**3. Transformation of the second variation of the functional by means of modified variant of matrix impulse method (when studying singular (in the sense of Definition 2.1) of controls)**

Let conditions (A1) and (A2) be fulfilled and along the singular control 0 <sup>⋅</sup> the equality (2.9) hold. Use Proposition 2.1. Let the variation <sup>=</sup> 0 , + 1, have the form:

$$\delta\_{\alpha}p\left(t\right) = \begin{cases} \underline{\xi}, & t \in \left[\theta, \theta + \varepsilon\right), \; \varepsilon \in \left(0, \varepsilon\_{0}\right), \\ 0, & t \in I\_{\downarrow} \backslash \left[\theta, \theta + \varepsilon\right), \end{cases} \qquad \delta q\left(t\right) = 0, t \in I,\tag{3.1}$$

where 0, <sup>0</sup>, 1 , and the number 0 was defined in Proposition 2.1.

Along the singular control 0 <sup>⋅</sup> <sup>=</sup> <sup>⋅</sup> , <sup>⋅</sup> satisfying condition (2.9), taking into account (3.1), formula (2.12) takes the form:

$$\operatorname{\mathcal{S}}^2 \mathcal{S} \{ \mu^0; \operatorname{\mathcal{S}u}(\cdot) \} = \operatorname{\mathcal{S}u}^T(t\_1) \varphi\_{\text{xx}} \left( \mathbf{x}^0(t\_1) \middle| \operatorname{\mathcal{S}u}(t\_1) - \Delta\_1^\* - 2\Delta\_2^\* \right), \tag{3.2}$$

where

where

(2.4) becomes

then 0 < 1 <sup>−</sup> <sup>ℎ</sup> and (2) if

200 Nonlinear Systems - Design, Analysis, Estimation and Control

d

ï

0, 1 and 0, 0 , with the number 0 0, ℎ be such that (1) if

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

*xt f t xt f t pt f t qt f t pt h f t qt h t t*

*xp qp*

<sup>ì</sup> =+ ++- ïï

*xt t t pt qt t t h*

ï = Î = = Î- î

*q*

í + -Î

qd

d

(b) the following representation is valid for the second variation (2.3):

(2.9), (2.11), and the definition of the variation <sup>⋅</sup> <sup>=</sup> 0 <sup>⋅</sup> , <sup>⋅</sup> .

hold. Use Proposition 2.1. Let the variation <sup>=</sup> 0 , +

, , , 0, ,

( ) [ ) ()

qq e

1

ï Î + î

 qq e e

*t*

x

**the sense of Definition 2.1) of controls)**

0

d

ddd

%&

0 0 0

**Proof**. To prove (a), it suffices to consider the definition of the variation <sup>⋅</sup> <sup>=</sup> 0 <sup>⋅</sup> , <sup>⋅</sup> in (2.4). The proof of (b) follows directly from (2.3), in view of (2.6),

**3. Transformation of the second variation of the functional by means of modified variant of matrix impulse method (when studying singular (in**

Let conditions (A1) and (A2) be fulfilled and along the singular control 0 <sup>⋅</sup> the equality (2.9)

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

 d

 e

0, , 0, \ , ,

*p t qt t I t I*

ìï Î+ Î <sup>=</sup> <sup>í</sup> = Î

 d

0, , , 0, 0, , ;

0 0 1

dd

%

, ,,

 q

<sup>1</sup> − ℎ, 1 , then 0 < 1 <sup>−</sup> . Then, (a) the variational system

q

<sup>0</sup>, 1 − ℎ ,

(2.11)

(2.12)

1, have the form:

(3.1)

$$
\Delta\_1^\* = \bigwedge\_{\theta}^{\wedge} \delta \mathbf{x}^T(t) H\_{\text{xx}}(t) \delta \mathbf{x}(t) dt,\\
\Delta\_2^\* = \bigwedge\_{\theta}^{\theta \wedge \mathcal{E}} \left[ \delta \mathbf{x}^T(t) H\_{\text{xy}}(t) + \delta \mathbf{x}^T(t+h) H\_{\text{xy}}(t+h) \right] \sharp dt,
$$

where , is the solution of the system (2.11).

By the Cauchy formula, we have

$$\delta \mathbf{x}(t) = \begin{cases} \frac{1}{\rho} \mathbb{A}(s,t) \Big[ \int\_{\rho} f\_{\rho}(s) \delta\_{0} p(s) + f\_{\rho}(s) \delta\_{0} p(s - h) \Big] ds, & t \in (\theta, t\_{1}] \\\\ 0, & t \in [t\_{0}, \theta], \end{cases} \tag{3.3}$$

where , , , × is the solution of the system

$$
\mathcal{A}\_{\mathbf{r}}\left(\mathbf{s},t\right) = f\_{\mathbf{x}}\left(t\right)\mathcal{A}\left(\mathbf{s},t\right), \quad t\_0 \le \mathbf{s} < t \le t\_1,\tag{3.4}
$$

, = 0, > , , = ( is a unit × matrix).

As (A2) and 0 <sup>⋅</sup> + 1, are fulfilled, then by (3.1) and (3.4) and for all 0, 1 , from (3.3) we get

$$
\delta \mathbf{x}(t) = \begin{cases} 0, & t \in [t\_0, \theta], \\ (t - \theta) \lambda(\theta, t) f\_\rho \left(\theta\right) \xi + o\left(t - \theta\right), & t \in (\theta, \theta + \varepsilon), \\ \varepsilon \lambda(\theta, t) f\_\rho \left(\theta\right) \xi + o\left(\varepsilon\right), & t \in [\theta + \varepsilon, \theta + h) \cap I, \\ \varepsilon \lambda(\theta, t) f\_\rho \left(\theta\right) \xi + \left(t - \theta - h\right) \chi(\theta) \lambda(\theta + h, t) f\_\rho \left(\theta + h\right) \xi \\ + o\left(t - \theta - h\right), & t \in [\theta + h, \theta + h + \varepsilon) \cap I, \\ \varepsilon \left[\lambda\left(\theta, t\right) f\_\rho\left(\theta\right) + \chi\left(\theta\right) \lambda\left(\theta + h, t\right) f\_\rho\left(\theta + h\right)\right] \xi + o\left(\varepsilon\right), & t \in \left[\theta + h + \varepsilon, t\_1\right] \cap I \end{cases} \tag{3.5}
$$

where ⋅ is the characteristic function of the set 0, 1 − ℎ ; / <sup>0</sup>, as <sup>0</sup>.

By (2.6) and (3.5) and taking into account , = and , = 0 for >, we calculate separate terms of (3.2). As a result, after simple reasoning, we get

$$\begin{split} \mathsf{Ric}^{T}(\mathfrak{t}\_{1})\mathfrak{g}\_{\text{xx}}\Big(\mathsf{x}^{0}(\mathfrak{t})\big)\mathsf{fix}(\mathfrak{t}\_{1}) &= \mathsf{z}^{2}\mathsf{z}^{T}\Big[f\_{p}^{T}(\theta)\mathsf{z}^{T}(\theta,\mathfrak{t}\_{1})\mathsf{g}\_{\text{xx}}\Big(\mathsf{x}^{0}(\mathfrak{t})\big)\mathsf{z}(\theta,\mathfrak{t}\_{1})f\_{p}(\theta) + \\ &+ 2\mathsf{z}(\theta)f\_{p}^{T}(\theta)\mathsf{z}^{T}(\theta,\mathfrak{t}\_{1})\mathsf{g}\_{\text{xx}}\Big(\mathsf{x}^{0}(\mathfrak{t}\_{1})\big)\mathsf{z}(\theta+h,\mathfrak{t}\_{1})f\_{\overline{p}}(\theta+h) + \\ &+ \mathsf{z}(\theta)f\_{\overline{p}}^{T}(\theta+h)\mathsf{z}^{T}(\theta+h,\mathfrak{t}\_{1})\mathsf{g}\_{\text{xx}}\Big(\mathsf{x}^{0}(\mathfrak{t}\_{1})\big)\mathsf{z}(\theta+h,\mathfrak{t}\_{1})f\_{\overline{p}}(\theta+h)\Big]\mathsf{z} + o\Big(\mathsf{z}^{2}\Big), \end{split} \tag{3.6}$$

$$\begin{split} \Lambda\_{1}^{\*} = & \varepsilon^{2} \tilde{\xi}^{\top} \Big[ \Big[ \int f\_{\rho}^{\top} \Big( \theta \right) \dot{\lambda}^{\top} \Big( \theta, t \big) H\_{\infty} \big( t \big) \dot{\lambda} \Big( \theta, t \big) f\_{\rho} \Big( \theta \Big) \\ & + 2 \chi \Big( \theta \big) f\_{\rho}^{\top} \Big( \theta \big) \dot{\lambda}^{\top} \Big( \theta, t \big) H\_{\infty} \big( t \big) \dot{\lambda} \Big( \theta + h, t \big) f\_{\rho} \Big( \theta + h \Big) \\ & + \chi \Big( \theta \big) f\_{\rho}^{\top} \Big( \theta + h \big) \dot{\lambda}^{\top} \Big( \theta + h, t \big) H\_{\infty} \big( \big$$

( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( )( )) () () () () () ( ) ( ) () ( ) ( ) ( ) \* 2 2 2 2 , , , 2 , <sup>2</sup> . *T TT T T p xp p xp T T p xp T T T T p xp p xp T p xp f tH t t f t hH t h f h h t h H t h t dt o f H f hH h f hH h o* q e q x ql q q cq e ql q q lq q xe e x q q c q q l qq q cq q q x e + D = <sup>é</sup> - + + + <sup>ë</sup> + + ++ + - + ù û = + ++ é ë + + ++ ù û ò % % % %% % (3.8)

Following [10, 14, 17], we consider the matrix functions

$$\Psi^{\rm I}(\mathbf{s},\tau) = \int\_{t\_0}^{t\_1} \boldsymbol{\lambda}^{\rm T}(\mathbf{s},t) \boldsymbol{H}\_{\rm x}\left(t\right) \boldsymbol{\lambda}\left(\tau,t\right) dt - \boldsymbol{\lambda}^{\rm T}\left(\mathbf{s},t\_1\right) \boldsymbol{\rho}\_{\rm x}\left(\mathbf{x}^{\rm 0}\left(t\_1\right)\right) \boldsymbol{\lambda}\left(\tau,t\_1\right), \quad \left(\mathbf{s},\tau\right) \in I \times I,\tag{3.9}$$

$$M\_0\left[p,\tilde{p}\right](\mathbf{s},\tau) = f\_p^{\tau}\left(\mathbf{s}\right)\boldsymbol{\uplambda}^{\tau}\left(\mathbf{s},\tau\right)H\_{\boldsymbol{u}\boldsymbol{\upbeta}}\left(\tau\right) + f\_p^{\tau}\left(\mathbf{s}\right)\boldsymbol{\upPsi}\left(\mathbf{s},\tau\right)f\_{\boldsymbol{\upbeta}}\left(\tau\right),\,\left(\mathbf{s},\tau\right) \in I \times I. \tag{3.10}$$

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

Thus, substituting (3.6)–(3.8) in (3.2), allowing for (3.9), (3.10) and equality , = 0, for > , , ∈ × , we get the validity of the following statement.

**Proposition 3.1**. Let conditions (A1) and (A2) be fulfilled, and the admissible control 0 <sup>⋅</sup> <sup>=</sup> <sup>⋅</sup> , <sup>⋅</sup> be singular (in the classic sense) and the condition (2.9) be fulfilled along it. Then, for each 0, 1 and for all 0 the following expansion is valid:

where ⋅ is the characteristic function of the set

202 Nonlinear Systems - Design, Analysis, Estimation and Control

1

q

ò

ë

cq

*T T*

*T*

ë

0

 l

t

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

*t*

t

*t*

 q

cq

*TT T*

 q l q

*T T*

 q l q

 q

 ql q

 lq

 q

*T T*

*t*

\* 2 1

\* 2

2

cq

e x

q e

+

x

q

q

ex

D = é

separate terms of (3.2). As a result, after simple reasoning, we get

( ) ( ) () ( ) ( )

*p xx p*

*f tH t t f*

+ + +

, ,

 lq

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

*p xx p*

( ) ( ) ( )( )) ()

*p xp p xp*

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

 l

Y = *s s t H t t dt s t x t t s I I*

<sup>0</sup> [ ] , , ( ) () ( ) () () ( ) () ( ) , , ,, . *T T <sup>T</sup> M pp s f s s H f s s f s I I <sup>p</sup> xp <sup>p</sup> <sup>p</sup>* %

Thus, substituting (3.6)–(3.8) in (3.2), allowing for (3.9), (3.10) and equality , = 0, for

 t

*xx xx*

1 11 ,, ,, ,, , ,

 j

 x

û

= + ++ é

*T TT T T*

*f h h t h H t h t dt o*

 c q

+ + ++ + - + ù

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

*f H f hH h*

 q l qq

 e

q cq e ql q

<sup>ò</sup> %% %

D = <sup>é</sup> - + + + <sup>ë</sup>

2

.

%% %

*f t H t ht f h*

 lq

+ + + + ++ ù

*p xx p*

2 ,,

 l q

() ( ) ( ) ( )

 q

*T T*

 lt

 lt

> , , ∈ × , we get the validity of the following statement.

*f hH h o*

 q

Following [10, 14, 17], we consider the matrix functions

2 , <sup>2</sup>

*T T T T*

,

*p xp*

*p xp*

+ + ++ ù

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

lq

 q

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

, ,

*f tH t t f t hH t h*

q

 xe

%% %(3.8)

0


 lt

 t

= % % + Y Î ´ (3.10)

 t  t

 t

û

*p xp p xp*

 q

*f h h t H t h t f h dt o*

,, ,

 q

0, 1 − ℎ ; / <sup>0</sup>, as <sup>0</sup>.

2

 e

 x

û

2

 q (3.6)

(3.7)

By (2.6) and (3.5) and taking into account , = and , = 0 for >, we calculate

$$\begin{split} \delta^{2}S\Big(\boldsymbol{\mu}^{0};\delta\boldsymbol{u}\big(\cdot\big)\big) &= -\varepsilon^{2}\xi^{T}\Big(M\_{0}\big[\boldsymbol{p},\boldsymbol{p}\big]\big(\theta,\theta\big)+2\chi\big(\theta\big)M\_{0}\big[\boldsymbol{p},\boldsymbol{\tilde{p}}\big]\big(\theta,\theta+\boldsymbol{h}\big) \\ &+\chi\big(\theta\big)M\_{0}\big[\boldsymbol{\tilde{p}},\boldsymbol{\tilde{p}}\big]\big(\theta+\boldsymbol{h},\theta+\boldsymbol{h}\big)\big)\big(\boldsymbol{\tilde{\xi}}+o\big(\boldsymbol{\varepsilon}^{2}\big),\ \forall\boldsymbol{\varepsilon}\in\big(0,\varepsilon\_{0}\big), \end{split} \tag{3.11}$$

where the number 0 was defined above (see Proposition 2.1), <sup>⋅</sup> is the characteristic function of the set 0, 1 − ℎ and matrix functions 0 , , , 0 , , + ℎ , 0 , + ℎ, + ℎ that are defined by (3.10).
