**4. Transformation of the second variation of the functional by means of modified variant of variations transformation method**

**4.1. Expansion of the second variation** 2 0 ; ⋅ **in Kelley-type variation (first-order transformation)**

Let 0 <sup>⋅</sup> be a singular control satisfying condition (2.9), and assumptions (A1), (A3), and (A4) be fulfilled. Now, we proceed to generalize and apply the variation transformation method [13].

Introduce the following set dependent on the admissible control 0 <sup>⋅</sup> :

( ) ( ) { [ ) ( ) } { [ ) ( ) } \* 0 0 1 1 0 1 0 : , : the derivative is continuous or continuous from the right at the point and , : the derivative is continuous or continuous from the right at the points and . *I I u t ht u h tt h u h* q qq q q q = ×= Î - × - Î- È × ± & & (4.1)

The following properties are obvious: (1) \\* is a finite set and <sup>1</sup> \*; (2) for every \*, there exists a sufficiently small number > 0 such that , + ∪ + ℎ, +ℎ+ ∩ \*; and (3) by (1.2), (1.3), and (2.5), the derivatives ˙ <sup>0</sup> <sup>⋅</sup> , ˙0 <sup>⋅</sup> are continuous or continuous from the right at every \*. These properties are important for our further reasoning, and we call them properties of the set \*.

Require that the variation <sup>⋅</sup> <sup>=</sup> 0 <sup>⋅</sup> , <sup>⋅</sup> satisfies additionally the following conditions as well:

$$\int\_{\theta}^{\theta+\varepsilon} \mathcal{S}\_0 p\left(t\right) dt = 0, \theta \in I^\*, \mathcal{S}\_0 p\left(t\right) = 0, \; \mathcal{S}q\left(t\right) = 0, \; t \in I\_1 \; \bigvee \left[\theta, \theta + \varepsilon\right), \; \varepsilon \in \left(0, \varepsilon^\*\right), \tag{4.2}$$

where \* = min 0, , and 0, were defined above.

Make a passage from the variation <sup>=</sup> 0 , , 1, satisfying (4.2), to a new variation 1 <sup>=</sup> 1 , , 1, where

$$\mathcal{S}\_{\mathsf{t}}p\left(t\right) = \bigcup\_{\theta} \mathcal{S}\_{\mathsf{0}}p\left(\mathsf{\tau}\right)d\mathsf{\tau}, \ t \in I\_{\mathsf{t}}.\tag{4.3}$$

Obvious,

$$\mathcal{S}\_1 p\left(t\right) = 0, \ t \in I\_1 \lor \left(\theta, \theta + \varepsilon\right). \tag{4.4}$$

Transform the variation of the trajectory as well: in place of , , consider the function 1 , :

$$\delta \mathcal{S} \mathbf{x}(t) = \delta \mathbf{x}(t) - \mathbf{g}\_0 \left[ p \right](t) \delta\_t p(t) - \mathbf{g}\_0 \left[ \tilde{p} \right](t) \delta\_t p\left(t - h\right), t \in I,\tag{4.5}$$

where

$$\log\_0\{\mu\}(t) \coloneqq f\_{\mu}(t), \ t \in I, \ \mu \in \{p, \tilde{p}\}.\tag{4.6}$$

As assumptions (A3) and (A4) are fulfilled, then by virtue of property of the set \* we easily have: the function 1 , is continuous and 1˙ , .

By direct differentiation, allowing for (A3), (A4) and (2.11), (4.3), (4.4) from (4.5) we obtain that 1 , is the solution of the system

$$\begin{split} \delta \dot{\varsigma} \dot{\varsigma}(t) &= f\_{\times}(t) \delta\_{!} \mathbf{x}(t) + \mathbf{g}\_{1} \big[ p \big](t) \delta\_{!} p\big(t \big) + \mathbf{g}\_{1} \big[ \tilde{p} \big](t) \delta\_{!} p\big(t - h \big) \\ &+ f\_{\neq}(t) \delta q\big(t \big) + f\_{\neq}(t) \delta q\big(t - h \big), t \in \left[ \theta, t\_{1} \right], \end{split} \tag{4.7}$$

$$\delta \mathcal{S} \ge \begin{pmatrix} t \end{pmatrix} = 0, \ t \in \left[ t\_0, \theta \right], \ \delta \not\ge p\left(t\right) = 0, \delta q\left(t\right) = 0, \quad t \in \left[t\_0 - h, \theta\right), \tag{4.8}$$

where

( ) () () [ ) ( ) \* \* 00 1 *p t dt I p t q t t I* 0, , 0, 0, \ , , 0, ,

Make a passage from the variation <sup>=</sup> 0 , , 1, satisfying (4.2), to a new

10 1 () ( ) , .

1 1 *pt t I* () ( ) =Î + 0, \ , .

qq e

 d<sup>1</sup>*xt xt g p t pt g p t pt h t I* () () = - - -Î 01 01 [ ]() () [ ]% () ( ), , (4.5)

<sup>1</sup> , ,,

& % (4.7)

 q  d

q

) (4.8)

Transform the variation of the trajectory as well: in place of , , consider the function

*t pt p d t I*

d

d

m

have: the function 1 , is continuous and 1˙ , .

*q q*

dd

qd

+ + -Î %

q

 d

m

*g t f t t I pp* <sup>0</sup> [ ]() () { } : , , ,.

= ÎÎ

As assumptions (A3) and (A4) are fulfilled, then by virtue of property of the set \* we easily

By direct differentiation, allowing for (A3), (A4) and (2.11), (4.3), (4.4) from (4.5) we obtain that

() () () [ ]() () [ ]() ( ) () () () ( ) [ ] 1 111 11

 d

=+ + -

*xt f t xt g p t pt g p t pt h f t qt f t qt h t t*

<sup>1</sup>*xt t t pt qt t t h* ( ) = Î = = Î- 0, , , 0, 0, , , [ ] 0 1

 d() () [ <sup>0</sup>

 m

 d tt

=Î = =Î + Î ò (4.2)

qq e e

<sup>=</sup> <sup>Î</sup> ò (4.3)

% (4.6)

 e

(4.4)

 d

q e

+

q

Obvious,

1 , :

where

d

 qd

204 Nonlinear Systems - Design, Analysis, Estimation and Control

where \* = min 0, , and 0, were defined above.

variation 1 <sup>=</sup> 1 , , 1, where

dd

1 , is the solution of the system

d

*x*

dd

$$\log\left[\mu\right](t) \coloneqq f\_x(t)\mathbf{g}\_0\left[\mu\right](t) - \frac{d}{dt}\mathbf{g}\_0\left[\mu\right](t), \ t \in I, \ \mu \in \left\{p, \tilde{p}\right\}.\tag{4.9}$$

Now, let us write down the second variation (2.12) in terms of new variables. By (4.4) from (4.5), we have <sup>1</sup> <sup>=</sup> 1 <sup>1</sup> . According to this property and (4.2)–(4.6), for any 0, \* the second variation (2.12), after simple reasoning takes a new form

$$\mathcal{S}^2 S\Big(\boldsymbol{\mu}^0; \mathcal{S}\boldsymbol{u}\big(\cdot\big)\big) = \sum\_{i=1}^4 \boldsymbol{\Delta}\_i,\tag{4.10}$$

where

() () () ( ) () () () ( ) () [ ]() ( ) ( ) [ ]( ) () ( ) () ( ) ( ) () 1 0 1 1 1 11 1 1 1 1 01 0 1 1 1 : 2 2 , *t T T xx xx T T xx xx T T xq xq x t x t x t x t H t x t dt x x H t g p t x t h H t h g p t h p t dt x x H t x t h H t h q t dt* q q e q q e q dj d d d dd d dd d + + D = - - é ù ++ + + - ë û - é ù ++ + ë û ò ò ò % % (4.11)

$$\begin{split} \boldsymbol{\Delta}\_{\text{2}} & \coloneqq -\int\_{\boldsymbol{\theta}}^{\boldsymbol{\theta}+\boldsymbol{\varepsilon}} \Big[ \boldsymbol{\delta}\_{\text{p}} \boldsymbol{p}^{\top}(\boldsymbol{t}) \Big] \Big\mathbf{g}\_{\text{o}}^{\top} \Big[ \boldsymbol{p} \big] \Big( \boldsymbol{t} \big) \boldsymbol{H}\_{\text{uv}} \Big( \boldsymbol{t} \big) \mathbf{g}\_{\text{o}} \big[ \boldsymbol{p} \big] \Big( \boldsymbol{t} \big) + \boldsymbol{g}\_{\text{o}}^{\top} \Big[ \boldsymbol{p} \big] \Big( \boldsymbol{t} + \boldsymbol{h} \big) \boldsymbol{H}\_{\text{uv}} \Big( \boldsymbol{t} + \boldsymbol{h} \big) \Big\| \boldsymbol{\varepsilon} \big] \Big\| \boldsymbol{t} + \boldsymbol{h} \Big] \Big\| \boldsymbol{\delta}\_{\text{v}} \big( \boldsymbol{t} \big) \\ + 2 \boldsymbol{\delta}\_{\text{i}} \boldsymbol{p}^{\top} \Big( \boldsymbol{t} \big) \Big[ \boldsymbol{g}\_{\text{o}}^{\top} \Big( \boldsymbol{p} \big) \big( \boldsymbol{t} \big) + \boldsymbol{g}\_{\text{v}}^{\top} \Big[ \boldsymbol{p} \big] \big( \boldsymbol{t} + \boldsymbol{h} \big) \boldsymbol{H}\_{\text{u}\hat{\boldsymbol{q}}} \Big( \boldsymbol{t} + \boldsymbol{h} \big) \Big\| \boldsymbol{\delta}\_{\text{u}} \big( \boldsymbol{t} \big) \\ + 2 \boldsymbol{\delta}\_{\text{o}} \boldsymbol{p}^{\top} \Big( \boldsymbol{t} \big) \Big[ \boldsymbol{H}\_{\text{vq}} \Big( \boldsymbol{t} \big) + \boldsymbol{H}\_{\text{p}\hat{\boldsymbol{q}}} \Big( \boldsymbol{t} + \boldsymbol{h} \big) \Big[ \boldsymbol{\delta}\_{\text{uq}} \big$$

$$\Delta\_3 \coloneqq -2 \int\_{\theta}^{\theta \wedge \pi} \left[ \mathcal{S}\_l \mathbf{x}^r \left( t \right) H\_{\imath \mathbf{p}} \left( t \right) + \mathcal{S}\_l \mathbf{x}^r \left( t + h \right) H\_{\imath \theta} \left( t + h \right) \right] \delta\_0 p \left( t \right) dt,\tag{4.13}$$

$$\Delta\_{4} \coloneqq -2\int\_{\boldsymbol{\theta}}^{\boldsymbol{\theta}\cdot\boldsymbol{\varepsilon}} \boldsymbol{\delta}\_{1} \boldsymbol{p}^{\boldsymbol{\prime}}\left(\boldsymbol{t}\right) \Big[\boldsymbol{\mathcal{g}}\_{0}^{\boldsymbol{\prime}}\Big[\boldsymbol{p}\big]\Big(\boldsymbol{t}\big)\boldsymbol{H}\_{\boldsymbol{\varepsilon}\boldsymbol{p}}\left(\boldsymbol{t}\right) + \boldsymbol{\mathcal{g}}\_{0}^{\boldsymbol{\prime}}\Big[\boldsymbol{\tilde{p}}\big]\Big(\boldsymbol{t}+\boldsymbol{h}\big)\boldsymbol{H}\_{\boldsymbol{\varepsilon}\boldsymbol{p}}\left(\boldsymbol{t}+\boldsymbol{h}\right)\Big]\boldsymbol{\delta}\_{0}\boldsymbol{p}\left(\boldsymbol{t}\right)d\boldsymbol{t}.\tag{4.14}$$

In the obtained representation, taking into account (A3), (A4), (4.2), (4.3), (4.7), (4.8), (4.13), (4.14) and the property of the set \*, we transform 3, 4 by integration by parts. Then, we have

{ ( ) ( ) ( ) () () ( ) ( ) ( ) ( ) ( ) () ( ) [ ]() () [ ]( ) ( ) () () () () ( ) ( ) () ( ) [ ]( ) [ ] 3 1 1 1 1 1 1 1 1 40 0 0 : 2 2 2 , : *T T xp x xp T T xp x xp T T T xp xp T T xp q xp q T <sup>d</sup> x t H t f tH t dt <sup>d</sup> x t h H t h f t h H t h p t dt dt p t g p t H t g p t h H t h p t dt p t H t f t H t h f t h q t dt p t Qpt Qpt* q e q q e q q e q q e q d d d d d d d d + + + + é ù D = <sup>+</sup> ê ú ë û é ùü + + ++ + + ê úý ë ûþ <sup>+</sup> é ù +++ ë û <sup>+</sup> é ù ++ + ë û D = + + ò ò ò ò % % % % % % % ( ) () ( ) ( [ ]() ()) ( [ ]( )( )) ( ) 1 1 0 <sup>0</sup> <sup>1</sup> , *T T <sup>T</sup> xp xp h p t dt d d p t g p t H t g p t h H t h p t dt dt dt* q e q d d d + é ù ë û é ù <sup>+</sup> + ++ ê ú ë û <sup>ò</sup> % %

where 1 <sup>⋅</sup> , , is defined by (2.19),

$$\mathcal{Q}\_0[\mu](t) \coloneqq \mathbf{g}\_0^\top[\mu](t)H\_{\times\mu}(t) - H\_{\times\mu}^\top(t)\mathbf{g}\_0[\mu](t), \quad t \in I, \,\mu \in \{p, \tilde{p}\}.\tag{4.15}$$

By substituting these relations in (4.10), after elementary transformations considering (4.11) and (4.12), we arrive at the validity of the following statement.

**Proposition 4.1**. Let assumptions (A1), (A3), (A4), and conditions (2.6) be fulfilled. Also, let the functions 0 <sup>⋅</sup> , 1 <sup>⋅</sup> , 0 <sup>⋅</sup> be defined by (4.6), (4.9), and (4.15), respectively, and 1 , be the solution of the system (4.7) and (4.8). Then along the singular control 0 <sup>⋅</sup> , satisfying condition (2.9), and on the variations <sup>=</sup> 0 , , 1 satisfying (4.2), (4.3), the following representation (first-order transformation) is valid:

$$\Delta^2 S\left(\mu^0; \delta\mathfrak{u}\left(\cdot\right)\right) = \Delta\_1^{(2)} S\left(\mu^0; \delta\_1 p, \delta q, \delta\_1 \mathfrak{x}, \varepsilon\right) + \Delta\_2^{(2)} S\left(\mu^0; \delta\_0 p, \delta\_1 p, \delta q, \varepsilon\right), \forall \varepsilon \in \left(0, \varepsilon^\*\right). \tag{4.16}$$

Here

( ) ( ) () () ( ) ( ) () () () { ( ) [ ]() ( ) [ ]( ) () () () ( ) ( ) ()} 1 2 0 0 1 1 1 1 1 1 11 1 1 11 1 1 1 1 1 ; ,, , 2 *t T T xx xx T T T T xq xq S u p q x x t x t x t x t H t x t dt x tG p t xt hG p t h pt x t H t x t h H t h q t dt* q q e q d dd e d j d d d dd d dd d + D= - - é ù ++ + ë û + ++ + é ù ë û ò ò % % (4.17)

#### Conditions for Optimality of Singular Controls in Dynamic Systems with Retarded Control http://dx.doi.org/10.5772/64225 207

$$\begin{split} \Delta\_{z}^{(2)}S(\boldsymbol{u}^{0};\delta\_{0}\boldsymbol{p},\delta\_{1}\boldsymbol{p},\delta\_{0}\boldsymbol{\varrho},\boldsymbol{\varepsilon}) &= \int\_{0}^{\theta\_{\text{eff}}} \delta\_{1} \boldsymbol{p}^{\top}(t) \Big[ L\_{1} \Big[ \boldsymbol{p} \big](t) + L\_{1} \Big[ \tilde{\boldsymbol{p}} \big](t+h) \Big] \delta\_{1} \boldsymbol{p}(t) \\ &+ 2\delta\_{1} \boldsymbol{p}^{\top}(t) \Big[ \boldsymbol{P}\_{1} \big[ \boldsymbol{p},\boldsymbol{q} \big](t) + \boldsymbol{P}\_{1} \big[ \tilde{\boldsymbol{p}},\tilde{\boldsymbol{q}} \big](t+h) \Big] \delta q(t) + \\ &\delta\_{0} \boldsymbol{p}^{\top}(t) \Big[ \boldsymbol{Q}\_{0} \big[ \boldsymbol{p} \big](t) + \boldsymbol{Q}\_{0} \big[ \tilde{\boldsymbol{p}} \big](t+h) \Big] \delta\_{1} \boldsymbol{p}(t) \\ &- 2\delta\_{0} \boldsymbol{p}^{\top}(t) \Big[ \boldsymbol{H}\_{\rho\_{0}} \big( t) + \boldsymbol{H}\_{\tilde{\rho}\_{0}}(t+h) \Big] \delta q(t) \\ &- \delta q^{\top}(t) \Big[ \boldsymbol{H}\_{\rho\_{0}} \big( t) + \boldsymbol{H}\_{\tilde{\rho}}(t+h) \Big] \delta q(t) \Big] dt, \end{split} \tag{4.18}$$

where \* was defined above (see (4.2)),

{ ( ) ( ) ( ) () ()

*<sup>d</sup> x t H t f tH t dt*

*T T*

é ù D = <sup>+</sup> ê ú

*T T*

*T T T*

*T T*

D = + +

1 1

<sup>+</sup> é ù +++ ë û

+ + ++ + + ê úý ë ûþ

3 1

q

ò

d

q e

+

206 Nonlinear Systems - Design, Analysis, Estimation and Control

: 2

2

d

q e

+

q

ò

+

q

ò

q e

:

q e

+

q

1

d

d

q e

+

q

d

m

d

Here

 d ò

d

( ) [ ]( ) [ ]

<sup>+</sup> é ù ++ + ë û

*p t Qpt Qpt*

40 0 0

 m = *x x* m

and (4.12), we arrive at the validity of the following statement.

d dd e

2

D= -

q e

+

q

ò

2 0 0

; ,, ,

d dd e d

*T*

where 1 <sup>⋅</sup> , , is defined by (2.19),

( ) ( ) ( ) ( ) ( ) ()

*<sup>d</sup> x t h H t h f t h H t h p t dt dt*

*xp x xp*

% %

*xp x xp*

ë û é ùü

() () () ( ) ( ) ()

*p t H t f t H t h f t h q t dt*

é ù <sup>+</sup> + ++ ê ú ë û ò % %

0 0 [ ]( ): [ ]() () () <sup>0</sup> [ ]( ), , ,. { } *T T Q t g tH t H t g t t I pp*

 m-

By substituting these relations in (4.10), after elementary transformations considering (4.11)

**Proposition 4.1**. Let assumptions (A1), (A3), (A4), and conditions (2.6) be fulfilled. Also, let the functions 0 <sup>⋅</sup> , 1 <sup>⋅</sup> , 0 <sup>⋅</sup> be defined by (4.6), (4.9), and (4.15), respectively, and 1 , be the solution of the system (4.7) and (4.8). Then along the singular control

0 <sup>⋅</sup> , satisfying condition (2.9), and on the variations <sup>=</sup> 0 , , 1 satisfying

 d d de

{ ( ) [ ]() ( ) [ ]( ) ()

*x tG p t xt hG p t h pt*

%

11 1 1 1

1

q

ò

 d

*t*

*xx xx*


() () ( ) ( ) ()}

*x t H t x t h H t h q t dt*

*xq xq*

+ ++ + é ù ë û

( ) ( ) ( ) ( ) ( ) ( ) () 2 0 2 2 <sup>0</sup> <sup>0</sup> \* 1 1 1 2 01

( ) ( ) () () ( ) ( ) () () ()

*S u p q x x t x t x t x t H t x t dt*

*T T*

 d

*T T*

dd

1 1 1 1 1 1 11 1 1

 j

1 1

dd

*T T*

*Su u Su p q x Su p p q* ; × =D ; , , , ; , , , , 0, .

(4.2), (4.3), the following representation (first-order transformation) is valid:

2 ,

*xp q xp q*

é ù ë û

1 1 1 1

( ) [ ]() () [ ]( ) ( ) ()

*p t g p t H t g p t h H t h p t dt*

*xp xp*

%

%% %

% ( ) ()

1

d

*h p t dt*

( ) ( [ ]() ()) ( [ ]( )( )) ( )

m

1 0 <sup>0</sup> <sup>1</sup> , *T T <sup>T</sup> xp xp*

*d d p t g p t H t g p t h H t h p t dt dt dt*

 d

 d  d

 m  d

 e

 d

 d  d

% (4.17)

+ D " Î (4.16)

 e

Î Î % (4.15)

$$\mathcal{G}\_{l}[\mu](t) \coloneqq H\_{\infty}(t)\mathbf{g}\_{0}[\mu](t) - f\_{\text{x}}^{\text{T}}(\mathbf{x})H\_{\text{x}\mu}(t) - \frac{d}{dt}H\_{\text{x}\mu}(t), \ t \in I, \mu \in \{p, \tilde{p}\}, \tag{4.19}$$

$$\begin{aligned} P\_1[p,q](t) &:= H\_{\ge p}^{\mathbb{T}}(t) f\_q(t) - \mathbf{g}\_0^{\mathbb{T}}[p](t) H\_{\ge q}(t), t \in I, \\ P\_1[\tilde{p}, \tilde{q}](t) &:= H\_{\ge 0}^{\mathbb{T}}(t) f\_{\tilde{q}}(t) - \mathbf{g}\_0^{\mathbb{T}}[\tilde{p}](t) H\_{\ge \overline{q}}(t), t \in I, \end{aligned} \tag{4.20}$$

$$\begin{split} \mathcal{L}\_{1}[\mu](t) & \coloneqq -\mathsf{g}\_{0}^{\top}[\mu](t)H\_{\scriptscriptstyle\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny} \begin{split} & \mu \left[ \mu \right](t)H\_{\scriptscriptstyle\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny} \mathcal{D}\_{0}[\mu](t) + 2\mathsf{g}\_{1}^{\top}[\mu](t)H\_{\scriptscriptstyle\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny} \left( t \right)H\_{\scriptscriptstyle\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny\tiny} & \end{split} \tag{4.21} \end{split} \tag{4.21}$$

#### **4.2. Higher-order transformation**

Let 0 <sup>⋅</sup> , 0 <sup>⋅</sup> be some process, where 0 <sup>⋅</sup> is a singular control satisfying condition (2.9), and assumptions (A1), (A5), and (A6) be fulfilled. Introduce the matrix functions calculated along the process 0 <sup>⋅</sup> , 0 <sup>⋅</sup> and determined by the following recurrent formulas:

$$\begin{aligned} \mathbf{g}\_{i+1}[\mu](t) &= f\_{\times}(t)\mathbf{g}\_{i}[\mu](t) - \frac{d}{dt}\mathbf{g}\_{i}[\mu](t),\\ \mathbf{g}\_{0}[\mu](t) &:= f\_{\mu}(t), \ t \in I, \ \mu \in \{p, \tilde{p}\}, i = 0, 1, \ldots, \end{aligned} \tag{4.22}$$

$$\begin{aligned} G\_{l+1}[\mu](t) &= H\_{\infty}(t) \lg[\mu](t) - f\_{\times}^{\top}(t) G\_{i}[\mu](t) - \frac{d}{dt} G\_{i}[\mu](t), \\ G\_{0}[\mu](t) &:= H\_{\times \mu}(t), \; t \in I, \; \mu \in \{p, \tilde{p}\}, \; i = 0, 1, \dots \end{aligned} \tag{4.23}$$

Furthermore, similar to (4.15), (4.20), (4.21), and (3.10), consider the functions

$$\begin{aligned} P\_{i+1}[p,q](t) &= G\_i^\top \left[ p \right](t) f\_q(t) - \mathbf{g}\_i^\top \left[ p \right](t) H\_{i\eta}(t), P\_0[p,q](t) \coloneqq H\_{\eta q}(t), \; t \in I, i = 0, 1, \dots, \\ P\_{i+1}[\tilde{p}, \tilde{q}](t) &= G\_i^\top \left[ \tilde{p} \right](t) f\_{\tilde{q}}(t) - \mathbf{g}\_i^\top \left[ \tilde{p} \right](t) H\_{i\eta}(t), P\_0[\tilde{p}, \tilde{q}](t) \coloneqq H\_{\tilde{p}\zeta}(t), \; t \in I, \end{aligned} \tag{4.24}$$

$$\underline{\mathbf{G}}\_{i}[\boldsymbol{\mu}](t) = \mathbf{g}\_{i}^{\mathrm{r}}[\boldsymbol{\mu}](t)G\_{i}[\boldsymbol{\mu}](t) - G\_{i}^{\mathrm{r}}[\boldsymbol{\mu}](t)\mathbf{g}\_{i}[\boldsymbol{\mu}](t), \ \boldsymbol{\mu} \in \{\boldsymbol{p}, \boldsymbol{\tilde{p}}\}, t \in I, i = 0, 1, \ldots, \tag{4.25}$$

$$\begin{split} L\_{i+1}[\mu](t) &= -\mathbf{g}\_{i}^{\top}[\mu](t)H\_{\text{av}}(t)\mathbf{g}\_{i}[\mu](t) + 2\mathbf{g}\_{i+1}^{\top}[\mu](t)G\_{i}[\mu](t) + \frac{d}{dt}(\mathbf{g}\_{i}^{\top}[\mu](t)G\_{i}[\mu](t)),\\ L\_{0}[\mu](t) &:= H\_{\mu\mu}(t), \ \mu \in \{p, \tilde{p}\}, t \in I, i = 0, 1, \ldots, \end{split} \tag{4.26}$$

$$\begin{aligned} M\_i[\boldsymbol{p}, \boldsymbol{\tilde{p}}](\boldsymbol{s}, \boldsymbol{\tau}) &\coloneqq \mathbf{g}\_i^r[\boldsymbol{p}](\boldsymbol{s}) \boldsymbol{\lambda}^T(\boldsymbol{s}, \boldsymbol{\tau}) G\_i[\boldsymbol{\tilde{p}}](\boldsymbol{\tau}) \\ + \mathbf{g}\_i^r[\boldsymbol{p}](\boldsymbol{s}) \boldsymbol{\Psi}(\boldsymbol{s}, \boldsymbol{\tau}) \mathbf{g}\_i[\boldsymbol{\tilde{p}}](\boldsymbol{\tau}), (\mathbf{s}, \boldsymbol{\tau}) &\in I \times I, \quad i = 0, 1, \dots, \end{aligned} \tag{4.27}$$

where ⋅ and ⋅ are determined by (3.4) and (3.9), respectively.

Similar to \*, we introduce the set \* \* when assumption (A6) is fulfilled:

 \* \*: = 0 <sup>⋅</sup> <sup>=</sup> <sup>1</sup> − ℎ, 1 : the admissible control 0 <sup>⋅</sup> is sufficiently smooth or sufficiently smooth from the right at the points *θ* and ℎ ∪ <sup>0</sup>, 1 − ℎ : the admissible control 0 <sup>⋅</sup> is sufficiently smooth or sufficiently smooth

$$\{\text{from the right at the points }\theta\text{ and }\theta \pm h\}. \tag{4.28}$$

The following obvious properties hold: (1) \ \* \* is a finite set, and <sup>1</sup> \* \*, also \* \* ⊂ \*; (2) for every \* \* there exists a sufficiently small number > 0, such that , furthermore, (3) by (A5), (A6), (1.2), (1.3), and (2.5), the functions are continuous and sufficiently smooth or sufficiently smooth from the right at every point \* \*. These properties are important at the next reasoning and we call them the properties of the set \* \*.

Let us consider a variation <sup>=</sup> 0 , , 1 that in addition satisfies the following conditions as well:

$$\begin{aligned} \delta \delta\_0 p\left(t\right) &= 0, \; \delta q\left(t\right) = 0, \; t \in I\_1 \; \bigvee \theta, \; \theta + \varepsilon \; \big|, \\\delta \delta\_1 p\left(t\right) &= \dots = \delta\_k p\left(t\right) = 0, \; t \in I\_1 \; \bigvee \left(\theta, \theta + \varepsilon\right), \end{aligned} \tag{4.29}$$

where

Conditions for Optimality of Singular Controls in Dynamic Systems with Retarded Control http://dx.doi.org/10.5772/64225 209

$$\mathcal{S}\_i p\left(t\right) = \bigcup\_{\theta} \mathcal{S}\_{i-1} p\left(\tau\right) d\tau, \ t \in I\_1, \ i = 1, 2, \ldots, k, \ k \in \{1, 2, \ldots\}, \tag{4.30}$$

\* \*, 0, \* \* , \* \* = min 0, , (0, , were defined above).

According to (4.30), we have

$$\delta\_i \delta\_j p\left(t\right) = \bigcup\_{\theta} \frac{\left(t - \tau\right)^{i-1}}{\left(i - 1\right)!} \delta\_o p\left(\tau\right) d\tau, \theta \in I^{\*\*}, \ t \in I\_1, \ i = 1, 2, \dots, k, \ k \in \{1, 2, \dots\}. \tag{4.31}$$

The following statement is valid.

**Proposition 4.2**. Let assumptions (A1), (A5), (A6), and condition (2.6) be fulfilled. Furthermore, let the functions <sup>⋅</sup> , <sup>⋅</sup> , , <sup>⋅</sup> , , <sup>⋅</sup> , <sup>⋅</sup> and <sup>⋅</sup> , where , , = 0, 1, ..., be defined by (4.22)–(4.26), and the set \* \* be defined by (4.28). Then along the singular control 0 <sup>⋅</sup> , satisfying condition (2.9), and on the variations <sup>=</sup> 0 , , 1 satisfying (4.29) and (4.30), the following representation (-th order transformation, where 1, 2, ... ) is valid:

$$\Delta^2 S\left(\mu^0; \delta\mu\right) = \Delta\_1^2 S\left(\mu^0; \delta\_\lambda p, \delta q, \delta\_\mu \mathbf{x}, \boldsymbol{\varepsilon}\right) + \Delta\_2^2 S\left(\mu^0; \delta\_\mathbf{q} p, \dots, \delta\_\lambda p, \delta q, \boldsymbol{\varepsilon}\right). \tag{4.32}$$

Here

[ ]( ) [ ]() () [ ]() () [ ]() () [ ]( ) [ ]() () [ ]() () [ ]() ()

*P pq t G p t f t g p t H t P pq t H t t Ii P pq t G p t f t g p t H t P pq t H t t I*

*i i q i xq pq*

*i i q i xq pq*

 m

+ + = - ++

, , , : , , 0,1,...,

%% % <sup>=</sup> % - = % %% % % % <sup>Î</sup> (4.24)

mm

, , , , 0,1,...,

q

 q

2 ,

 t

<sup>1</sup> − ℎ, 1 : the admissible control 0 <sup>⋅</sup> is sufficiently smooth or

, furthermore, (3) by (A5), (A6), (1.2), (1.3), and (2.5), the

% (4.27)

mm

0, 1 − ℎ : the admissible

<sup>1</sup> \* \*, also \* \* ⊂ \*; (2)

(4.29)

± *h*} (4.28)

1 that in addition satisfies the follow-

(4.26)

 mm= - Î Î= % (4.25)

= - = Î=

, ,, : ,,

[ ]( ) [ ]( ) [ ]( ) [ ]( ) [ ]() { } , , , , 0,1,..., *T T Q t g tG t G t g t pp t Ii i ii ii*

[ ]( ) [ ]() () [ ]( ) [ ]( ) [ ]( ) ( [ ]( ) [ ]( ))

*T TT i i xx i i i i i <sup>d</sup> L t g tH tg t g tG t g tG t dt*

[ ]( ) [ ]() ( ) [ ]( )

+ Y δ = % %

*T T ii i*

 lt

 tt

from the right at the points and .

for every \* \* there exists a sufficiently small number > 0, such that

functions are continuous and sufficiently smooth or sufficiently smooth from the right at every point \* \*. These properties are important at the next reasoning and we call

> () () [ ) () () ( )

= =Î + == = Î +

0, 0, \ , , ... 0, \ , , *<sup>k</sup>*

qq e

> qq e

0 1 1 1

 d

 d

*pt qt t I pt pt t I*

[ ]() ( ) [ ]( )( ) , ,: ,

Similar to \*, we introduce the set \* \* when assumption (A6) is fulfilled:

t

where ⋅ and ⋅ are determined by (3.4) and (3.9), respectively.

sufficiently smooth from the right at the points *θ* and ℎ ∪

The following obvious properties hold: (1) \ \* \* is a finite set, and

=

*M pp s g p s s G p g ps s gp s I I i*

 m

1 0 1 0

> m

*T T*

+ +

0

m

m

m

\* \*: = 0 <sup>⋅</sup> <sup>=</sup>

them the properties of the set \* \*.

ing conditions as well:

where

*T T*

1 1

= Î Î= %

: , , , , 0,1,...,

*i i*

control 0 <sup>⋅</sup> is sufficiently smooth or sufficiently smooth

Let us consider a variation <sup>=</sup> 0 , ,

d

d

t

 m

208 Nonlinear Systems - Design, Analysis, Estimation and Control

[ ]() () { }

*L t H t pp t Ii* mm

 m

*T*

 m

$$\begin{split} \Delta\_{\mathbf{i}}^{2}S(\cdot) &= \delta\_{\mathbf{i}}\mathbf{x}^{\top}\left(\mathbf{t}\_{1}\right)\boldsymbol{\uprho}\_{\boldsymbol{\infty}}\left(\mathbf{x}^{\top}\left(\mathbf{t}\_{1}\right)\right)\boldsymbol{\updelta}\_{\mathbf{i}}\mathbf{x}\left(\mathbf{t}\_{1}\right) - \int\_{\boldsymbol{\theta}}^{\boldsymbol{\delta}} \boldsymbol{\delta}\_{\mathbf{i}}\mathbf{x}^{\top}\left(\mathbf{t}\right)H\_{\boldsymbol{\infty}}\left(\mathbf{t}\right)\boldsymbol{\updelta}\_{\mathbf{i}}\mathbf{x}\left(\mathbf{t}\right)d\mathbf{t} \\ &- 2\int\_{\boldsymbol{\theta}}^{\boldsymbol{\delta}\_{\mathbf{i}\mathbf{x}}} \left\{ \left[\int\_{\boldsymbol{\delta}} \boldsymbol{\delta}\_{\mathbf{i}}\mathbf{x}^{\top}\left(\mathbf{t}\right)G\_{\boldsymbol{\delta}}\left[\boldsymbol{\eta}\right]\left(\mathbf{t}\right) + \boldsymbol{\updelta}\_{\mathbf{i}}\mathbf{x}^{\top}\left(\mathbf{t}+\boldsymbol{h}\right)G\_{\boldsymbol{\delta}}\left[\tilde{\boldsymbol{\mu}}\right]\left(\mathbf{t}+\boldsymbol{h}\right)\right\}\delta\_{\mathbf{i}}\mathbf{p}\left(\mathbf{t}\right) \\ &+ \left[\int\_{\boldsymbol{\delta}} \boldsymbol{\delta}\_{\mathbf{i}}\mathbf{x}^{\top}\left(\mathbf{t}\right)H\_{\boldsymbol{\delta}\mathbf{q}}\left(\mathbf{t}\right) + \boldsymbol{\updelta}\_{\mathbf{i}}\mathbf{x}^{\top}\left(\mathbf{t}+\boldsymbol{h}\right)H\_{\boldsymbol{\delta}\mathbf{q}}\left(\mathbf{t}+\boldsymbol{h}\right)\right] \delta\mathbf{q}\left(\mathbf{t}\right) \right\}dt, \end{split} \tag{4.33}$$

$$\begin{split} \Delta\_{2}^{2}S(\cdot) &= \int\_{\boldsymbol{\theta}}^{\boldsymbol{\theta}\cdot\boldsymbol{\xi}} \Big[ \sum\_{i=0}^{k-1} \Big[ \mathcal{S}\_{i+1} \boldsymbol{p}^{\top} \Big( t \big) \Big( L\_{i+1} \big[ \boldsymbol{p} \big] \big( t \big) + L\_{i+1} \big[ \tilde{\boldsymbol{p}} \big] \big( t + h \big) \Big) \mathcal{S}\_{i+1} \boldsymbol{p} \big( t \big) \, dt \\ &+ 2 \mathcal{S}\_{i+1} \boldsymbol{p} \big( t \big) \Big( P\_{i+1} \big[ \boldsymbol{p}, \boldsymbol{q} \big] \big( t \big) + P\_{i+1} \big[ \tilde{\boldsymbol{p}}, \tilde{\boldsymbol{q}} \big] \big( t + h \big) \Big) \delta \boldsymbol{q} \big( t \big) \\ &+ \delta\_{i} \boldsymbol{p}^{\top} \big( t \big) \big( \boldsymbol{Q}\_{i} \big[ \boldsymbol{p} \big] \big( t \big) + \boldsymbol{Q}\_{i} \big[ \tilde{\boldsymbol{p}} \big] \big( t + h \big) \Big) \delta\_{i+1} \boldsymbol{p} \big( t \big) \Big] \\ &- 2 \delta\_{0} \boldsymbol{p}^{\top} \big( t \big) \big( P\_{0} \big[ \boldsymbol{p}, \boldsymbol{q} \big] \big( t \big) + P\_{0} \big[ \tilde{\boldsymbol{p}}, \tilde{\boldsymbol{q}} \big] \big( t + h \big) \Big) \delta \boldsymbol{q} \big( t \big) \\ &- \delta q^{\top} \big( t \big) \Big( H\_{q\varsigma} \big( t \big) + H\_{p,q} \big( t + h \big) \Big) \delta \boldsymbol{q} \big( t \big) \Big) \, dt, \end{split} \tag{4.34}$$

where \* \*, 0, \* \* (the number \* \* was defined above), , is the solution of the system

$$\begin{aligned} \delta\_{\boldsymbol{x}} \dot{\mathbf{x}}(t) &= f\_{\boldsymbol{x}}(t) \delta\_{\boldsymbol{x}} \mathbf{x}(t) + \mathbf{g}\_{\boldsymbol{\epsilon}} \left[ p \right](t) \delta\_{\boldsymbol{x}} p(t) + \mathbf{g}\_{\boldsymbol{\epsilon}} \left[ \tilde{p} \right](t) \delta\_{\boldsymbol{x}} p(t - h) \\ &+ f\_{\boldsymbol{q}}(t) \delta q(t) + f\_{\boldsymbol{q}}(t) \delta q(t - h), \boldsymbol{t} \in \left[ \theta, t\_{1} \right] \\ \delta\_{\boldsymbol{\epsilon}} \mathbf{x}(t) &= 0, \ \boldsymbol{t} \in \left[ t\_{0}, \theta \right], \ \delta\_{\boldsymbol{\epsilon}} p(t) = 0, \ \delta q(t) = 0, \ \boldsymbol{t} \in \left[ t\_{0} - h, \theta \right), \ k \in \{1, 2, ...\}, \end{aligned} \tag{4.35}$$

**Proof**. We carry out the proof of Proposition 4.2 by induction. For = 1, Proposition 4.2 was completely proved at item 4 (see Proposition 4.1). Assume that Proposition 4.2 is valid for all the cases to 1 inclusively, 2 . We prove the validity of representation (4.32) for the case . Let the variation <sup>=</sup> 0 , , 1 satisfies the conditions (4.29) and (4.30). Then by assumption the following representation is valid:

$$\delta^2 S(\mu^0; \delta \mu) = \Lambda\_1^2 S(\mu^0; \delta\_{k-1} p, \delta q, \delta\_{k-1} \ge \varepsilon) + \Lambda\_2^2 S(\mu^0; \delta\_0 p, \dots, \delta\_{k-1} p, \delta q, \varepsilon). \tag{4.36}$$

Here

( ) () () ( ) ( ) () () () { ( ) [ ]() ( ) [ ]( ) ( ) () () ( ) ( ) ()} 1 2 0 0 1 1 1 1 1 1 11 1 1 11 1 1 1 1 1 ; ,, , 2 , *t T T k k k xx k k xx k T T kk k k k T T k xq k xq S u p q x x t x t x t x t H t x t dt x tG p t x t hG p t h pt x t H t x t h H t h q t dt* q q e q d dd e d j d d d dd d dd d -- - -- - + -- - - - - - D= - - é ù ++ + ë û + +++ é ù ë û ò ò % % (4.37)

$$\begin{split} \Delta\_{2}^{\Delta}S(\boldsymbol{u}^{0};\delta\_{0}\boldsymbol{p},\ldots,\delta\_{k-1}\boldsymbol{p},\delta\boldsymbol{q},\boldsymbol{\varepsilon}) &= \\ &\int\_{\boldsymbol{\varrho}}^{\boldsymbol{\varrho}\_{+2}} \Big[ \sum\_{l=0}^{k-2} \Big[ \delta\_{l+1} \boldsymbol{p}^{\intercal}(t) \Big(L\_{l+1} \Big[\boldsymbol{p}\big](t) + L\_{l+1} \Big[\tilde{\boldsymbol{p}}\big](t+h) \Big) \delta\_{l+1} \boldsymbol{p}(t) \Big] \\ &+ 2\delta\_{l+1} \boldsymbol{p}^{\intercal}(t) \Big( P\_{i+1} \Big[\boldsymbol{p},\boldsymbol{q}\big](t) + P\_{i+1} \Big[\tilde{\boldsymbol{p}},\tilde{\boldsymbol{q}}\big](t+h) \Big) \delta\boldsymbol{q}(t) \Big. \\ &+ \delta\_{i} \boldsymbol{p}^{\intercal}(t) \Big( Q\_{i} \big[\boldsymbol{p}\big](t) + Q\_{i} \Big[\tilde{\boldsymbol{p}}\big](t+h) \Big) \delta\_{i+1} \boldsymbol{p}(t) \Big] \Big] - \\ &2\delta\_{0} \boldsymbol{p}^{\intercal}(t) \Big(P\_{0} \big[\boldsymbol{p},\boldsymbol{q}\big](t) + P\_{0} \big[\tilde{\boldsymbol{p}},\tilde{\boldsymbol{q}}\big](t+h) \Big) \delta\boldsymbol{q}(t) \\ &- \delta q^{\intercal}(t) \Big( H\_{\mathcal{H}}(t) + H\_{\mathcal{H}}(t+h) \Big) \delta\boldsymbol{q}(t) \Big) \Big] dt, \end{split} \tag{4.38}$$

where <sup>⋅</sup> , , <sup>⋅</sup> , , <sup>⋅</sup> , <sup>⋅</sup> , <sup>⋅</sup> , , , = 0, 1, ... are defined by (4.23)-(4.26), and 1 , is the solution of the system:

Conditions for Optimality of Singular Controls in Dynamic Systems with Retarded Control http://dx.doi.org/10.5772/64225 211

$$\begin{aligned} \boldsymbol{\delta}\_{k-1} \dot{\mathbf{x}}(t) &= f\_x(t) \boldsymbol{\delta}\_{k-1} \mathbf{x}(t) + \mathbf{g}\_{k-1} [p](t) \, \boldsymbol{\delta}\_{k-1} p(t) + \mathbf{g}\_{k-1} [\tilde{p}](t) \, \boldsymbol{\delta}\_{k-1} p(t - h) \\ &+ f\_q(t) \, \boldsymbol{\delta} q(t) + f\_q(t) \, \boldsymbol{\delta} q(t - h), \\ \boldsymbol{\delta}\_{k-1} \mathbf{x}(t) &= \mathbf{0}, \, t \in [t\_0, \theta], \, \boldsymbol{\delta}\_{k-1} p(t) = \mathbf{0}, \, \delta q(t) = \mathbf{0}, \, t \in [t\_0 - h, \theta), \, k \ge 2. \end{aligned} \tag{4.39}$$

Apply the modified variant of variations transformations method [13] to the system for <sup>1</sup> , and representation (4.36). According to the technique of the previous item (see item 4.1), we introduce a new variation in the following way:

$$\mathcal{S}\_k \mathbf{x}(t) = \mathcal{S}\_{k-l} \mathbf{x}(t) - \mathbf{g}\_{k-l} \begin{bmatrix} p \\ \end{bmatrix} (t) \, \mathcal{S}\_k p(t) - \mathbf{g}\_{k-l} \begin{bmatrix} \vec{p} \\ \end{bmatrix} (t) \, \mathcal{S}\_k p(t-h), t \in I. \tag{4.40}$$

According to (4.22), (4.30), (4.31), and (4.39) from (4.40) by direct differentiation, we get the system (4.35) for , . Furthermore, as \* \*, then by (4.40) we get <sup>1</sup> = 1 <sup>1</sup> . Taking into account this equality and by (4.29), (4.30), and (4.40) in (4.37), let us transform the representation (4.36) into new variables <sup>⋅</sup> , <sup>⋅</sup> , <sup>⋅</sup> . Then,

$$\begin{split} \delta^{2}S\Big(\boldsymbol{u}^{0};\delta\boldsymbol{u}\Big) &= \delta\_{\boldsymbol{k}}\boldsymbol{x}^{\boldsymbol{r}}\Big(t\_{1}\big)\boldsymbol{\rho}\_{\boldsymbol{xx}}\Big(\boldsymbol{x}^{0}\big(t\_{1}\big)\big)\delta\_{\boldsymbol{k}}\boldsymbol{x}\big(t\_{1}\big) \\ &- \Delta\_{\boldsymbol{l}1} - \Delta\_{\boldsymbol{l}2} - \Delta\_{\boldsymbol{l}3} + \Delta\_{\boldsymbol{z}}^{2}S\Big(\boldsymbol{u}^{0};\delta\_{\boldsymbol{0}}\boldsymbol{p},...,\delta\_{\boldsymbol{k}-1}\boldsymbol{p},\boldsymbol{\delta}\boldsymbol{q},\boldsymbol{x}\big), \end{split} \tag{4.41}$$

where 2 2 <sup>⋅</sup> is determined by formula (4.38) as well as 1, = 1, 2, 3 by (4.22), (4.29), (4.30), (4.35), (2.6) are calculated in the following way:

$$\begin{split} \Delta\_{\scriptscriptstyle{\rm I}} &= \mathop{\rm l}\_{\scriptscriptstyle{\rm I}} \mathop{\rm s}\_{\scriptscriptstyle{\rm I}}^{\scriptscriptstyle{\rm I}} \{t\} H\_{\scriptscriptstyle{\rm x}} \{t\} \delta\_{\scriptscriptstyle{\rm I}} \mathbf{x}(t) dt \\ &+ 2 \mathop{\rm l}\_{\scriptscriptstyle{\rm I}}^{\boldsymbol{\rm I} + \boldsymbol{t}} \left[ \boldsymbol{\delta}\_{\scriptscriptstyle{\rm I}} \mathbf{x}^{\intercal} \{t\} H\_{\scriptscriptstyle{\rm x}} \{t\} \mathbf{g}\_{\scriptscriptstyle{\rm I} - 1} \right] \boldsymbol{p} \} (t) + \boldsymbol{\delta}\_{\scriptscriptstyle{\rm I}} \mathbf{x}^{\intercal} \{t + h\} H\_{\scriptscriptstyle{\rm x}} (t + h) \mathbf{g}\_{\scriptscriptstyle{\rm I} - 1} \left[ \widetilde{\boldsymbol{p}} \right] (t + h) \right] \delta\_{\scriptscriptstyle{\rm I}} p (t) \\ &+ \mathop{\rm l}\_{\scriptscriptstyle{\rm I}}^{\boldsymbol{\rm I} + \boldsymbol{t}} \boldsymbol{\delta}\_{\scriptscriptstyle{\rm I}} p^{\scriptscriptstyle \rm T} \{t\} \left[ \boldsymbol{\varrho}\_{\scriptscriptstyle \rm I}^{\Gamma} \left[ \boldsymbol{p} \right] (t) H\_{\scriptscriptstyle{\rm x}} (t) \mathbf{g}\_{\scriptscriptstyle \rm$$

$$\begin{split} \boldsymbol{\Lambda}\_{12} &= \mathbf{2} \int\_{\boldsymbol{\theta}}^{\boldsymbol{\theta} \cdot \boldsymbol{\varepsilon}} \Big[ \left[ \boldsymbol{\delta}\_{k} \mathbf{x}^{\top} \left( \boldsymbol{t} \right) + \boldsymbol{\delta}\_{k} \boldsymbol{p}^{\top} \left( \mathbf{t} \right) \mathbf{g}\_{k-1}^{\top} \left[ \boldsymbol{p} \right] \big( \mathbf{t} \big) \mathbf{G}\_{k-1} \big[ \boldsymbol{p} \big] \big( \boldsymbol{t} \big) \\ &+ \Big[ \boldsymbol{\delta}\_{k} \mathbf{x}^{\top} \left( \mathbf{t} + \boldsymbol{h} \right) + \boldsymbol{\delta}\_{k} \boldsymbol{p}^{\top} \left( \mathbf{t} \big) \mathbf{g}\_{k-1}^{\top} \left[ \tilde{\boldsymbol{p}} \big] \big( \mathbf{t} + \boldsymbol{h} \big) \big] \mathbf{G}\_{k-1} \big[ \tilde{\boldsymbol{p}} \big] \big( \mathbf{t} + \boldsymbol{h} \big) \big| \boldsymbol{\delta}\_{k-1} \mathbf{p} \big( \mathbf{t} \big) \mathbf{d} \mathbf{t} =: \boldsymbol{\Lambda}\_{12}^{\text{\*}} + \boldsymbol{\Delta}\_{12}^{\text{\*\*}}, \end{split} \tag{4.43} \end{split} \tag{4.43}$$

where

() () () [ ]() () [ ]() ( )

=+ + -

 d

*xt f t xt g p t pt g p t pt h*

*xt t t pt qt t t h k*

*k xk k k k k*

& %

0 0

 d

, ,

 q

= Î = = Î- Î

( ) [ ] () () [ ){} 1

() ( ) ( ) 20 20 2 0 1 1 1 20 1 ; ; , , , ; ,..., , , . *k k <sup>k</sup>*

( ) () () ( ) ( ) () () ()

*S u p q x x t x t x t x t H t x t dt*


dd

*k k k xx k k xx k*

1 1 1 1 1 1 11 1 1

 j

1 1


2

*<sup>k</sup> <sup>T</sup>*

d

0

*i T*

=

d

+ -

ì

ò å

d

d

d

(4.23)-(4.26), and 1 , is the solution of the system:

q e

q

*T*

*T T*

dd

*T T k xq k xq*

 e

*T T*

 d

*T T*

+ +++ é ù ë û

0, , , 0, 0, , , 1,2,... ,

**Proof**. We carry out the proof of Proposition 4.2 by induction. For = 1, Proposition 4.2 was completely proved at item 4 (see Proposition 4.1). Assume that Proposition 4.2 is valid for all the cases to 1 inclusively, 2 . We prove the validity of representation (4.32) for the case . Let the variation <sup>=</sup> 0 , , 1 satisfies the conditions (4.29) and (4.30).

 d

> d

{ ( ) [ ]() ( ) [ ]( ) ( )


*kk k k k*

11 1 1 1

*x tG p t x t hG p t h pt*

( )( [ ]( ) [ ]( )) ( )

% %

%

*p t L p t L p t h pt*

1

+

 d

,

 d

( )( [ ]( ) [ ]( )) ( )

 d

11 1 1

++ + +

*ii i i*

*p t P pq t P pq t h qt*

+ +

% %

( )( [ ]( ) [ ]( )) ( ) ( )( [ ]( ) [ ]( )) ( )

%

*p t Q p t Q p t h pt p t P pq t P pq t h qt*

+ ++ -ù

<sup>í</sup> <sup>é</sup> + + <sup>ë</sup> <sup>î</sup>

() () ( ) ( ) ( )}

% %

*q t H t H t h q t dt*

11 1

*ii i i*

+ ++

2 ,,

++ +

*ii i*

*qq qq*

where <sup>⋅</sup> , , <sup>⋅</sup> , , <sup>⋅</sup> , <sup>⋅</sup> , <sup>⋅</sup> , , , = 0, 1, ... are defined by

00 0


2 ,,

1

q

ò

 d

*t*

() () ( ) ( ) ()}

*x t H t x t h H t h q t dt*


*Su u Su p q x Su p p q* = D - - + D - (4.36)

 d  de

> d

 d ,

 d

 d

(4.38)

û

%% (4.37)

 d

q

(4.35)

() () () ( ) [ ]

*f t qt f t qt h t t*

qd

Then by assumption the following representation is valid:

 d  dd

*q q k k*

210 Nonlinear Systems - Design, Analysis, Estimation and Control

dd

+ + -Î

%

dd

d

d

Here

 d

2 0 0

( )

 d

D =

*Su p p q*

; ,..., , ,

*k*


 de

20 1

d

 e d

2

q e

D= -

+

q

ò

; ,, ,

 dd

d

2 0

( ) [ ]() ( ) [ ]( ) () ( ) [] []( ) [ ]( ) [ ]( ) () { () () [ ]( ) ( ) ( ) () [ ]() ()} ( ) ( ) \* 12 1 1 1 \*\* 12 1 1 1 1 1 13 1 1 : 2 , : 2 , 2 2 *T kk k k k T T T k kk k k k T TT k k k xq T T T k k k xq T k x x t G p t x t h G p t h p t dt p t g p G p t g p t h G p t h p t dt x t p tg p t H t x t h p t g p t h H t h q t dt x tH* q e q q e q q e q dd d d d d d d d d d + - - - + -- - - - + - - D = é ù ++ + ë û D = é ù ++ + ë û D = é ù <sup>+</sup> ë û + ++ é ù + + ë û = ò ò ò % % % % % () ( ) ( ) () 2 . ( ) 1 1 [ ]() () [ ]( ) ( ) () *T q k xq T T T k k xq k xq t x t h H t h q t dt p t g p t H t g p t h H t h q t dt* q e q q e q d d d d + + - é ù ++ + ë û <sup>+</sup> é ù + ++ ë û ò ò % % % (4.44)

Taking into account (A5), (A6), (4.29), (4.30), (4.35), and the properties of the set \* \*, let us calculate 12\* , 12\* \*. Then, applying the method of integration by parts, we have

$$\begin{split} \boldsymbol{\Delta}\_{12}^{\*} &= 2\int\_{0}^{\theta\_{\mathsf{s}}} \Big[ \Big[ \Big( \int\_{\mathbb{R}} \boldsymbol{\delta}\_{k} \boldsymbol{x}^{\top} \Big( \boldsymbol{t} \Big) \Big( \int\_{\mathbb{R}} \boldsymbol{f}\_{\boldsymbol{x}} \Big( \boldsymbol{t} \Big) \boldsymbol{G}\_{k-1} \Big[ \Big[ \boldsymbol{p} \Big] \Big( \boldsymbol{t} \Big) + \frac{d}{dt} \Big( \boldsymbol{G}\_{k-1} \Big[ \Big[ \boldsymbol{p} \Big] \Big) \Big) \Big) \Big. \\ & \quad + \delta\_{k} \boldsymbol{x}^{\top} \Big( \boldsymbol{t} + \boldsymbol{h} \Big) \Big( \int\_{\mathbb{R}} \boldsymbol{f}\_{\boldsymbol{x}}^{\top} \Big( \boldsymbol{t} + \boldsymbol{h} \Big) \boldsymbol{G}\_{k-1} \Big[ \Big[ \Big( \boldsymbol{t} \Big) \Big] + \frac{d}{dt} \Big( \boldsymbol{G}\_{k-1} \Big[ \Big[ \Big( \boldsymbol{t} \Big) \Big) \Big) \Big) \Big] \Big} \Big\Big] \delta\_{k} \boldsymbol{p} \Big( \boldsymbol{t} \Big) \\ & \quad + \delta\_{k} \boldsymbol{p}^{\top} \Big( \boldsymbol{t} \Big) \Big[ \Big{\Big} \boldsymbol{g}\_{\boldsymbol{x}}^{\top} \Big[ \Big{(} \boldsymbol{p} \Big] \Big{(} \Big) \Big{(} \boldsymbol{t} \Big) + \operatorname{\boldsymbol{q}}\_{k-1}^{\top} \Big[ \Big{(} \bar{\boldsymbol{p}} \Big{(} \Big) \Big{(} \boldsymbol{t} + \boldsymbol{h} \Big) \Big{\Big{)}} \Big{\Big{.}} \Big{(} \boldsymbol{p} \Big{(}$$

At first, we substitute the last expression 12\* , 12\* \* in (4.43), and then (4.42)–(4.44) in (4.41). Then by (4.23)–(4.26), (4.33), (4.34), and (4.38), it is easy to get representation (4.32). Consequently, we get the proof for . This completes the proof of Proposition 4.2.

#### **5. Optimality conditions**

Based on Propositions 3.1, 4.1, and 4.2, we prove the following theorem.

**Theorem 5.1**. Let conditions (A1), (A5), and (A6) be fulfilled, and the matrix functions , <sup>⋅</sup> , , <sup>⋅</sup> , <sup>⋅</sup> , <sup>⋅</sup> , , <sup>⋅</sup> , , , = 0, 1, ... be defined as in (4.24)–(4.27). Let also the set \* \* be defined as in (4.28) and along the singular (in the classical sense) control 0 <sup>⋅</sup> the following equalities be fulfilled:

$$\mathbb{E}L\_i[p](t) + \chi(t)L\_i[\tilde{p}](t+h) = 0, \ \forall t \in I^{\*\*}, i = 0, 1, \ldots, k, \ k \in \{0, 1, \ldots\}, \tag{5.1}$$

where ⋅ is the characteristic function of the set 0, 1 − ℎ .

Then for the optimality of the admissible control 0 <sup>⋅</sup> , it is necessary that the relations

$$P\_i[\underline{p},q](\theta) + \chi(t)P\_i[\tilde{p},\tilde{q}](\theta+h) = 0, \; i = 0,1,\dots,k,\tag{5.2}$$

$$\begin{aligned} \xi^{r} \left\{ M\_{i} \left[ p, p \right] (\theta, \theta) + 2 \chi \left( t \right) M\_{i} \left[ p, \tilde{p} \right] (\theta, \theta + h) \right\} \\ + \chi \left( \theta \right) M\_{i} \left[ \tilde{p}, \tilde{p} \right] (\theta + h, \theta + h) \right\} \xi \le 0, \ i = 0, 1, \ldots, k, \end{aligned} \tag{5.3}$$

$$\mathbb{E}\left[Q\right](\theta) + \mathbb{X}(\theta)\mathbb{Q}[\tilde{p}](\theta + h) = 0, \ i = 0, 1, \ldots, k,\tag{5.4}$$

$$\begin{split} \mathcal{L}\_{k+l}(\boldsymbol{\theta},\boldsymbol{\xi},\boldsymbol{\eta}) & \coloneqq \xi^{T} (\mathcal{L}\_{k+l}[p](\boldsymbol{\theta}) + \mathcal{\chi}(\boldsymbol{\theta})\mathcal{L}\_{k+l}[\widetilde{p}](\boldsymbol{\theta}+h))\xi + \\ & + 2\xi^{T} (P\_{k+l}[p,q](\boldsymbol{\theta}) + \mathcal{\chi}(\boldsymbol{\theta})P\_{k+l}[\widetilde{p},\widetilde{q}](\boldsymbol{\theta}+h))\eta - \\ & \qquad - \eta^{T} (H\_{qq}(\boldsymbol{\theta}) + \mathcal{\chi}(\boldsymbol{\theta})H\_{\widetilde{q}\widetilde{q}}(\boldsymbol{\theta}+h))\eta \geq 0, \end{split} \tag{5.5}$$

be fulfilled for all \* \*, <sup>0</sup> and 1.

**Proof**. Let 0 <sup>⋅</sup> be an optimal control. We will prove the theorem by induction. Let = 0, that is, = 0. Then, according to (4.24) and (2.10) we get the proof of optimality condition (5.2) for = 0. The proof of optimality condition (5.3) for = 0 directly follows from (3.11) allowing for (2.1) (see Proposition 3.1). Now, based on Proposition 4.1 prove the optimality conditions (5.4) and (5.5) for = 0.

We first prove the validity of (5.4) for *k*=0.

Suppose that

( ) [ ]() ( ) [ ]( ) ()


*x t G p t x t h G p t h p t dt*

: 2 ,

%

*k kk k k k*

12 1 1 1 1 1

() ( ) ( ) ()

*t x t h H t h q t dt*

%

*p t g p t H t g p t h H t h q t dt*

() () [ ]( ) ( ) [ ]( )

*<sup>d</sup> x t f tG p t G p t dt*

+ + + ++ + ç ÷ú è øû

<sup>+</sup> é ù ++ + ë û

*k xk k k*

( ) [ ]() () [ ]( ) ( ) ()}

*p t G p t f t G p t h f t h q t dt*

*kk k*

é ù + + ë û

quently, we get the proof for . This completes the proof of Proposition 4.2.

Based on Propositions 3.1, 4.1, and 4.2, we prove the following theorem.

*kkk k k k*

()() [ ]( ) ( [ ]( )) ( )

æ öù


*<sup>d</sup> x t h f t hG p t h G p t h pt dt*


*p t g p tG p t g p t hG p t h pt*

%

[ ]( ) [ ]( ) ()

%

é ù - + ++ ê ú

*Q p t Q p t h p t dt*


1 1

% %

( ) [ ]( ) [ ]( ) [ ]( ) [ ]( ) ()

1 1

2 . ( ) 1 1 [ ]() () [ ]( ) ( ) ()

D = é ù ++ + ë û

: 2 ,

*kk k k k*

{ () () [ ]( ) ( )

*k k k xq*

*x t p tg p t H t*

%

*T TT*

 d

D = é ù <sup>+</sup> ë û

dd

*k k k xq*

d

*T T T*

12 1 1

*T T T*

*T T T*

*T T*

1 1


*k k qk q*

<sup>+</sup> é ù + ++ ë û

1 1


*T T T*

+ ++ é ù + + ë û

*T T T*

D = é ù ++ + ë û

1

*T q k xq*

*k k xq k xq*

<sup>+</sup> é ù + ++ ë û

*k xk k*

ìïé æ ö D =- <sup>í</sup> <sup>+</sup> <sup>ê</sup> ç ÷ ïîë è ø



é ù ++ + ë û

( )

*x tH*

*T k x*

13 1

*T T T*

 d

( )

At first, we substitute the last expression 12

*T k*

*p t*

*T T*

d

*T*

212 Nonlinear Systems - Design, Analysis, Estimation and Control

\*

\*\*

2

q e

+

q

ò

=

calculate 12

\*

\*\*

D =-

12 1

q e

+

q

ò

q

ò

d

**5. Optimality conditions**

q e

+

d


d

q e

+

q

ò

d

2

q e

q

ò

d

\* , 12

2

d

d

d

q e

+

q

ò

q e

+

q

ò

+

q

ò

+

d

d

q e

( ) () [ ]() ()} ( )


*x t h p t g p t h H t h q t dt*

12 1 1 1

( ) [] []( ) [ ]( ) [ ]( ) ()


 d

% %

Taking into account (A5), (A6), (4.29), (4.30), (4.35), and the properties of the set \* \*, let us

\* \*. Then, applying the method of integration by parts, we have

%

d

% %

 d

> d

> > ;

\* \* in (4.43), and then (4.42)–(4.44) in (4.41).

 d

 d

 d

 d

11 1 1 .

% %

%

d

( ) ( [ ]( ) [ ]( )) ( [ ]( ) [ ]( )) ( )

*d d p t g p t G p t g p t h G p t h p t dt dt dt*

ë û

*k kk k k k*


\* , 12

Then by (4.23)–(4.26), (4.33), (4.34), and (4.38), it is easy to get representation (4.32). Conse-

**Theorem 5.1**. Let conditions (A1), (A5), and (A6) be fulfilled, and the matrix functions , <sup>⋅</sup> , , <sup>⋅</sup> , <sup>⋅</sup> , <sup>⋅</sup> , , <sup>⋅</sup> , , , = 0, 1, ... be defined as in

% %

 d

(4.44)

*p t g p G p t g p t h G p t h p t dt*

$$\left(\mathcal{S}\_0 p\_m \left(t\right) = 0, \,\forall t \in I\_1, \forall m \in \{1, 2, \ldots, r\_0\} \backslash \{i, j\}\right) \tag{5.6}$$

$$\begin{aligned} \delta\_0 p\_i \left( t \right) &= \begin{cases} \alpha I\_1 \Big( \frac{2 \left( t - \theta \right)}{\varepsilon} - 1 \Big), & t \in \left[ \theta, \theta + \varepsilon \right), \ \varepsilon \in \left( 0, \varepsilon^{\*\*} \right), \\ 0, & t \in I\_1 \big( \left[ \theta, \theta + \varepsilon \right), \end{cases} \\ \delta\_0 p\_{\big{\prime}} \left( t \right) &= \begin{cases} \beta I\_2 \Big( \frac{2 \left( t - \theta \right)}{\varepsilon} - 1 \Big), & t \in \left[ \theta, \theta + \varepsilon \right), \ \varepsilon \in \left( 0, \varepsilon^{\*\*} \right), \\ 0, & t \in I\_1 \big{\prime} \left[ \theta, \theta + \varepsilon \right), \end{cases} \end{aligned}$$

$$
\delta q\left(t\right) = 0, \ t \in I\_1
$$

where , are arbitrary fixed points of the set 1, 2, ..., 0 and 0 <sup>⋅</sup> is the -th coordinate of the vector 0 <sup>⋅</sup> ; , and \* \* are arbitrary fixed points, the functions <sup>1</sup> = , 2 <sup>=</sup> <sup>3</sup> 2 <sup>2</sup> <sup>−</sup> <sup>1</sup> 2, −1, 1 are the Legendre polynomials.

It is clear that the variation , defined by (5.6) satisfies the condition (4.2) and, according to (5.6) the function 1 , 1, defined by (4.3) is of order , and the solution 1 , of the system (4.7), (4.8) is of order 2. Also, according to (4.15) it is easy to see that for every the matrix 0 + 0 + ℎ is skew-symmetric. Therefore, by Proposition 4.1 and condition (2.6), considering (2.1), (4.3), (4.17), (4.18), and the properties of the set \* \*, along the singular optimal control 0 <sup>⋅</sup> , we have

( ) ( ) ( ) [ ]() () [ ]( ) () ( ) ( ) () () () ( ) () () () ( ) ( ) ( ) ( ) ( ) () () ( ) ( ) ( ) ( ) ( ) () ( ) 2 0 2 00 0 1 0 0 2 0 1 0 1 2 1 0 0 2 1 2 1 1 2 0 0 2 \*\* ; 4 0, 0, <sup>30</sup> *T ij ijji ji s ij ji ij ji S u u p t Q p t t Q p t h p t dt o q t p t p t q t p t p t dt o q q l s l d ds o qq o* q e q q e q dd d c de dd d d e e ab q q tt e e ab q q e ee + + - - × = é ù + ++ ë û = ++ é ù ë û =- + é ù ë û = - é ù - + ³ "Î ë û ò ò ò ò %

where <sup>0</sup> , <sup>0</sup> are the elements of the matrix 0 + 0 + ℎ .

Then, we conclude from the arbitrariness of , , \* \* and , 1, 2, ..., 0 , that the skew-symmetric matrix 0 + 0 + ℎ is also symmetric. Consequently, for every \* \* we have 0 + 0 + ℎ = 0. This completes the proof of the optimality condition (5.4) for = 0.

To prove statement (5.5) for = 0, under the conditions (4.2) and (4.3), we write down the vector components of the variation <sup>⋅</sup> <sup>=</sup> 0 <sup>⋅</sup> , <sup>⋅</sup> in the following form:

$$\begin{split} \delta\_{0}p\left(t\right) &= \begin{cases} \frac{\varepsilon}{\varepsilon}l\_{1}\left(\frac{2\left(t-\theta\right)}{\varepsilon}-1\right), t \in \left[\theta, \theta+\varepsilon\right), \\ 0 & t \in I\_{1}\left(\left\{\theta, \theta+\varepsilon\right\}, \varepsilon \in \left(0, \varepsilon^{\star\star}\right), \end{cases} \\ \delta q\left(t\right) &= \begin{cases} \eta\left|l\_{1}\left(\frac{2\left(s-\theta\right)}{\varepsilon}-1\right)ds, t \in \left[\theta, \theta+\varepsilon\right), \\ 0, & t \in I\_{1}\left(\left\{\theta, \theta+\varepsilon\right\}, \varepsilon \in \left(0, \varepsilon^{\star\star}\right), \end{cases} \end{split} \tag{5.7}$$

where <sup>1</sup> = , −1, 1 is a Legendre polynomial, 0, 1, \* \* are arbitrary fixed points.

According to (4.2), (4.3), (4.7), (4.8), and (5.7), it is easy to prove that

( )

0

d

214 Nonlinear Systems - Design, Analysis, Estimation and Control

*i*

*p t*

0

d

<sup>1</sup> = , 2 <sup>=</sup> <sup>3</sup>

2 <sup>2</sup> <sup>−</sup> <sup>1</sup> 1

a 2

*t l t*

e

q

q

2

*t l t*

e

d

2

and the solution 1 , of the system (4.7), (4.8) is of order

*T*

2 1

q e

+

q

ò

dd

 d

*ij ji*

ë û

 q

*ij ji*

q

*qq o*

 q

=- + é ù

 q

= - é ù - + ³ "Î ë û

2

e ab

4

<sup>0</sup> ,

where

q

ò

e ab

q e

+

;

dd

properties of the set \* \*, along the singular optimal control 0 <sup>⋅</sup> , we have

b

( )

*j*

*p t*

( ) [ ) ( )

qq e e

1

0, \, ,

<sup>ì</sup> æ- ö <sup>ï</sup> ç ÷ -Î+ Î <sup>=</sup> <sup>í</sup> è ø <sup>ï</sup> Î + <sup>î</sup>

*t I*

0, \, ,

<sup>ì</sup> æ- ö <sup>ï</sup> ç ÷ -Î+ Î <sup>=</sup> <sup>í</sup> è ø <sup>ï</sup> Î + <sup>î</sup>

*t I*

( ) <sup>1</sup>

*qt t I* = Î 0,

where , are arbitrary fixed points of the set 1, 2, ..., 0 and 0 <sup>⋅</sup> is the -th coordinate of the vector 0 <sup>⋅</sup> ; , and \* \* are arbitrary fixed points, the functions

2, −1, 1 are the Legendre polynomials.

It is clear that the variation , defined by (5.6) satisfies the condition (4.2) and, according to (5.6) the function 1 , 1, defined by (4.3) is of order ,

it is easy to see that for every the matrix 0 + 0 + ℎ is skew-symmetric. Therefore, by Proposition 4.1 and condition (2.6), considering (2.1), (4.3), (4.17), (4.18), and the

( ) ( ) ( ) [ ]() () [ ]( ) () ( )

tt

c

× = é ù + ++ ë û

 d

 ee

<sup>0</sup> are the elements of the matrix 0 + 0 + ℎ .

Then, we conclude from the arbitrariness of , , \* \* and , 1, 2, ..., 0 , that the skew-symmetric matrix 0 + 0 + ℎ is also symmetric. Consequently, for

 e

%

 e

de

2 0 2 00 0 1

*S u u p t Q p t t Q p t h p t dt o*

( ) () () () ( ) () () () ( )

 d

0 0 2 0 1 0 1

*s*

( ) ( ) ( ) ( ) () () ( )

0 0 2 1 2 1 1


ò ò

*q q l s l d ds o*

*q t p t p t q t p t p t dt o*

( ) ( ) ( ) ( ) () ( )

 e

0 0 2 \*\*

0, 0, <sup>30</sup>

*ij ijji ji*

= ++ é ù ë û

1

( ) [ ) ( )

qq e e

qq e

[ )

1 , , , 0, ,

\*\*

 e

\*\*

2. Also, according to (4.15)

 e

[ )

qq e

1 , , , 0, ,

$$
\delta\_\restriction p\left(t\right) \sim \varepsilon, \delta\!\!\!q\left(t\right) \sim \varepsilon, \; t \in I\_\restriction, \; \delta\!\!\!\!\!x\left(t\right) \sim \varepsilon^2, \; t \in I.
$$

In view of the last relations and above proved condition (5.4) (for the case *k* = 0) taking into account the properties of the set \* \* and the relations (2.1), (4.3), (4.17), (4.18), and (5.7) from (4.16), we obtain the following relation along the singular optimal control 0 ,1:

( ) ( ) { ( ) [ ]() () [ ]( ) ( ) ( ) [ ]() () [ ]( ) () () () () ( ) ()} ( ) { [ ]() () [ ]( ) [ ]() () [ ]( ) () () ( ) } ( ) 2 0 11 1 1 1 1 1 3 3 11 1 1 1 1 ; 2, , 2, , <sup>8</sup> *T T T qq qq T T t T qq qq Su u p t L p t tL p t h pt p t P pq t t P pq t h qt q t H t t H t h q t dt o L p L p h P pq P pq h H H h ld* q e q dd d c d d cd d c de e x q cq q x x q cq q h h q cq q h t t + - × = é ù + + ë û + ++ é ù ë û - ++ + é ù ë û = + ++ + + é ù é ù ë û ë û æ ö -+ + é ù ç ÷ ë û <sup>ç</sup> è ø ò ò % % % % % % % % % % () ( ) <sup>2</sup> <sup>1</sup> 3 \* 1 *dt o* e ee 0, 0, . - + ³ "Î ò <sup>÷</sup>

Hence, taking into account the arbitrariness of \*, <sup>0</sup> and 1, we easily get the validity of the optimality condition (5.5) for = 0.

Now suppose that all the statements of Theorem 5.1 are valid for = 1, 2, ..., 1 2 as well. Prove statements (5.2)–(5.5), for = . By assumption, the inequality *<sup>k</sup>* , , 0 (see (5.5) for the case *k-1*) is valid for all \* \*, <sup>0</sup> and 1. Hence, taking into account (5.1), we have

$$\begin{aligned} &2\xi\Big(P\_{k}\Big[p,q\Big]\big(\theta\big)+\mathcal{X}\big(\theta\big)P\_{k}\Big[\tilde{p},\tilde{q}\Big]\big(\theta+h\Big)\Big)\eta \\ &-\eta^{r}\Big(H\_{\neq\overline{q}}\Big(\theta\big)+\mathcal{X}\big(\theta\big)H\_{\overline{q}\overline{q}}\Big(\theta+h\Big)\Big)\eta \geq 0, \forall \theta\in I^{\text{\textquotedblleft}}, \forall \tilde{\xi}\in R^{\text{\textquotedblleft}}, \forall \eta\in R^{\text{\textquotedblleft}}, \end{aligned}$$

From this inequality, we easily get that , + , + ℎ = 0, that is, we get the validity of optimality condition (5.2) for = .

Now, prove the validity of condition (5.3) for = . In formula (4.32), we put

$$\delta\_0 p\left(t\right) = \begin{cases} \left[\xi I\_k\left(\frac{2\left(t-\theta\right)}{\sigma} - 1\right)\right], & t \in \left[\theta, \theta + \varepsilon\right), \\ 0 & , t \in I\_1 \backslash \left[\theta + \varepsilon\right), \end{cases} \qquad \delta q\left(t\right) = 0, \, t \in I\_1,\tag{5.8}$$

where , 1, 1 is the -th Legendre polynomial , 0, \* \* which the number \* \* is defined above (see (4.30)) and \* \*, 0.

Obviously, conditions (4.29) and (4.30) are fulfilled for variation (5.8).

As the conditions + = 0, \* \*, = 0, and <sup>+</sup> + ℎ = 0, \* \*, = 0, 1, are fulfilled, then by (4.33), (4.34), and (5.8), formula (4.32) takes the form:

$$\mathcal{S}^{2}S\{\boldsymbol{\mu}^{0};\boldsymbol{\delta\mu}\} = \mathcal{S}\_{k}\boldsymbol{\chi}^{T}(\boldsymbol{t}\_{1})\boldsymbol{\rho}\_{xx}\{\boldsymbol{\chi}^{0}(\boldsymbol{t}\_{1})\}\boldsymbol{\mathcal{S}}\_{k}\boldsymbol{x}(\boldsymbol{t}\_{1}) - \boldsymbol{\Delta}\_{1k}^{\*} - 2\boldsymbol{\Delta}\_{1k}^{\*\*},\tag{5.9}$$

where

$$
\Delta\_{1k}^\* = \bigwedge\_{\theta}^{\iota\_l} \delta\_k \ge^r \left( t \right) H\_{\times x} \left( t \right) \delta\_k \ge \left( t \right) dt,\tag{5.10}
$$

$$
\Delta\_{1\ell}^{\ast\ast} = \int\_{\theta}^{\theta \ast \varepsilon} \left[ \mathcal{S}\_k \mathbf{x}^\top \left( t \right) G\_k \left[ p \right] \left( t \right) + \mathcal{S}\_k \mathbf{x}^\top \left( t + h \right) G\_k \left[ \tilde{p} \right] \left( t + h \right) \right] \mathcal{S}\_k p \left( t \right) dt. \tag{5.11}
$$

Here, by (4.31), (4.35), (5.8), and the Cauchy formula, <sup>⋅</sup> and <sup>⋅</sup> are determined as follows:

$$\mathcal{S}\_k p\left(t\right) = \begin{cases} \frac{t}{\xi} \Big| \frac{\left(t - s\right)^{k-1}}{\left(k - 1\right)!} l\_k \Big( \frac{2\left(s - \theta\right)}{\varepsilon} - 1\Big) ds, & t \in \left[\theta, \theta + \varepsilon\right), \\ 0, & t \in I\_1 \lor \left[\theta, \theta + \varepsilon\right), \quad \varepsilon \in \left(0, \varepsilon^\*\right), \end{cases} \tag{5.12}$$

$$\delta\_k \mathbf{x}(t) = \begin{cases} \int\_{\theta} \mathbb{A}(\tau, t) \Big[ \mathbf{g}\_k \Big[ p \big](\tau) \delta\_k p(\tau) + \mathbf{g}\_k \Big[ \bar{p} \big](\tau) \delta\_k p(\tau - h) \Big] d\tau, & t \in (\theta, t\_i],\\ 0, & t \in [t\_0, \theta], \end{cases} \tag{5.13}$$

where ⋅ is the solution of the system (3.4).

By considering (5.12) in (5.13), we calculate , . As \* \*, then by the properties of the set \* \*, we have

$$
\delta\_{i}\mathbf{x}(t) = \begin{cases} 0, & t \in [t\_{0}, \theta], \\ \lambda(\theta, t) \operatorname{g}\_{i}\left[\boldsymbol{p}\right](\theta) \boldsymbol{\xi} \big{{}^{\prime}c\_{i}\left(\tau\right)} \operatorname{d}\tau + o\left(\boldsymbol{\varepsilon}^{k+1}\right), & t \in (\theta, \theta + \varepsilon), \\ \lambda(\theta, t) \operatorname{g}\_{i}\left[\boldsymbol{p}\right](\theta) \boldsymbol{\xi} \big{{}^{\prime}c\_{i}\left(\tau\right)} \operatorname{c}\_{i}\left(\tau\right) \operatorname{d}\tau + o\left(\boldsymbol{\varepsilon}^{k+1}\right), & t \in [\theta + \varepsilon, \theta + h\right) \cap I, \\ \lambda(\theta, t) \operatorname{g}\_{i}\left[\boldsymbol{p}\right](\theta) \boldsymbol{\xi} \big{{}^{\prime}c\_{i}\left(\tau\right)} \operatorname{d}\tau + o\left(\boldsymbol{\varepsilon}^{k+1}\right), & t \in [\theta + \varepsilon, \theta + h\right) \cap I, \\ \lambda(\theta, t) \operatorname{g}\_{i}\left[\boldsymbol{p}\right](\theta) \boldsymbol{\xi} \big{{}^{\prime}c\_{i}\left(\tau\right)} \operatorname{d}\tau + \chi\left(\theta\right) \boldsymbol{\lambda}\left(\theta + h, t\right) \operatorname{g}\_{i}\left[\boldsymbol{p}\right](\theta + h) \boldsymbol{\xi} \times \\ \operatorname{b}\_{i}\left[\boldsymbol{\lambda}\left(\theta, t\right) \operatorname{d}\tau + o\left(\boldsymbol{\varepsilon}^{k+1}\right), & t \in [\theta + h, t + h\right) \cap I, \\ \left[\lambda\left(\theta, t\right) \operatorname{g}\_{i}\left[\boldsymbol{p}\right]\left[\boldsymbol{\theta}\right] \boldsymbol{\$$

where

Now suppose that all the statements of Theorem 5.1 are valid for = 1, 2, ..., 1 2 as well. Prove statements (5.2)–(5.5), for = . By assumption, the inequality *<sup>k</sup>* , , 0 (see

( () () ( )) \*\* <sup>0</sup> <sup>1</sup>

*H Hh I R R*

 h

> q

*T r r*

From this inequality, we easily get that , + , + ℎ = 0, that is, we get the

[ ) <sup>0</sup> ( ) <sup>1</sup>

, 1, 1 is the -th Legendre polynomial , 0, \* \* which the number \* \*

0.

As the conditions + = 0, \* \*, = 0, and <sup>+</sup> + ℎ = 0, \* \*, = 0, 1, are fulfilled, then by (4.33), (4.34), and (5.8), formula (4.32) takes the form:

() () () <sup>1</sup>

 d

D = é ù ++ + ò ë û % (5.11)

ò (5.10)

 d

<sup>1</sup> ,

*k k xx k x t H t x t dt*

( ) [ ]() ( ) [ ]( ) () \*\* <sup>1</sup> . *T T k kk k k k x t G p t x t h G p t h p t dt*

1, , ,

*p t qt t I*

q e

qq e

<sup>ì</sup> æ- ö <sup>ï</sup> ç ÷ -Î+ <sup>=</sup> <sup>í</sup> è ø = Î

 q

> h


<sup>0</sup> and

0, , , .

 h

0, ,

(5.8)

(5.9)

 x

 d 1. Hence, taking into account

(5.5) for the case *k-1*) is valid for all \* \*,

216 Nonlinear Systems - Design, Analysis, Estimation and Control

x

h

( )

d

where

where

2, ,

 q cq

 q cq

validity of optimality condition (5.2) for = .

2

*t l t*

e

q

*k*

is defined above (see (4.30)) and \* \*,

x

*k k*

*qq qq*

( [ ]() () [ ]( ))

 q

Now, prove the validity of condition (5.3) for = . In formula (4.32), we put

( ) [ )

*t I*

0 ,\ ,

Obviously, conditions (4.29) and (4.30) are fulfilled for variation (5.8).

\*

q e

+

q

D =

dd

*t*

q

d

*T*

<sup>ï</sup> Î + <sup>î</sup>

1

+ +

% %

*P pq P pq h*

(5.1), we have

$$c\_k\left(\boldsymbol{\tau}\right) = \int\_{\boldsymbol{\theta}}^{\boldsymbol{\tau}} \frac{\left(\boldsymbol{\tau} - \boldsymbol{s}\right)^{k-1}}{\left(k - 1\right)!} I\_k\left(\frac{2\left(\boldsymbol{s} - \boldsymbol{\theta}\right)}{\boldsymbol{\varepsilon}} - 1\right) d\boldsymbol{s}, \; \boldsymbol{\tau} \in \left[\boldsymbol{\theta}, \boldsymbol{\theta} + \boldsymbol{\varepsilon}\right], \; \boldsymbol{\varepsilon} \in \left(0, \boldsymbol{\varepsilon}^{\*\*}\right). \tag{5.15}$$

As , −1, 1 is the -th Legendre polynomial, then it is easy to get

$$\int\_{\theta}^{\theta+\varepsilon} c\_k\left(\tau\right)d\tau = \frac{\varepsilon^{k+1}}{k!\cdot 2^{k+1}} \Big| \left(1-\tau\right)^k l\_k\left(\tau\right)d\tau = \frac{\varepsilon^{k+1} \left(-1\right)^k}{k!\cdot 2^{k+1}} \int\_{-1}^1 \tau^k l\_k\left(\tau\right)d\tau \neq 0. \tag{5.16}$$

Taking into account (5.12)–(5.16) and the fact that , = 0 for > we calculate separately each terms of (5.9). As a result, after simple reasoning we get

() () ( ) ( ) [ ]() ( ) () ( ) ( ) [ ]( ) ( ) [ ]() ( ) () ( ) ( ) [ ]( ) ( ) [ ]( ) ( ) () ( ) ( ) [ ]( ) () ( ) [ ]( ) ( ) () ( ) [ ]( ) ( ) [ ]( ) 1 0 0 1 11 1 11 0 11 1 0 1 1 2 2 2 1 \* 1 , , 2, , , , , , , 2 *T TT T k xx k k xx k T T k xx k T T xx k k k t TT T k k xx k T k x t x t xt g p t x t t g p g p t x t ht g p h g p h ht x t ht g p h c d o g p tH t tg p g p* q e q q d j d x q l q j lq q cq q l q j lq q cq q l q j l q q x tt e x q l q lq q cq q + + = é ë + + + + ++ æ ö ´+ + ùç ÷ + û è ø D = é ë + ò ò % % % ( ) () ( ) [ ]( ) ( ) [ ]( ) ( ) () ( ) [ ]( ) () ( ) [ ]( ) [ ]() () [ ]() ( ) [ ]( ) ( ) [ ]( ) [ ]( ) () ( ) 2 2 2 \*\* 1 2 2 2 , , , , , <sup>1</sup> 2 , <sup>2</sup> *T xx k T T k k xx k k T T T T k kk k k T k kk k t H t ht g p h g p h h t H t h t g p h dt c d o g p Gp g p hG p h g p hG p h c d o* q e q q e q l q lq q cq q l q l q q x tt e x q q c q q l qq q cq q q x t t e + + + + + + æ ö + ++ + + <sup>ù</sup> ç ÷ <sup>+</sup> <sup>û</sup> è ø D = <sup>é</sup> <sup>+</sup> + + <sup>ë</sup> æ ö + ++ + <sup>ù</sup> ç ÷ <sup>û</sup> è ø ò ò % % % % % % (5.17)

Substitute (5.15)–(5.17) in (5.9). Then by (3.9), (4.27), and (5.14), we have

$$\begin{split} \delta^{2}S\Big(\mu^{0};\delta\mu\Big) &= -\xi^{T}\Big\{M\_{\boldsymbol{k}}\left[p,p\right]\Big(\theta,\theta\Big) + 2\,\mathcal{X}\Big(\theta\Big)M\_{\boldsymbol{k}}\left[p,\tilde{p}\right]\Big(\theta,\theta\Big) + \\ &+ \mathcal{X}\Big(\theta\Big)M\_{\boldsymbol{k}}\left[\tilde{p},\tilde{p}\right]\Big(\theta+\boldsymbol{h},\theta+\boldsymbol{h}\Big)\Big\}\xi\frac{\mathcal{E}^{2\boldsymbol{k}+2}}{\left(k!\right)^{2}4^{k+1}}\Big{\left(\int\_{-1}^{1}\boldsymbol{\tau}^{\boldsymbol{k}}l\_{\boldsymbol{k}}\left(\boldsymbol{\tau}\right)d\boldsymbol{\tau}\right)^{2}} \\ &+ o\left(\varepsilon^{2\boldsymbol{k}+2}\right),\ \forall\,\theta\in I^{\star},\ \forall\,\boldsymbol{\xi}\in R^{\diamond}. \end{split}$$

Hence, taking into account the inequality in (2.1), it is easy to complete the proof of optimality condition (5.3) for =.

Continuing the proof of Theorem 5.1, we prove also the validity of optimality condition (5.4) for =. Based on Proposition 4.2, let us consider the + 1 -th order transformation. As the equalities

$$\begin{aligned} &L\_i[p](t) + \chi(t)L\_i[\tilde{p}](t+h) = 0, \, P\_i[p,q](t) + \chi(t)P\_i[\tilde{p},\tilde{q}](t+h) = 0, \, t \in I^{\*\*}, \, i = \overline{0,k}, \\ &\underline{Q}\_i[p](t) + \chi(t)\underline{Q}\_i[\tilde{p}](t+h) = 0, \, t \in I^{\*\*}, \, i = \overline{0,k-1} \end{aligned}$$

taking into account (2.6), we have

( ) ( ) () ( ) ( ) <sup>1</sup> 1 1 <sup>1</sup> 1 1 1 1

 tt

+ + - -

*<sup>k</sup> <sup>k</sup> <sup>k</sup> <sup>k</sup> <sup>k</sup> k kk k k c d l d l d k k*

Taking into account (5.12)–(5.16) and the fact that , = 0 for > we calculate separately

2 2

+

*k*

 q

q

( ) [ ]( ) ( ) () ( ) [ ]( ) () ( )

%

*g p h h t H t h t g p h dt c d o*

+ +

*T T k*

2

2 2

2 1

 c q

+ + -

+

 e

( ) { [ ]( ) () [ ]( )

*k k k*

= -+ +

e

; ,,2 ,,

*Su u M pp M pp h*

( ) [ ]( )} ( ) ( )

+ + + ç ÷

*k k k*

*M pp h h l d k*

 x

Hence, taking into account the inequality in (2.1), it is easy to complete the proof of optimality

Continuing the proof of Theorem 5.1, we prove also the validity of optimality condition (5.4) for =. Based on Proposition 4.2, let us consider the + 1 -th order transformation. As the

 qq

, , ! 4

, , ,

q

q e

+

 x

%

<sup>2</sup> 2 2 <sup>1</sup>

 qq

%

*k*

æ ö

è ø

1

 t t t

ò

q

ò

 q  tt

æ ö

[ ]( ) [ ]() () [ ]() ( ) [ ]( )

 t t

æ ö

 q l qq

*g p Gp g p hG p h*

 e e


, ,

q

%

 lq  q

2

2 2

+

(5.17)

 e

! 2 ! 2

t

() () ( ) ( ) [ ]() ( ) () ( ) ( ) [ ]( )

2

+ + +

each terms of (5.9). As a result, after simple reasoning we get

( ) [ ]() ( ) () ( ) ( ) [ ]( )

*g p t x t ht g p h*

*xx*

æ ö

è ø

 tt

 lq

( ) () ( ) [ ]( )

q e

+

q

Substitute (5.15)–(5.17) in (5.9). Then by (3.9), (4.27), and (5.14), we have

*T*

 x ò

*t H t ht g p h*

*k xx k k*

+ ++ + + <sup>ù</sup> ç ÷ <sup>+</sup> <sup>û</sup> è ø

l q

*xx k*

0 11 1 0 1 1

 j

0 0 1 11 1 11

*k xx k k xx k*

*x t x t xt g p t x t t g p*

 lq

 q l q j

[ ]( ) ( ) () ( ) [ ]( )

, ,

 lq

% %

( ) [ ]( ) [ ]( ) () ( )

*T k*

 x

( ) <sup>0</sup>

*o IR*

*k r*

qx

 q q

,, .

2 2 \*

% %

+ "Î "Î

 c q

D = <sup>é</sup> <sup>+</sup> + + <sup>ë</sup>

<sup>1</sup> 2 , <sup>2</sup>

*g p hG p h c d o*

*T T T T k kk k k*

 q

*kk k*

2 0

cq

e

+

 d

d

+ ++ + <sup>ù</sup> ç ÷ <sup>û</sup> è ø

 q

, ,

, ,

q

ò

q e

q

*T T*

 q l q j

*T T*

%

 q

+ ++

%

*TT T*

 q

 q

 q

condition (5.3) for =.

equalities

 q l q

*T*

 l q

l q

> q

% %

( ) [ ]( )

*g p*

*T k*

1

q

x

ò

*t*

x

D = é ë

cq

cq

cq

 j

cq

cq

\* 1

l q

d

2

\*\* 1

+

1

t t

218 Nonlinear Systems - Design, Analysis, Estimation and Control

( ) [ ]( ) ( ) () ( )

 l q

q

*g p h ht x t*

*k k*

´+ + ùç ÷ + û

*k k xx k*

2, ,

*T TT T*

 d

( ) [ ]( ) () ( )

 x

*g p tH t tg p*

*ht g p h c d o*

,

q e

+

*k xx k*

 x

= é ë + + +

e

1 1 0.

 t tt

$$\begin{split} \delta^{2}S\Big(\boldsymbol{u}^{0};\delta\boldsymbol{u}\big) &= \Delta\_{1}^{2}S\Big(\boldsymbol{u}^{0};\delta\_{k+1}\boldsymbol{p},\delta\boldsymbol{q},\delta\_{k+1}\boldsymbol{x},\boldsymbol{\varepsilon}\big) + \int\_{\boldsymbol{\theta}}^{\boldsymbol{\theta}\cdot\boldsymbol{x}} \Big[\mathcal{S}\_{k+1}\boldsymbol{p}^{\top}\Big(\boldsymbol{t}\big)\Big(\boldsymbol{L}\_{k+1}\big[\boldsymbol{p}\big]\big(\boldsymbol{t}\big) + \boldsymbol{L}\_{k+1}\big[\boldsymbol{p}\big]\big(\boldsymbol{t}+\boldsymbol{h}\big)\Big] \\ \times\delta\boldsymbol{\delta}\_{k+1}\boldsymbol{p}\big(\boldsymbol{t}\big) &+ 2\delta\_{k+1}\boldsymbol{p}^{\top}\big(\boldsymbol{t}\big)\Big(\boldsymbol{P}\_{k+1}\big[\boldsymbol{p}\big]\big(\boldsymbol{t}+\boldsymbol{h}\big)\Big) + \boldsymbol{P}\_{k+1}\big[\tilde{\boldsymbol{p}}\big]\big(\boldsymbol{t}+\boldsymbol{h}\big)\Big)\delta\boldsymbol{q}\big(\boldsymbol{t}\big) \\ &+ \delta\_{k}\boldsymbol{p}^{\top}\big(\boldsymbol{t}\big)\Big(\boldsymbol{Q}\_{k}\big[\boldsymbol{p}\big]\big(\boldsymbol{t}\big) + \boldsymbol{Q}\_{k}\big[\tilde{\boldsymbol{p}}\big]\big(\boldsymbol{t}+\boldsymbol{h}\big)\Big)\delta\_{k+1}\boldsymbol{p}\big(\boldsymbol{t}\big) \\ - \delta\boldsymbol{q}^{\top}\big(\boldsymbol{t}\big)\Big(\boldsymbol{H}\_{\operatorname{op}}\big(\boldsymbol{t}\big) + \boldsymbol{H}\_{\tilde{\boldsymbol{q}}\big]\big(\boldsymbol{t}+\boldsymbol{h}\big)\Big)\delta\boldsymbol{q}\big(\boldsymbol{t}\big)\Big]dt,\delta\in\{0,\mathcal{E}^{\star}\},\end{split} \tag{5.18}$$

where 1 2 0 <sup>⋅</sup> ; + 1, + 1, are determined similarly to (4.33) by changing the index by + 1, and + 1 is the solution of the system (similar to (4.35))

$$\begin{aligned} \boldsymbol{\delta}\_{k+1} \dot{\mathbf{x}}(t) &= f\_x(t) \boldsymbol{\delta}\_{k+1} \mathbf{x}(t) + \mathbf{g}\_{k+1} [\boldsymbol{p}](t) \boldsymbol{\delta}\_{k+1} p(t) + \mathbf{g}\_{k+1} [\tilde{p}](t) \boldsymbol{\delta}\_{k+1} p(t - h) \\ &+ f\_q(t) \boldsymbol{\delta} q(t) + f\_q(t) \boldsymbol{\delta} q(t - h), \ t \in [\boldsymbol{\theta}, t\_l], \\ \boldsymbol{\delta}\_{k+1} \mathbf{x}(t) &= \mathbf{0}, \ t \in [t\_0, \boldsymbol{\theta}], \ \boldsymbol{\delta}\_{k+1} p(t) = \mathbf{0}, \ \boldsymbol{\delta} q(t) = \mathbf{0}, \ t \in [t\_0 - h, \boldsymbol{\theta}]. \end{aligned} \tag{5.19}$$

Choose the variation <sup>=</sup> 0 , , 1 in the following way:

$$\begin{aligned} \delta\_o p\_n \left( t \right) = 0, \; t \in I\_1, \; m \in \{1, 2, \ldots, r\_0\} \backslash \{i, j\}, \; i, j \in \{1, 2, \ldots, r\_0\}, \; i \neq j, \\\ \delta\_o p\_i \left( t \right) = \begin{cases} \alpha l\_{k+1} \left( \frac{2\left(t - \theta\right)}{\varepsilon} - 1 \right), \; t \in \left[\theta, \theta + \varepsilon\right), \\\ 0, \; \; t \in I\_1 \cup \left[\theta, \theta + \varepsilon\right), \end{cases} \end{aligned} \tag{5.20}$$

$$\begin{aligned} \delta\_0 p\_\ne(t) &= \begin{cases} \beta \, I\_{k \cdot 2} \left( \frac{2\left(t - \theta\right)}{\varepsilon} - 1 \right), & t \in \left[\theta, \theta + \varepsilon\right), \\ 0, & t \in I\_1 \backslash \left[\theta, \theta + \varepsilon\right), \end{cases} \\ \delta q\left(t\right) &= 0, \; t \in I\_1, \end{aligned}$$

where , + 1, + 2 is a Legendre polynomials , , \* \*, 0, \* \* .

Obviously, by (5.20), the variation <sup>⋅</sup> <sup>=</sup> 0 <sup>⋅</sup> , <sup>⋅</sup> defined in (5.20) satisfies conditions (4.29), (4.30) for + 1. Taking into account (5.20), by means of (4.30), (4.31), (4.33), and (5.19), it is easy to calculate

$$\begin{split} \mathcal{S}\_k p(t) &\sim \varepsilon^k, \mathcal{S}\_{k+1} p(t) \sim \varepsilon^{k+1}, t \in I\_1, \mathcal{S}\_{k+1} \mathbf{x}(t) \sim \varepsilon^{k+2}, t \in I, \\ \Delta\_t^2 \mathcal{S} \left( \mu^0; \mathcal{S}\_{k+1} p, \delta q, \mathcal{S}\_{k+1} \mathbf{x}, \varepsilon \right) &\sim \varepsilon^{2k+4}. \end{split} \tag{5.21}$$

By (5.20) and (5.21), from (5.18) we get

$$
\delta^2 S(\boldsymbol{u}^0; \boldsymbol{\delta}\boldsymbol{u}) = \int\_{\boldsymbol{\theta}}^{\boldsymbol{\theta} \cdot \boldsymbol{\varepsilon}} \mathcal{S}\_{\boldsymbol{\theta}} \boldsymbol{p}^{\boldsymbol{\tau}}(\boldsymbol{t}) \Big(\mathcal{Q}\_{\boldsymbol{\theta}}\Big[\boldsymbol{p}\big](\boldsymbol{t}) + \mathcal{Q}\_{\boldsymbol{\theta}}\Big[\tilde{\boldsymbol{p}}\big](\boldsymbol{t} + \boldsymbol{h})\Big) \mathcal{S}\_{\boldsymbol{\theta} + 1} \boldsymbol{p}\big(\boldsymbol{t}\big) d\boldsymbol{t} + o\Big(\boldsymbol{\varepsilon}^{2k + 2}\Big),
$$

where <sup>⋅</sup> , , is determined in (4.25).

Hence, taking into account the skew symmetry of the matrix + + ℎ , and the properties of the set \* \*, and also by (2.1), (4.30), and (5.20), we have

$$\begin{split} &\delta^{2}S\Big(\boldsymbol{u}^{0};\delta\boldsymbol{u}\Big) = \Big[q\_{\boldsymbol{\upupupup}}^{(k)}\left(\boldsymbol{\uptheta}\right) - q\_{\boldsymbol{\upupup}}^{(k)}\left(\boldsymbol{\uptheta}\right)\Big] \int\_{\boldsymbol{\upupup}}^{\theta+\varepsilon} \delta\_{k}p\_{i}\Big(\boldsymbol{\uppi}\big)\delta\_{k+1}p\_{j}\Big(\boldsymbol{\uppi}\big)\delta\boldsymbol{\uppi} + o\left(\varepsilon^{2k+2}\right) \\ &= 4\Big(k+1\big)\Big(k+2\big)\Big(\frac{\varepsilon}{2}\Big)^{2k+2}\alpha\beta\delta ab \Big[q\_{\boldsymbol{\upup}}^{(k)}\left(\boldsymbol{\uptheta}\right) - q\_{\boldsymbol{\upup}}^{(k)}\left(\boldsymbol{\uptheta}\right)\Big] \int\_{-1}^{1} \tau^{2}\left(\tau^{2}-1\right)^{2k+1}d\tau + o\left(\varepsilon^{2k+2}\right) \ge 0. \end{split}$$

where \* \*, = 1 + 1 ! 2 + 1, = <sup>1</sup> + 2 ! 2 + 2, and , are the elements of the matrix + + ℎ .

From the last inequality, by arbitrariness of \* \*, , and , 1, 2, ..., 0 it follows that for each \* \*, the skew-symmetric matrix + + ℎ is also symmetric. Consequently, + + ℎ = 0, that is, condition (5.4) is proved for *i=k*.

At last, let us prove optimality condition (5.5). Choose the variation <sup>=</sup> 0 , , 1 in the following way:

Conditions for Optimality of Singular Controls in Dynamic Systems with Retarded Control http://dx.doi.org/10.5772/64225 221

$$\delta\_0 p\left(t\right) = \begin{cases} \left[ \mathcal{\widetilde{s}} \ I\_{k+1} \left( \frac{\mathcal{Q}\left(t-\theta\right)}{\mathcal{E}} - 1 \right), \ t \in \left[ \theta, \theta + \varepsilon \right), \\ 0, \quad t \in I\_1 \backslash \left[ \theta, \theta + \varepsilon \right), \end{cases} \tag{5.22}$$

$$\delta q\left(t\right) = \begin{cases} \eta \Big/ \left(\frac{t-s}{k}\right)^k I\_{k+1}\left(\frac{2\left(s-\theta\right)}{\varepsilon} - 1\right) ds, & t \in \left[\theta, \theta + \varepsilon\right), \\ 0, & t \in I\_1 \backslash \left[\theta, \theta + \varepsilon\right), \end{cases} \tag{5.23}$$

where + 1 , −1, 1 is the 1 + -th Legendre polynomial, <sup>0</sup>, <sup>1</sup>, \* \*, , \* \* .

Obviously, the variation <sup>=</sup> 0 , , 1defined in (5.22) satisfies the conditions (4.29) and (4.30) for = 1, 2... + 1

By (4.30), (4.31), (5.12), (5.19), (5.22), and (5.23), the following relations hold:

Obviously, by (5.20), the variation <sup>⋅</sup> <sup>=</sup> 0 <sup>⋅</sup> , <sup>⋅</sup> defined in (5.20) satisfies conditions (4.29), (4.30) for + 1. Taking into account (5.20), by means of (4.30), (4.31), (4.33), and

~, ~ , , ~ , ,

*k*

+

( ) ( )( [ ]( ) [ ]( )) ( ) ( ) 2 0 2 2 <sup>1</sup> ; , *<sup>T</sup> <sup>k</sup>*

Hence, taking into account the skew symmetry of the matrix + + ℎ ,

( )( ) ( ) ( ) ( ) ( ) ( ) ()

æ ö =+ + é ù - -+ ³ ç ÷ ë û è ø

*ij ji*

*k k ab q q d o*

 q

*k k k*

41 2 1 0

1

+

 q t t

+ 2 ! 2 + 2, and

From the last inequality, by arbitrariness of \* \*, , and , 1, 2, ..., 0 it follows that for each \* \*, the skew-symmetric matrix + + ℎ is also symmetric. Consequently, + + ℎ = 0, that is, condition (5.4) is proved for

At last, let us prove optimality condition (5.5). Choose the variation

2 2 <sup>1</sup> 2 1 2 2 2 2 1

<sup>+</sup> <sup>+</sup> <sup>+</sup> -

ò

 t dt

 e

,

 t

+

 e

are the elements of

*<sup>k</sup> <sup>k</sup> k k <sup>k</sup>*

and the properties of the set \* \*, and also by (2.1), (4.30), and (5.20), we have

( ) ( ) ( ) ( ) () () () ( )

<sup>=</sup> é ù - + ë û

 d t d

2 0 2 2

q

ò

q e

+

*ij ij ki k j*

*Su u q q p p o*

 q

ab

+ 1 ! 2 + 1, = <sup>1</sup>

<sup>+</sup> <sup>=</sup> ò ++ + <sup>+</sup> %

*kk k k S u u p t Q p t Q p t h p t dt o*

1 2

+ +

 e

d

 e

Î Î <sup>D</sup> (5.21)

 d

() () ( )

 e

 e e

1 1 1

+ +

*k k k*

*pt pt t I xt t I*

( )

+ +

*k k*

 dd

111

 ed

d

q e

+

q

where <sup>⋅</sup> , , is determined in (4.25).

2

1

<sup>=</sup> 0 , , 1 in the following way:

e  q

 d

*Su p q x*

2 0 2 4

*k k k*

; ,, ,~ .

(5.19), it is easy to calculate

d

220 Nonlinear Systems - Design, Analysis, Estimation and Control

By (5.20) and (5.21), from (5.18) we get

dd

;

 d

the matrix + + ℎ .

d

where \* \*, =

*i=k*.

$$\delta\_{k+1}p\left(t\right) = \begin{cases} \frac{i}{\varepsilon} \int\_{\theta}^{t} \frac{\left(t - s\right)^{k}}{k!} I\_{k+1}\left(\frac{2\left(s - \theta\right)}{\varepsilon} - 1\right) ds, & t \in \left[\theta, \theta + \varepsilon\right), \\ 0, & t \in I\_{1}\left(\left[\theta, \theta + \varepsilon\right)\right), \end{cases} \tag{5.24}$$

$$\begin{aligned} \delta \mathcal{S}\_{k+1} p(t) &\sim \varepsilon^{k+1}, \; t \in I\_1, \; \delta q(t) \sim \varepsilon^{k+1}, \; t \in I\_1, \; \mathcal{S}\_{k+1} \ge \left(t\right) \sim \varepsilon^{k+2}, \; t \in I, \\\ \Delta\_1^2 S \left(\mu^0; \mathcal{S}\_{k+1} p, \mathcal{S} q, \mathcal{S}\_{k+1} \ge \varepsilon^{2k+4}\right. \tag{5.25} \end{aligned} \tag{5.25}$$

Taking into account (5.23)–(5.25) and validity of the equality + + ℎ = 0, \* \* (see (5.4)), from (5.18), we get

$$\begin{split} &\delta^{2}S\Big(u^{0};\delta u\Big) = \left(\frac{\varepsilon}{2}\right)^{2k+3} \Big[\mathbb{\hat{z}}^{\operatorname{T}}\Big(L\_{k+1}\big[p\big](\theta)+\mathbbm{x}\big(\theta\big)L\_{k+1}\big[\big|\hat{p}\big](\theta+h)\Big)\Big\|^{2} \\ &+2\mathbb{\hat{z}}^{\operatorname{T}}\Big(P\_{k+1}\big[p,q\big]\big(\theta\big)+\mathbbm{x}\big(\theta\big)P\_{k+1}\big[\big|\hat{p},\hat{q}\big]\big(\theta+h\big)\Big)\eta \\ &-\eta^{\operatorname{T}}\Big(H\_{qq}\big(\theta\big)+\mathbbm{x}\big(\theta\big)H\_{\overset{\operatorname{q}\overline{q}}}\big(\theta+h\big)\Big)\eta\Big\|\frac{1}{\left(k\big)^{2}}\Big\|\_{-1}^{1}\Big\|\_{-1}^{1}\Big(\int\_{-1}^{1}\big(t-s\big)^{k}\big\big]p\_{k+1}\big(s\big)ds\Big\Big)^{2}\Big. \end{split}$$

From this expansion, taking into account (2.1), it follows inequality (5.5).

Therefore, Theorem 5.1 is completely proved.

**Corollary 5.1**. Let all the conditions of Theorem 5.1 be fulfilled. Let, in addition, the following equalities hold:

$$L\_i[\,\_p\text{I}](t) + \mathcal{X}(t)L\_i[\,\_p\text{I}](t+h) = 0, \,\,\forall t \in I^{\*\*}, \,i = 0, 1, \dots$$

Then, for optimality of the singular control 0 <sup>⋅</sup> , it is necessary that the relations

$$\begin{aligned} &P\_i[\boldsymbol{p},\boldsymbol{q}](\boldsymbol{\theta}) + \chi(\boldsymbol{\theta})P\_i[\boldsymbol{\tilde{p}},\boldsymbol{\tilde{q}}](\boldsymbol{\theta}+\boldsymbol{h}) = 0, \; i = 0, 1, \ldots; \\ &\xi^T \left(M\_i[\boldsymbol{p},\boldsymbol{p}](\boldsymbol{\theta},\boldsymbol{\theta}) + 2\chi(\boldsymbol{\theta})M\_i[\boldsymbol{p},\boldsymbol{\tilde{p}}](\boldsymbol{\theta},\boldsymbol{\theta}+\boldsymbol{h})\right) \\ &+ \chi(\boldsymbol{\theta})M\_i[\boldsymbol{\tilde{p}},\boldsymbol{\tilde{p}}](\boldsymbol{\theta}+\boldsymbol{h},\boldsymbol{\theta}+\boldsymbol{h})\Big|\,\xi \le 0, \; i = 0, 1, \ldots; \\ &Q\_i[\boldsymbol{p}](\boldsymbol{\theta}) + \chi(\boldsymbol{\theta})Q\_i[\boldsymbol{\tilde{p}}](\boldsymbol{\theta}+\boldsymbol{h}) = 0, \; i = 0, 1, \ldots \end{aligned}$$

be fulfilled for all \* \*, 0.

The proof of the corollary follows immediately from Theorem 5.1.

**Remark 5.1**. As is seen (see Proposition 3.1 and (4.6), (4.15), and (4.24)), for validity of optimality conditions (5.2)–(5.4), for = 0 it is sufficient that assumptions (A1) and (A2) be fulfilled.

**Remark 5.2**. It is clear that (see Proposition 4.1) for validity of optimality conditions (5.5), for = 0 it is sufficient that assumptions (A1), (A3), and (A4) be fulfilled.

**Remark 5.3**. If in Definition 2.1 a special plot is some interval , ⊂ , then very easily similar to the proof of Theorem (5.1) we can prove that conditions (5.2)–(5.5) as optimality conditions are valid for all , <sup>∩</sup> \* \* and 0, 1.
