A Hybrid Time-Scaling Transformation for Time-Delay Optimal Control Problems

Abstract

In this paper, we consider a class of nonlinear time-delay optimal control problems with canonical equality and inequality constraints. We propose a new computational approach, which combines the control parameterization technique with a hybrid time-scaling strategy, for solving this class of optimal control problems. The proposed approach involves approximating the control variables by piecewise constant functions, whose heights and switching times are decision variables to be optimized. Then, the resulting problem with varying switching times is transformed, via a new hybrid time-scaling strategy, into an equivalent problem with fixed switching times, which is much preferred for numerical computation. Our new time-scaling strategy is hybrid in the sense that it is related to two coupled time-delay systems—one defined on the original time scale, in which the switching times are variable, the other defined on the new time scale, in which the switching times are fixed. This is different from the conventional time-scaling transformation widely used in the literature, which is not applicable to systems with time-delays. To demonstrate the effectiveness of the proposed approach, we solve four numerical examples. The results show that the costs obtained by our new approach are lower, when compared with those obtained by existing optimal control methods.

Appendix

1.1 Proof of Theorem 3.1

Proof

Let ${\varvec{\delta }}$ and $r\in \{1,\ldots ,q\}$ be arbitrary but fixed, and let $\varvec{e}^{r}$ be the rth unit vector in $\mathbb {R}^{q}$. Then,

$$\begin{aligned} \frac{\partial \bar{\varvec{x}}(s)}{\partial \theta _{r}}=\lim _{\xi \rightarrow 0}\frac{{\bar{\varvec{x}}} (s|{\varvec{\delta }},\varvec{\theta }^{\xi })-{\bar{\varvec{x}}}(s|{\varvec{\delta }},{\varvec{\theta }})}{\xi }, \end{aligned}$$

(41)

where $\varvec{\theta }^{\xi }=\varvec{\theta }+\xi \varvec{e}^{r}$, $s_{\text {delay}}^{\xi }$ and $t_{\text {delay}}^{\xi }$ are the corresponding delayed time points in the new time horizon and the original time horizon such that

$$\begin{aligned} \mu \left( s_{\text {delay}}^{\xi }|{\varvec{\theta }}^{\xi }\right) =t_{\text {delay}}^{\xi } = \mu (s|{\varvec{\theta }}^{\xi })-h, \quad s\in [0,q]. \end{aligned}$$

Now, we will prove the theorem in the following steps.

Step 1: Preliminaries

For each real number $\xi \in \mathbb {R}$, let ${\bar{\varvec{x}}}^{\xi }$ denote the function ${\bar{\varvec{x}}}\big (\cdot |\varvec{\delta },\varvec{\theta }^{\xi }\big )$. Then, it follows from (20) that, for each $\xi \in \mathbb {R}$,

$$\begin{aligned} {\bar{\varvec{x}}}^{\xi }(s) = {\bar{\varvec{x}}}(s)+ \displaystyle \int _{0}^{s} \varvec{F}^{\xi }(t)\hbox {d}t,\quad s\in [0,q], \end{aligned}$$

(42)

where ${\bar{\varvec{x}}}(s)$ denotes the function ${\bar{\varvec{x}}}\big (\cdot |\varvec{\delta },\varvec{\theta }\big )$, and $\varvec{F}^{\xi }$ is defined as follows:

$$\begin{aligned} \varvec{F}^{\xi }(s):=\left\{ \begin{array}{l@{\quad }r} \theta _{\lfloor s\rfloor +1}^{\xi } \varvec{f}\big ( {\bar{\varvec{x}}}(s|\varvec{\delta },\varvec{\theta }^{\xi }), {\bar{\varvec{x}}}\left( s_{\text {delay}}^{\xi }|\varvec{\delta },\varvec{\theta }^{\xi }\right) ,{\varvec{\delta }} \big ), &{}\text {if }s_{\text {delay}}\ge 0,\\ \\ \theta _{\lfloor s\rfloor +1}^{\xi } \varvec{f}\big ({\bar{\varvec{x}}}(s|\varvec{\delta },\varvec{\theta }^{\xi }), {\bar{\varvec{x}}}\left( s_{\text {delay}}^{\xi }|\varvec{\delta },\varvec{\theta }^{\xi }\right) ,{\varvec{\delta }}, \varphi \left( t_{\text {delay}}^{\xi }\right) \big ),&{}\text {if }s_{\text {delay}}<0. \end{array}\right. \end{aligned}$$

Define

$$\begin{aligned} \varGamma ^{\xi }(s)= & {} {\bar{\varvec{x}}}(s|{\varvec{\delta }},\varvec{\theta }^{\xi })-{\bar{\varvec{x}}}(s|{\varvec{\delta }},{\varvec{\theta }})\nonumber \\= & {} \displaystyle \int _{0}^{s}\big (\varvec{F}^{\xi }- \varvec{F}^{0}\big )\hbox {d}t. \end{aligned}$$

(43)

Applying the mean value theorem, we have, for $s\in [0,q]$,

$$\begin{aligned} \displaystyle \varvec{F}^{\xi }(s)- \varvec{F}^{0}(s)= & {} \displaystyle \int _{0}^{1}\bigg \{ \displaystyle \frac{\partial \varvec{f}\big (\bar{\varvec{x}}+\eta \varGamma ^{\xi }(s),{\bar{\varvec{x}}}_{d}+\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}}\varGamma ^{\xi }(s)\nonumber \\&+\, \displaystyle \frac{\partial \varvec{f}\big (\bar{\varvec{x}}+\eta \varGamma ^{\xi }(s),{\bar{\varvec{x}}}_{d}+\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{d}}({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d})\nonumber \\&+\,\displaystyle \frac{\partial \varvec{f}\big (\bar{\varvec{x}}+\eta \varGamma ^{\xi }(s),{\bar{\varvec{x}}}_{d}+\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )}{\partial \theta _{r}}\xi \bigg \}\hbox {d}\eta , \end{aligned}$$

(44)

and

$$\begin{aligned} {\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}^{0}= & {} {\bar{\varvec{x}}}\left( s_{\text {delay}}^{\xi }|\varvec{\delta },\varvec{\theta }^{\xi }\right) -{\bar{\varvec{x}}}\left( s_{\text {delay}}|\varvec{\delta },\varvec{\theta }\right) \\= & {} {\bar{\varvec{x}}}\left( s_{\text {delay}}^{\xi }|\varvec{\delta },\varvec{\theta }^{\xi }\right) -{\bar{\varvec{x}}}\left( s_{\text {delay}}|\varvec{\delta },\varvec{\theta }^{\xi }\right) +{\bar{\varvec{x}}}\left( s_{\text {delay}}|\varvec{\delta },\varvec{\theta }^{\xi }\right) -{\bar{\varvec{x}}}\left( s_{\text {delay}}|\varvec{\delta },\varvec{\theta }\right) \\= & {} {\bar{\varvec{x}}}\left( s_{\text {delay}}^{\xi }|\varvec{\delta },\varvec{\theta }^{\xi }\right) -{\bar{\varvec{x}}}\left( s_{\text {delay}}|\varvec{\delta },\varvec{\theta }^{\xi }\right) +\varGamma ^{\xi }(s_{\text {delay}}) \end{aligned}$$

From Assumption $\mathbf A1$, it follows that the state set $\{{\bar{\varvec{x}}}^{\xi }(s): \xi \in [-a,a]\}$ is equibounded on [0, q], where $a>0$ is a fixed small real number. Hence, there exists a real number $C_{1}>0$ such that for each $\xi \in [-a,a]$,

$$\begin{aligned} {\bar{\varvec{x}}}^{\xi }(s)\in \mathcal {N}_{n}(C_{1}), \quad s\in [0,q], \end{aligned}$$

where $\mathcal {N}_{n}(C_{1})$ denotes the closed ball in $\mathbb {R}^{n}$ of radius $C_{1}$ centered at the origin. Note that, $\mathcal {N}_{n}(C_{1})$ is convex, thus, for each $\xi \in [-a,a]$,

$$\begin{aligned} {\bar{\varvec{x}}}(s)+\eta \varGamma ^\xi (s)\in \mathcal {N}_{n}(C_{1}),\quad s\in [0,q],~\eta \in [0,1]. \end{aligned}$$

Furthermore, it is easy to see that for each $\xi \in [-a,a]$,

$$\begin{aligned} \varvec{\theta }+\eta \xi \varvec{e}^{r}\in \mathcal {N}_{q}(C_{2}),\quad \eta \in [0,1], \end{aligned}$$

where $C_{2} :=|\varvec{\theta }|_{q}+a.$ Recall from Assumption $\mathbf A2$ that $\partial \varvec{f}/\partial \varvec{x}$ and $\partial \varvec{f}/\partial \theta _{r}$ are continuous. Hence, it follows from the compactness of [0, T], $\mathcal {V}$, $\mathcal {N}_{n}(C_{1})$ and $\mathcal {N}_{q}(C_{2})$ and the definitions of $z(s|\varvec{\theta })$ and $\phi$ that there exists a real number $C_{3}>0$ such that, for each $\xi \in [-a,a]$,

$$\begin{aligned} \bigg |\frac{\partial \varvec{f}^{\xi }_{\eta }}{\partial {\bar{\varvec{x}}}}\bigg |_{n\times n}\le & {} C_{3},\quad s\in [0,q],~\eta \in [0,1],\\ \bigg |\frac{\partial \varvec{f}^{\xi }_{\eta }}{\partial {\bar{\varvec{x}}}_{d}}\bigg |_{n\times n}\le & {} C_{3},\quad s\in [0,q],~\eta \in [0,1],\\ \bigg |\frac{\partial \varvec{f}^{\xi }_{\eta }}{\partial \theta _{r}}\bigg |_{n}\le & {} C_{3},\quad s\in [0,q],~\eta \in [0,1],\\ \bigg |\frac{\partial \phi _{\eta }^{\xi }}{\partial t}\bigg |_{n}\le & {} C_{3},\quad s\in [0,q],~\eta \in [0,1],\\ \end{aligned}$$

where $\varvec{f}^{\xi }_{\eta }$ denotes $\varvec{f}\big (\bar{\varvec{x}}+\eta \varGamma ^{\xi }(t),{\bar{\varvec{x}}}_{d}+\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )$, and $\phi _{\eta }^{\xi }$ denotes $\phi (\mu (t|\varvec{\theta }+\eta \xi \varvec{e}^{r})-h)$, and $| \cdot |$ denotes the Euclidian norm.

Step 2: The function $\varGamma ^{\xi }(s)$ is of order $\xi$

Let $\xi \in [-a,a]$ be arbitrary. When $s_{\text {delay}}<0$, taking the norm of both sides of (43) and applying the definition of $C_{3}$ gives

$$\begin{aligned} |\varGamma ^{\xi }(s)|_{n} = \bigg |\int _{0}^{s}\int _{0}^{1}\bigg \{\frac{\partial \varvec{f}^{\xi }_{\eta }}{\partial {\bar{\varvec{x}}}}\varGamma ^{\xi }(t)+ \frac{\partial \varvec{f}^{\xi }_{\eta }}{\partial \theta _{r}}\xi +\frac{\partial \varvec{f}^{\xi }_{\eta }}{\partial {\bar{\varvec{x}}}_{d}}\left( \phi ^{\xi }_{\eta }-\phi ^{0}_{\eta }\right) \bigg \}\hbox {d}\eta \hbox {d}t\bigg |_{n}, \end{aligned}$$

(45)

where

$$\begin{aligned} \phi ^{\xi }_{\eta }-\phi ^{0}_{\eta }= & {} \phi (\mu (t|\varvec{\theta }+\xi \varvec{e}^{r})-h) -\phi (\mu (t|\varvec{\theta })-h)\\= & {} \xi \frac{\partial \phi (\mu (t|\varvec{\theta }+\eta \xi \varvec{e}^{r})-h)}{\partial t}\frac{\partial \mu (t|\varvec{\theta }+\eta \xi \varvec{e}^{r})}{\partial \theta _{r}}, \quad \eta \in [0,1]. \end{aligned}$$

Thus, we have

$$\begin{aligned} |\varGamma ^{\xi }(s)|_{n}\le C_{3}|\xi |+C_{3}^{2}T|\xi |+\int _{0}^{s}C_{3}|\varGamma ^{\xi }(t)|_{n}\hbox {d}t, \quad s_{\text {delay}}<0. \end{aligned}$$

(46)

Applying Gronwall’s Lemma gives

$$\begin{aligned} |\varGamma ^{\xi }(s)|_{n}\le \left( C_{3}+C_{3}^{2}T\right) \exp {(C_{3}q)}|\xi |, \quad s\in [0,P_{1}[. \end{aligned}$$

(47)

When $s_{\text {delay}}\ge 0$,

where $P_{1}$ is a time point in the new time horizon such that

$$\begin{aligned} \mu (P_{1}|\varvec{\theta })=h. \end{aligned}$$

Since $s_{\text {delay}}\ge 0$, it follows from the definition of $s_{\text {delay}}$ that

$$\begin{aligned} \bigg |\int _{P_{1}}^{s}\int _{0}^{1}\frac{ \partial \varvec{f}^{\xi }_{\eta }}{\partial {\bar{\varvec{x}}}_{d}}\varGamma ^{\xi }(s_{\text {delay}}) \hbox {d}\eta \hbox {d}t \bigg |_{n} \le \int _{P_{1}}^{s}C_{3}|\varGamma ^{\xi }(s_{\text {delay}})|_{n}\hbox {d}t\le \int _{0}^{s}C_{3}|\varGamma ^{\xi }(t)|_{n}\hbox {d}t, \end{aligned}$$

and by the mean value theorem

$$\begin{aligned}&\displaystyle \bigg |\int _{P_{1}}^{s}\int _{0}^{1}\frac{ \partial \varvec{f}^{\xi }_{\eta }}{\partial {\bar{\varvec{x}}}_{z}}\big [\bar{\varvec{x}}(s_{\text {delay}}^{\xi }|\varvec{\delta },\varvec{\theta }^{\xi })-\bar{\varvec{x}}(s_{\text {delay}}|\varvec{\delta },\varvec{\theta }^{\xi })\big ] \hbox {d}\eta \hbox {d}t \bigg |_{n}\nonumber \\&\quad \le \displaystyle C_{3}\int _{P_{1}}^{s}\int _{0}^{1}\bigg |\frac{\partial {\bar{\varvec{x}}}(z(t,\varvec{\theta }+\iota \xi \varvec{e}^{r})|\varvec{\delta },\varvec{\theta }+\iota \xi \varvec{e}^{r})}{\partial z}\frac{\partial z}{\partial \theta _{r}}\xi \bigg |_{n}\hbox {d}\iota \hbox {d}t \le C_{3}^{3}q|\xi |, \end{aligned}$$

(48)

where $\iota \in [0,1].$ Again, by applying Gronwall’s Lemma, we have

$$\begin{aligned} |\varGamma ^{\xi }(s)|_{n}\le & {} \left( C_{3}+C_{3}^{2}T\right) \exp {(C_{3}q)}|\xi |+C_{3}|\xi | +C_{3}^{3}q|\xi |+\int _{0}^{s}2C_{3}|\varGamma ^{\xi }(t)|_{n}\hbox {d}t\nonumber \\\le & {} \left( C_{3}+C_{3}^{2}T\right) \exp {(C_{3}q)}|\xi | +\left( C_{3}+C_{3}^{3}q\right) \exp {(2C_{3}q)}|\xi |. \end{aligned}$$

(49)

Since $\xi \in [-a,a]$ is arbitrary, the function $\varGamma ^{\xi }(s)$ is of order $\xi$.

Step 3: The definition of $\rho$ and its properties

For each $\xi \in \mathbb {R}$, define the corresponding functions $\lambda ^{1,\xi }{:}\,[0,q]\rightarrow \mathbb {R}^{n},$ $\lambda ^{2,\xi }{:}\,[0,q]\rightarrow \mathbb {R}^{n}$ and $\lambda ^{3,\xi }{:}\,[0,q]\rightarrow \mathbb {R}^{n}$ as follows:

$$\begin{aligned} \lambda ^{1,\xi }(t):= & {} \int _{0}^{1}\bigg \{\frac{\partial \varvec{f}\big (\bar{\varvec{x}}+\eta \varGamma ^{\xi }(t),{\bar{\varvec{x}}}_{d} +\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }} +\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}}\nonumber \\&\qquad \qquad -\,\frac{\partial \varvec{f}\big (\bar{\varvec{x}},{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}} \bigg \}\varGamma ^{\xi }(t)\hbox {d}\eta \end{aligned}$$

(50)

$$\begin{aligned} \lambda ^{2,\xi }(t):= & {} \int _{0}^{1}\bigg \{\frac{\partial \varvec{f}\big (\bar{\varvec{x}}+\eta \varGamma ^{\xi }(t),{\bar{\varvec{x}}}_{d} +\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r}, {\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{d}}\nonumber \\&\qquad \qquad -\,\frac{\partial \varvec{f}\big (\bar{\varvec{x}},{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{d}} \bigg \}({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d})\hbox {d}\eta \end{aligned}$$

(51)

$$\begin{aligned} \lambda ^{3,\xi }(t):= & {} \int _{0}^{1}\bigg \{\frac{\partial \varvec{f}\big (\bar{\varvec{x}}+\eta \varGamma ^{\xi }(t), {\bar{\varvec{x}}}_{d}+\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}), {\varvec{\theta }}+\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )}{\partial \theta _{r}}\nonumber \\&\qquad \qquad -\,\frac{\partial \varvec{f}\big (\bar{\varvec{x}},{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial \theta _{r}} \bigg \}\xi \hbox {d}\eta \end{aligned}$$

(52)

In addition, let the function $\rho {:}\,[-a,0[\cup ]0,a]\rightarrow \mathbb {R}$ be defined as follows:

$$\begin{aligned} \rho (\xi ):=|\xi |^{-1}\int _{0}^{q}\{|\lambda ^{1,\xi }(t)|_{n} +|\lambda ^{2,\xi }(t)|_{n}+|\lambda ^{3,\xi }(t)|_{n}\}\hbox {d}t \end{aligned}$$

(53)

Since the function $\varGamma ^{\xi }(s)$ is of order $\xi$, it follows that

$$\begin{aligned}&{\bar{\varvec{x}}} +\eta \varGamma ^{\xi }(t)\rightarrow {\bar{\varvec{x}}} ,\quad \text {as}~ \xi \rightarrow 0, \end{aligned}$$

(54)

$$\begin{aligned}&{\bar{\varvec{x}}}_{d}+\eta \left( {\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}\right) \rightarrow {\bar{\varvec{x}}}_{d} ,\quad \text {as}~ \xi \rightarrow 0, \end{aligned}$$

(55)

uniformly with respective to $t\in [0,q]$ and $\eta \in [0,1]$. Meanwhile, it is obvious that

$$\begin{aligned} \varvec{\theta }+\eta \xi \varvec{e}^{r}\rightarrow \varvec{\theta }, \quad \text {as}~ \xi \rightarrow 0, \end{aligned}$$

(56)

uniformly with respect to $\eta \in [0,1]$. Since the convergences in (54) and (55) take place inside the ball $\mathcal {N}_{n}(C_{1})$, the convergence in (56) takes place inside of the ball $\mathcal {N}_{n}(C_{2})$, $\partial \varvec{f}/\partial {\bar{\varvec{x}}}$, $\partial \varvec{f}/\partial {\bar{\varvec{x}}}_{z}$ and $\partial \varvec{f}/\partial \theta _{r}$ are uniformly continuous on the compact set $[0,q]\times \mathcal {N}_{n}(C_{1})\times \mathcal {N}_{n}(C_{1})\times \mathcal {V}\times \mathcal {N}_{n}(C_{2})$,

$$\begin{aligned} \frac{\partial \varvec{f}\big (\bar{\varvec{x}}^{0}+\eta \varGamma ^{\xi }(s),{\bar{\varvec{x}}}_{d} +\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r}, {\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}}\rightarrow & {} \frac{\partial \varvec{f}\big (\bar{\varvec{x}}^{0},{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}},\quad \text {as}~\xi \rightarrow 0,\\ \frac{\partial \varvec{f}\big (\bar{\varvec{x}}^{0}+\eta \varGamma ^{\xi }(s),{\bar{\varvec{x}}}_{d}+\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{d}}\rightarrow & {} \frac{\partial \varvec{f}\big (\bar{\varvec{x}}^{0},{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{d}},\quad \text {as}~\xi \rightarrow 0,\\ \frac{\partial \varvec{f}\big (\bar{\varvec{x}}^{0}+\eta \varGamma ^{\xi }(s),{\bar{\varvec{x}}}_{d}+\eta ({\bar{\varvec{x}}}_{d}^{\xi }-{\bar{\varvec{x}}}_{d}),{\varvec{\theta }}+\eta \xi \varvec{e}^{r},{\varvec{\delta }}\big )}{\partial \theta _{r}}\rightarrow & {} \frac{\partial \varvec{f}\big (\bar{\varvec{x}}^{0},{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial \theta _{r}},\quad \text {as}~\xi \rightarrow 0,\\ \frac{\partial {\bar{\varvec{x}}}\left( s_{\text {delay}}^{\xi ,\iota }|\varvec{\delta },\varvec{\theta }+\iota \xi \varvec{e}^{r}\right) }{\partial s_{\text {delay}}}\rightarrow & {} \frac{\partial {\bar{\varvec{x}}}\left( s_{\text {delay}}|\varvec{\delta },\varvec{\theta }\right) }{\partial s_{\text {delay}}},\quad \text {as}~\xi \rightarrow 0,\\ \frac{\partial s_{\text {delay}}^{\xi ,\iota }}{\partial \theta _{r}}\rightarrow & {} \frac{\partial s_{\text {delay}}}{\partial \theta _{r}},\quad \text {as}~\xi \rightarrow 0, \end{aligned}$$

uniformly with respect to $t\in [0,q]$, $\eta \in [0,1]$ and $\iota \in [0,1]$, where $s_{\text {delay}}^{\xi ,\iota }$ is the corresponding delayed time of the control $\varvec{\theta }+\iota \xi \varvec{e}^{r}$. These results together with (49) indicate that $|\xi |^{-1}\lambda ^{1,\xi }\rightarrow \varvec{0}$, $|\xi |^{-1}\lambda ^{2,\xi }\rightarrow \varvec{0}$ and $|\xi |^{-1}\lambda ^{3,\xi }\rightarrow \varvec{0}$ uniformly on [0, q] as $\xi \rightarrow 0$. Thus,

$$\begin{aligned} \lim _{\xi \rightarrow 0}\rho (\xi )=\varvec{0}. \end{aligned}$$

(57)

Step 4: The final step

Let $\xi \in [-a,0[\cup ]0,a]$ be arbitrary but fixed. Then, it follows from (43) that

$$\begin{aligned} \varGamma ^{\xi }(s)= & {} \int _{0}^{s}\big [\lambda ^{1,\xi }(t)+\lambda ^{2,\xi }(t)+\lambda ^{3,\xi }(t)\big ]\hbox {d}t+\int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}}\varGamma ^{\xi }(t)\hbox {d}t\nonumber \\&+\int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{d}}\left[ \varGamma ^{\xi }(s_{\text {delay}}+\frac{\partial {\bar{\varvec{x}}}(s_{\text {delay}}^{\xi ,\iota }|(\varvec{\delta },\varvec{\theta }+\iota \xi \varvec{e}^{r}))}{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}^{\xi ,\iota }}{\partial \theta _{r}}\xi \right] \hbox {d}t\nonumber \\&+\int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial \theta _{r}}\xi \hbox {d}t \end{aligned}$$

(58)

Furthermore, integrating the auxiliary system gives

$$\begin{aligned} \varLambda ^{r}(s|\varvec{\delta ,\theta })= & {} \int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}}\varLambda ^{r}(t|\varvec{\delta ,\theta })\hbox {d}t\nonumber \\&+\int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{d}}\left[ \varLambda ^{r}(s_{\text {delay}}|\varvec{\delta ,\theta })+\frac{\partial {\bar{\varvec{x}}}(s_{\text {delay}}|(\varvec{\delta },\varvec{\theta }))}{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}}{\partial \theta _{r}}\right] \hbox {d}t\nonumber \\&+\int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial \theta _{r}}\hbox {d}t \end{aligned}$$

(59)

Multiplying (58) by $\xi ^{-1}$, and subtracting it from (59) yields

$$\begin{aligned}&\xi ^{-1}\varGamma ^{\xi }(s)-\varLambda ^{r}(s|\varvec{\delta ,\theta })\nonumber \\&\quad =\xi ^{-1}\int _{0}^{s}\big [\lambda ^{1,\xi }(t)+\lambda ^{2,\xi }(t)+\lambda ^{3,\xi }(t)\big ]\hbox {d}t\nonumber \\&\qquad +\int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}}(\xi ^{-1}\varGamma ^{\xi }(t)-\varLambda ^{r}(s|\varvec{\delta ,\theta }))\hbox {d}t\nonumber \\&\qquad +\int _{0}^{s}\frac{\partial \varvec{f}\big (\bar{\varvec{x}} ,{\bar{\varvec{x}}}_{d},{\varvec{\theta }},{\varvec{\delta }}\big )}{\partial {\bar{\varvec{x}}}_{z}}\bigg [\left( \xi ^{-1}\varGamma ^{\xi }(s_{\text {delay}})-\varLambda ^{r}(s_{\text {delay}}|\varvec{\delta ,\theta })\right) \nonumber \\&\qquad +\,\frac{\partial {\bar{\varvec{x}}}\left( s_{\text {delay}}^{\xi ,\iota }|\left( \varvec{\delta },\varvec{\theta }+\iota \xi \varvec{e}^{r}\right) \right. }{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}^{\xi ,\iota }}{\partial \theta _{r}}-\frac{\partial {\bar{\varvec{x}}}\left( s_{\text {delay}}|(\varvec{\delta },\varvec{\theta })\right) }{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}}{\partial \theta _{r}}\bigg ]\hbox {d}t \end{aligned}$$

(60)

Let

$$\begin{aligned} \bar{\rho }(\xi )= & {} \rho (\xi )+\int _{0}^{s}C_{3}\bigg |\frac{\partial {\bar{\varvec{x}}}(s_{\text {delay}}^{\xi ,\iota }|(\varvec{\delta },\varvec{\theta }+\iota \xi \varvec{e}^{r})}{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}^{\xi ,\iota }}{\partial \theta _{r}}\\&-\, \frac{\partial {\bar{\varvec{x}}}(s_{\text {delay}}|(\varvec{\delta },\varvec{\theta }))}{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}}{\partial \theta _{r}}\bigg |_{n}\hbox {d}t. \end{aligned}$$

Then, it is easy to see that $\rho ^{\prime }(\xi )\rightarrow 0,$ as $\xi \rightarrow 0$. Therefore,

$$\begin{aligned}&|\xi ^{-1}\varGamma ^{\xi }(s)-\varLambda ^{r}(s|\varvec{\delta ,\theta })|_{n}\\&\quad \le \rho (\xi )+\int _{0}^{s}C_{3}|\xi ^{-1}\varGamma ^{\xi }(t)-\varLambda ^{r}(t|\varvec{\delta ,\theta })|_{n}\hbox {d}t\\&\qquad +\int _{0}^{s}C_{3}|\xi ^{-1}\varGamma ^{\xi }(s_{\text {delay}})-\varLambda ^{r}(s_{\text {delay}}|\varvec{\delta ,\theta })|_{n}\hbox {d}t\\&\qquad +\int _{0}^{s}C_{3}\bigg |\frac{\partial {\bar{\varvec{x}}}(s_{\text {delay}}^{\xi ,\iota }|(\varvec{\delta },\varvec{\theta }+\iota \xi \varvec{e}^{r})}{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}^{\xi ,\iota }}{\partial \theta _{r}}-\frac{\partial {\bar{\varvec{x}}}(s_{\text {delay}}|(\varvec{\delta },\varvec{\theta }))}{\partial s_{\text {delay}}}\frac{\partial s_{\text {delay}}}{\partial \theta _{r}}\bigg |_{n}\hbox {d}t\\&\quad \le \bar{\rho }(\xi )+\int _{0}^{s}2C_{3}|\xi ^{-1}\varGamma ^{\xi }(t) -\varLambda ^{r}(t|\varvec{\delta ,\theta })|_{n}\hbox {d}t \end{aligned}$$

By Gronwall’s Lemma

$$\begin{aligned} |\xi ^{-1}\varGamma ^{\xi }(s)-\varLambda ^{r}(s|\varvec{\delta ,\theta })|_{n} \le \bar{\rho }(\xi )\exp {(2C_{3}q)},\quad s\in [0,q]. \end{aligned}$$

(61)

Noting that $\xi \in [-a,0[\cup ]0,a]$ is arbitrary, we can take the limit as $\xi \rightarrow 0$ in (61) and then apply (57) to get

$$\begin{aligned} \lim _{\xi \rightarrow 0}\xi ^{-1}\varGamma ^{\xi }(s)=\varLambda ^{r}(s|\varvec{\delta },\varvec{\theta }), \quad s\in [0,q]. \end{aligned}$$

$\square$