \(c=\left( \frac{b-a}{\pi} \right)\).
Then, there are two refined forms of the weighted sum of \(n\) Wirtinger inequalities, presented as follows: \[\sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx + \left\{ \begin{array}{r} q_{1}^{2}\\ q_{2}^{2} \end{array} \right\} \leq c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx, \tag{8}\label{eq:8}\] where \(\left\{ m_{i} \right\}_{i=1}^{n}>0\), \(M=\sum\limits_{i=1}^{n}m_{i}\) and \[ q_{1}^{2}= c^{2}\int_{a}^{b} y_{c}'(x)^{2}\,dx – \int_{a}^{b} \left( y_{c}(x)-\overline{y_{c}} \right)^{2}\,dx, \tag{9}\label{eq:9}\] \[q_{2}^{2}= \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( c^{2}\int_{a}^{b} r_{i}'(x)^{2}\,dx – \int_{a}^{b} \left( r_{i}(x)-\overline{r}_{i} \right)^{2}\,dx \right), \tag{10}\label{eq:10}\] \[y_{c}(x)= \frac{\sum\limits_{i=1}^{n} y_{i}(x) m_{i}}{M}, \tag{11}\label{eq:11}\] \[y_{c}'(x)= \frac{\sum\limits_{i=1}^{n} y_{i}'(x) m_{i}}{M}, \tag{12}\label{eq:12}\] \[r_{i}(x)= y_{i}(x)-y_{c}(x), \tag{13}\label{eq:13}\] \[r_{i}'(x)= y_{i}'(x)-y_{c}'(x). \tag{14}\label{eq:14} \]
Please note that \(\overline{r}_{i}=\frac{1}{b-a}\int_{a}^{b} r_{i}(x)\,dx\).
Proof. By substituting \(\overline{y}_{i}=\frac{1}{b-a}\int_{a}^{b} y_{i}(x)\,dx\) into Inequality (7), left side of Inequality (7) is rewritten as follows: \[\begin{aligned} \int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx &= \int_{a}^{b} y_{i}(x)^{2}\,dx – 2\overline{y}_{i}\int_{a}^{b} y_{i}(x)\,dx + \overline{y}_{i}^{2}\int_{a}^{b} dx \notag\\ &= \int_{a}^{b} y_{i}(x)^{2}\,dx – 2(b-a)\overline{y}_{i}^{2} + (b-a)\overline{y}_{i}^{2} \notag\\ &= \int_{a}^{b} y_{i}(x)^{2}\,dx – (b-a)\overline{y}_{i}^{2} \notag\\ &= \int_{a}^{b} y_{i}(x)^{2}\,dx – \frac{1}{b-a}\left( \int_{a}^{b} y_{i}(x)\,dx \right)^{2}. \end{aligned}\tag{15}\]
Therefore, the weighted sum of \(n\) Wirtinger inequalities can be rewritten as follows: \[\sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} y_{i}(x)^{2}\,dx – \frac{1}{b-a} \left( \int_{a}^{b} y_{i}(x)\,dx \right)^{2} \right) \leq c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx. \tag{16}\label{eq:16}\]
The proof of the Theorem consists of three steps: the left-hand side of Inequality (16) is derived in the first step, the right-hand side in the second, and finally, in the third step, the two expressions are subtracted.
Step 1: Derivation of the left-hand side
We assign \(S_{1}\) to represent the left-hand side of Inequality (16), as follows:
\[S_{1}=H_{1}-H_{2}, \tag{17}\label{eq:17}\] where \[H_{1}=\sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} y_{i}(x)^{2}\,dx \right), \tag{18}\label{eq:18}\] \[H_{2}=\frac{1}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} y_{i}(x)\,dx \right)^{2}. \tag{19}\label{eq:19}\]
First, we obtain the value of \(H_{1}\). Substituting \(y_{i}(x)\) from Eq. (13) into Eq. (18) yields:
\[\begin{aligned} H_{1} &= \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} \left( y_{c}(x)+r_{i}(x) \right)^{2} dx \right) \notag\\ &= \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} \left( y_{c}(x)^{2}+2y_{c}(x)r_{i}(x)+r_{i}(x)^{2} \right) dx \right) \notag\\ &= \frac{1}{M}\sum\limits_{i=1}^{n} m_{i} \int_{a}^{b} y_{c}(x)^{2}\,dx + \frac{2}{M}\sum\limits_{i=1}^{n} m_{i} \int_{a}^{b} y_{c}(x)r_{i}(x)\,dx + \frac{1}{M}\sum\limits_{i=1}^{n} m_{i} \int_{a}^{b} r_{i}(x)^{2}\,dx \notag\\ &= \int_{a}^{b} y_{c}(x)^{2}\,dx + \frac{2}{M} \int_{a}^{b} y_{c}(x) \left( \sum\limits_{i=1}^{n} m_{i} r_{i}(x) \right) dx + \frac{1}{M}\sum\limits_{i=1}^{n} m_{i} \int_{a}^{b} r_{i}(x)^{2}\,dx. \end{aligned}\tag{20}\]
From Eqs. (11) and (13), we have \(\sum\limits_{i=1}^{n} m_{i} r_{i}(x)=0\). Therefore, \[H_{1}=\int_{a}^{b} y_{c}(x)^{2}\,dx + \frac{1}{M}\sum\limits_{i=1}^{n} m_{i} \int_{a}^{b} r_{i}(x)^{2}\,dx. \tag{21}\label{eq:21}\]
Now, we obtain the value of \(H_{2}\). Substituting \(y_{i}(x)\) from Eq. (13) into Eq. (19) gives: \[\begin{aligned} H_{2} &= \frac{1}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} \left( y_{c}(x)+r_{i}(x) \right) dx \right)^{2} \notag\\ &= \frac{1}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} y_{c}(x)\,dx \right)^{2} + \frac{2}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} y_{c}(x)\,dx \right) \left( \int_{a}^{b} r_{i}(x)\,dx \right) \notag\\ &\quad + \frac{1}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} r_{i}(x)\,dx \right)^{2} \notag\\ &= \frac{1}{b-a} \left( \int_{a}^{b} y_{c}(x)\,dx \right)^{2} + \frac{2}{M(b-a)} \left( \int_{a}^{b} y_{c}(x)\,dx \right) \sum\limits_{i=1}^{n} m_{i} \int_{a}^{b} r_{i}(x)\,dx \notag\\ &\quad + \frac{1}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} r_{i}(x)\,dx \right)^{2} \notag\\ &= \frac{1}{b-a} \left( \int_{a}^{b} y_{c}(x)\,dx \right)^{2} + \frac{2}{M(b-a)} \left( \int_{a}^{b} y_{c}(x)\,dx \right) \int_{a}^{b} \left( \sum\limits_{i=1}^{n} m_{i} \right) r_{i}(x)\,dx \notag\\ &\quad + \frac{1}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} r_{i}(x)\,dx \right)^{2}. \notag \end{aligned}\]
Similar to Eq. (20), we have \(\sum\limits_{i=1}^{n} m_{i} r_{i}(x)=0\). Therefore, \[H_{2}=\frac{1}{b-a} \left( \int_{a}^{b} y_{c}(x)\,dx \right)^{2} + \frac{1}{b-a} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} r_{i}(x)\,dx \right)^{2}. \tag{22}\]
Substituting Eq. (21) and (22) into Eq. (17) gives: \[S_{1}=\int_{a}^{b} y_{c}(x)^{2}\,dx – \frac{1}{b-a} \left( \int_{a}^{b} y_{c}(x)\,dx \right)^{2} + \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( \int_{a}^{b} r_{i}(x)^{2}\,dx – \frac{1}{b-a} \left( \int_{a}^{b} r_{i}(x)\,dx \right)^{2} \right). \tag{23}\label{eq:23}\]
Step 2: Derivation of the right-hand side
We denote the right side of Inequality (16) by \(S_{2}\), defined as follows: \[S_{2}=c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx. \tag{24}\label{eq:24}\]
From Eq. (14), Eq. (24) can be rewritten as follows: \[\begin{aligned} S_{2} &= c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx \notag\\ &= c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} \left( y_{c}'(x)+r_{i}'(x) \right)^{2} dx \notag\\ &= c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{c}'(x)^{2}\,dx + 2c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{c}'(x) r_{i}'(x)\,dx + c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} r_{i}'(x)^{2}\,dx \notag\\ &= c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{c}'(x)^{2}\,dx + 2\frac{c^{2}}{M} \int_{a}^{b} \sum\limits_{i=1}^{n} m_{i} y_{c}'(x) r_{i}'(x)\,dx + c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} r_{i}'(x)^{2}\,dx \notag\\ &= c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{c}'(x)^{2}\,dx + 2\frac{c^{2}}{M} \int_{a}^{b} y_{c}'(x) \sum\limits_{i=1}^{n} m_{i} r_{i}'(x)\,dx + c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} r_{i}'(x)^{2}\,dx. \end{aligned}\tag{25}\]
From Eq. (12) and (14), we have \(\sum\limits_{i=1}^{n} m_{i} r_{i}'(x)=0\). Therefore, \[S_{2}=c^{2} \int_{a}^{b} y_{c}'(x)^{2}\,dx + c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} r_{i}'(x)^{2}\,dx. \tag{26}\label{eq:26}\]
Step 3: Derivation of \(S_{2}-S_{1}\)
From Eq. (26) and (23), we have:
\[\begin{aligned} S_{2}-S_{1} &= \left( c^{2}\int_{a}^{b} y_{c}'(x)^{2}\,dx – \int_{a}^{b} y_{c}(x)^{2}\,dx + \frac{1}{b-a} \left( \int_{a}^{b} y_{c}(x)\,dx \right)^{2} \right) \notag\\ &\quad + \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( c^{2} \int_{a}^{b} r_{i}'(x)^{2}\,dx – \int_{a}^{b} r_{i}(x)^{2}\,dx + \frac{1}{b-a} \left( \int_{a}^{b} r_{i}(x)\,dx \right)^{2} \right). \end{aligned}\tag{27}\]
From Eq. (15), Eq. (27) can be rewritten as follows: \[\begin{aligned} S_{2}-S_{1} &= \left( c^{2}\int_{a}^{b} y_{c}'(x)^{2}\,dx – \int_{a}^{b} \left( y_{c}(x)-\overline{y_{c}} \right)^{2}\,dx \right) \notag\\ &\quad + \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( c^{2} \int_{a}^{b} r_{i}'(x)^{2}\,dx – \int_{a}^{b} \left( r_{i}(x)-\overline{r}_{i} \right)^{2}\,dx \right). \end{aligned}\tag{28}\]
According to the Wirtinger inequality in Inequality (4), the value of \(S_{2}-S_{1}\) is equal to the sum of two positive terms \(q_{1}^{2}\) and \(q_{2}^{2}\). That is: \[S_{2}-S_{1}=c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx – \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx = q_{1}^{2}+q_{2}^{2}, \tag{29}\label{eq:29}\] where \[q_{1}^{2}=\left( c^{2}\int_{a}^{b} y_{c}'(x)^{2}\,dx – \int_{a}^{b} \left( y_{c}(x)-\overline{y_{c}} \right)^{2}\,dx \right),\] \[q_{2}^{2}=\sum\limits_{i=1}^{n} \frac{m_{i}}{M} \left( c^{2}\int_{a}^{b} r_{i}'(x)^{2}\,dx – \int_{a}^{b} \left( r_{i}(x)-\overline{r}_{i} \right)^{2}\,dx \right).\]
Therefore, two refinements for the weighted sum of \(n\) Wirtinger inequalities in the form presented in Inequality (7) can be written as follows: \[S_{2}-S_{1}=c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx – \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx \geq q_{1}^{2}, \tag{30}\label{eq:30}\] or \[S_{2}-S_{1}=c^{2} \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} y_{i}'(x)^{2}\,dx – \sum\limits_{i=1}^{n} \frac{m_{i}}{M} \int_{a}^{b} \left( y_{i}(x)-\overline{y}_{i} \right)^{2}\,dx \geq q_{2}^{2}. \tag{31}\label{eq:31}\]
The proof is complete. ◻
In this section, a simple example demonstrating the application of the proven refinements is presented.
Example 1. Consider the following nonlinear system: \[\dot{x}_{1}=f\left( x_{1},x_{2} \right), \tag{32-a}\label{eq:32a}\] \[\dot{x}_{2}=-f\left( x_{1},x_{2} \right)+i(t), \tag{32-b}\label{eq:32b}\] where \(f:R^{2} \rightarrow R\) is of class \(C^{1}\left( R^{2} \right)\), \(i(t)\) represents the input of the system, and the initial conditions are \(x_{1}(0)=0\) and \(x_{2}(0)=0\). Suppose that the energy of the system is defined as \(E=\dot{x}_{1}^{2}+\dot{x}_{2}^{2}\). Our aim is to find a lower bound for the time-averaged energy of the system that is independent of the system dynamics and depends only on the input \(i(t)\).
First, the Wirtinger inequalities are written for \(x_{1}(t)\) and \(x_{2}(t)\). From Inequality (7), we have: \[\int_{0}^{T} \left( x_{1}(t)-\overline{x}_{1} \right)^{2} \,dt \leq c^{2} \int_{0}^{T} \dot{x}_{1}(t)^{2} \,dt, \tag{33}\label{eq:33}\] \[\int_{0}^{T} \left( x_{2}(t)-\overline{x}_{2} \right)^{2} \,dt \leq c^{2} \int_{0}^{T} \dot{x}_{2}(t)^{2} \,dt. \tag{34}\label{eq:34}\]
Therefore, a weighted sum of Inequalities (33) and (34) can be written as follows: \[\frac{1}{2}\int_{0}^{T} \left( x_{1}(t)-\overline{x}_{1} \right)^{2} dt + \frac{1}{2}\int_{0}^{T} \left( x_{2}(t)-\overline{x}_{2} \right)^{2} dt \leq \frac{c^{2}}{2}\int_{0}^{T} \left( \dot{x}_{1}(t)^{2}+\dot{x}_{2}(t)^{2} \right) dt = \frac{c^{2}}{2}\int_{0}^{T} E\,dt. \tag{35}\label{eq:35}\]
Based on the theorem, Inequality (35) can be refined as follows: \[q^{2}+\frac{1}{2}\int_{0}^{T} \left( x_{1}(t)-\overline{x}_{1} \right)^{2} dt + \frac{1}{2}\int_{0}^{T} \left( x_{2}(t)-\overline{x}_{2} \right)^{2} dt \leq \frac{c^{2}}{2}\int_{0}^{T} E\,dt. \tag{36}\]
From Eq. (9), we have: \[q^{2}=c^{2}\int_{0}^{T} \dot{x}_{c}(t)^{2} dt – \int_{0}^{T} \left( x_{c}(t)-\overline{x}_{c} \right)^{2} dt. \tag{37}\label{eq:37}\]
Inequalities (33) and (34) are each multiplied by \(\frac{1}{2}\) and then added together. Therefore, from Eq. (11) and (12), we have: \[x_{c}(t)=\frac{x_{1}(t)+x_{2}(t)}{2}, \tag{38}\label{eq:38}\] \[\dot{x}_{c}(t)=\frac{\dot{x}_{1}(t)+\dot{x}_{2}(t)}{2}, \tag{39}\label{eq:39}\]
From Eqs. (32-a) and (32-b), we have: \[\dot{x}_{1}(t)+\dot{x}_{2}(t)=i(t), \tag{40}\label{eq:40}\] and \[x_{1}(t)+x_{2}(t)=\int_{0}^{t} i(s)\,ds. \tag{41}\label{eq:41}\]
Substituting Eqs. (40) and (41) into Eqs. (38) and (39) gives:
\[x_{c}(t)=\frac{1}{2}\int_{0}^{t} i(s)\,ds, \tag{42}\label{eq:42}\] \[\dot{x}_{c}(t)=\frac{1}{2}i(t), \tag{43}\label{eq:43}\] and \[\overline{x}_{c}=\frac{1}{2T}\int_{0}^{T} \int_{0}^{t} i(s)\,ds\,dt. \tag{44}\label{eq:44}\]
From Eq. (36), we can conclude that \[\frac{2q^{2}}{Tc^{2}} \leq \frac{1}{T}\int_{0}^{T} E\,dt = \overline{E}, \tag{45}\label{eq:45}\] where \(q^{2}\) is calculated from Eq. (37), and the functions \(x_{c}(t)\) and \(\dot{x}_{c}(t)\) can be obtained from Eqs. (42) and (43). Therefore, according to Inequality (45), the lower bound of the system’s average energy depends solely on the input function \(i(t)\).
The author declares that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Mitrinovic, D. S., Pecaric, J., & Fink, A. M. (2012). Inequalities Involving Functions and Their Integrals and Derivatives. Springer Science & Business Media.
Seuret, A., & Gouaisbaut, F. (2013). Wirtinger-based integral inequality: Application to time-delay systems. Automatica, 49(9), 2860-2866.
Liu, K., & Fridman, E. (2012). Wirtinger’s inequality and Lyapunov-based sampled-data stabilization. Automatica, 48(1), 102-108.
Gyurkovics, E. (2015). A note on Wirtinger-type integral inequalities for time-delay systems. Automatica, 61, 44-46.
Ding, S., Wang, Z., & Zhang, H. (2018). Wirtinger-based multiple integral inequality for stability of time-delay systems. International Journal of Control, 91(1), 12-18.
Kwon, O. M., Park, M. J., Park, J. H., Lee, S. M., & Cha, E. J. (2014). Improved results on stability of linear systems with time-varying delays via Wirtinger-based integral inequality. Journal of the Franklin Institute, 351(12), 5386-5398.
Zhang, L., Wang, S., Yu, W., & Song, Y. (2019). New absolute stability results for Lurie systems with interval time‐varying delay based on improved Wirtinger‐type integral inequality. International Journal of Robust and Nonlinear Control, 29(8), 2422-2437.
Suresh, R., & Manivannan, A. (2021). Robust stability analysis of delayed stochastic neural networks via Wirtinger-based integral inequality. Neural Computation, 33(1), 227-243.
Ding, S., Wang, Z., Wu, Y., & Zhang, H. (2016). Stability criterion for delayed neural networks via Wirtinger-based multiple integral inequality. Neurocomputing, 214, 53-60.
Choi, H. D., Ahn, C. K., Shi, P., Lim, M. T., & Song, M. K. (2015). L2-L\(^\infty\) Filtering for Takagi–Sugeno fuzzy neural networks based on Wirtinger-type inequalities. Neurocomputing, 153, 117-125.
Zhang, L., & Wang, S. (2018). Refined Wirtinger-type integral inequality. Journal of Inequalities and Applications, 2018(1), 109.
Zhang, X. M. (1996). A refinement of the discrete Wirtinger inequality. Journal of Mathematical Analysis and Applications, 200(3), 687-697.
Alzer, H. (1997). Note on Wirtinger’s inequality. In General Inequalities 7: 7th International Conference at Oberwolfach, November 13–18, 1995 (pp. 153-156). Basel: Birkhäuser Basel.
Xu, G., Liu, Z., & Lu, W. (2019). A kind of sharp Wirtinger inequality. Journal of Inequalities and Applications, 2019(1), 166.
Ricciardi, T. (2005). A sharp weighted Wirtinger inequality. arXiv preprint math/0501044.
Chueshov, I. (2015). Dynamics of Quasi-Stable Dissipative Systems (Vol. 1). Berlin: Springer.
