**2. The second variation of the objective functional and the definition of a singular (in the classical sense) control**

Let assumptions (A1) and (A2) be fulfilled, and 0 <sup>⋅</sup> , <sup>0</sup> <sup>⋅</sup> be some admissible process. If the process 0 <sup>⋅</sup> , 0 <sup>⋅</sup> is optimal, then, by using the known technique (see, e.g., [27, p. 51]), it is easy to get

$$\delta \left( \mathcal{S} \left( u^0; \delta u \left( \cdot \right) \right) \right) = 0, \mathcal{S}^2 \mathcal{S} \left( u^0; \delta u \left( \cdot \right) \right) \ge 0, \ \forall \delta u \left( \cdot \right) \in \tilde{C}^\* \left( I\_1, R' \right), \delta u \left( t \right) = 0, \ t \in I\_0. \tag{2.1}$$

Here

$$\begin{aligned} \delta \prescript{}{}{\delta} \left( u^0; \delta u \left( \cdot \right) \right) &:= \frac{\prescript{}{\iota}}{\prescript{}{\iota}} \Big[ H\_u^{\operatorname{T}} \left( t \right) \delta u \left( t \right) + H\_v^{\operatorname{T}} \left( t \right) \delta u \left( t - h \right) \Big] dt, \\ \delta u \left( \cdot \right) &\in \tilde{C}^+ \left( I\_1, R^\prime \right), \delta u \left( t \right) = 0, \; t \in I\_0, \end{aligned} \tag{2.2}$$

$$\begin{split} \delta \prescript{}{}{S} \Big( \mathbf{u}^{0}; \delta \mathbf{u} \big( \cdot \big) \big) := \delta \mathbf{x}^{\top} \left( t\_{1} \right) \boldsymbol{\upphi}\_{\scriptscriptstyle\mathbf{u}} \left( \mathbf{x}^{0} \big( t\_{1} \big) \right) \delta \mathbf{x} \big( t\_{1} \big) - \int\_{t\_{0}}^{t} \Big( \delta \mathbf{x}^{\top} \big( t \big) H\_{\scriptscriptstyle\mathbf{u}} \big( t \big) \delta \mathbf{x} \big( t \big) \\ + \delta \mathbf{u}^{\top} \big( t \big) H\_{\scriptscriptstyle\mathbf{u}} \big( t \big) \delta \mathbf{u} \big( t \big) + \delta \mathbf{u}^{\top} \big( t - h \big) H\_{\scriptscriptstyle\mathbf{u}} \big( t \big) \delta \mathbf{u} \big( t - h \big) + 2 \Big[ \delta \mathbf{x}^{\top} \big( t \big) H\_{\scriptscriptstyle\mathbf{u}} \big( t \big) \delta \mathbf{u} \big( t \big) \end{split} \tag{2.3}$$
 
$$\begin{split} \delta \mathbf{x}^{\top} \big( t \big) H\_{\scriptscriptstyle\mathbf{u}} \big( t \big) \delta \mathbf{u} \big( t - h \big) + \delta \mathbf{u}^{\top} \big( t \big) H\_{\scriptscriptstyle\mathbf{u}} \big( t \big) \delta \mathbf{u} \big( t - h \big) \Big] \end{split} \tag{2.4}$$
  $\delta \mathbf{u} \big( \cdot \big) \in \mathsf{C}^{+} \left( I\_{1}, R^{\prime} \right), \delta \mathbf{u} \big( t \big) = \mathbf{0}, \ t \in I\_{0},$ 

where 1 0 ; <sup>⋅</sup> and 2 0 ; ⋅ are, respectively, the first and the second variations of the functional at the point 0 <sup>⋅</sup> ; , , , , , ; ⋅ is the variation of the control 0 <sup>⋅</sup> , while <sup>⋅</sup> is the corresponding variation of the trajectory 0 , , which <sup>⋅</sup> is the solution of the system

$$\begin{aligned} \delta \delta \dot{x}(t) &= f\_x(t) \delta x(t) + f\_u(t) \delta u(t) + f\_v(t) \delta u(t-h), \; t \in I, \\ \delta \boldsymbol{x}(t\_0) &= 0 \; , \; \delta u(t) = 0, \; t = I\_0, \end{aligned} \tag{2.4}$$

where : = <sup>0</sup> , 0 , 0 <sup>ℎ</sup> , , and , , , while the vector function is the solution of the conjugate system

$$
\psi^0 \left( t \right) = -H\_\times \left( t \right), t \in I, \quad \psi^0 \left( t\_1 \right) = -\varphi\_\times \left( \mathbf{x}^0 \left( t\_1 \right) \right). \tag{2.5}
$$

Below, we consider that the following conditions are fulfilled:

$$H(t) = 0, \ H\_{\mu}(t) = 0, \ H\_{\mu\nu}(t) = 0, \text{ for } \iota > \iota\_{\iota} \text{ and } \mu, \nu \in \{\mathbf{x}, \mu, \nu\}\ . \tag{2.6}$$

If 0 <sup>⋅</sup> , 0 <sup>⋅</sup> is an optimal process, then, by definition of an admissible control and taking into consideration (2.2)–(2.4) from (2.1), proceeding the same way as in [27, p. 53], we obtain the classical necessary conditions of optimality (analogs of the Euler equation and Legendre-Clebsch condition) [10, 43], that is, the following relations are valid:

$$H\_{\boldsymbol{\nu}}(t) + \chi(t)H\_{\boldsymbol{\nu}}(t+h) = 0, \ \forall t \in I;\tag{2.7}$$

(2.2)

(2.3)

( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 0 2 0

( ) ( ) ( )

( ) ( ) () () ( ) ( ) () () { ( )

 d

*Su u x t x t xt x tH t xt*

*T T*

<sup>+</sup>

*u t*

*r*

*u C I R ut t I*

× Î = Î

 d

*xx xx t T T T*

+ + - -+ é

() () ( ) () () ( ) }

0 <sup>⋅</sup> , while <sup>⋅</sup> is the corresponding variation of the trajectory 0 , , which <sup>⋅</sup> is

u

& (2.4)

<sup>0</sup> , 0 , 0 <sup>ℎ</sup> , , and , , , while the vector function

1 1

. *x x t Htt I t xt* (2.5)

 j

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

*x u xt f t xt f t ut f t ut h t I*

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

y

& =- Î =- , ,

= + + -Î

*x t H t u t h u t H t u t h dt*

*x u*

+ - + - ù

dd

1

ò

*t*

0

%

1 11

1 0

*T T*

 dd

, , 0, ,

dd

Here

d

where 1 0

where

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d

( ) ( ) ( )

; <sup>⋅</sup> and 2 0

<sup>+</sup>

*r*

*u C I R ut t I*

× Î = Î

 d

of the functional at the point 0 <sup>⋅</sup> ;

( ) () 0 0

= = =

*xt ut t I*

Below, we consider that the following conditions are fulfilled:

 d

0 , 0, ,

dddd

u

× = -

1 0

198 Nonlinear Systems - Design, Analysis, Estimation and Control

2 0 0

 dj

d

d

%

; :

 d

d

the solution of the system

: =

d

is the solution of the conjugate system

y

dd

1 0 ; 0, ; 0, , , 0, . *<sup>r</sup>*

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ò

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, , 0, ,

; : ,

× =- é ù + - ë û

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u  d

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

 d

 d  d

*uu xu*

uu

 u

*u tH t ut u t hH t ut h x tH t ut*

2

û

; ⋅ are, respectively, the first and the second variations

,

,

, , , , ; ⋅ is the variation of the control

, ,

andfor (2.6)

 d

ë

 d

 d

 d

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

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*S u u H t u t H t u t h dt*

$$
\tilde{u}^{\top} \Big[ H\_{\text{uv}} \left( t \right) + \mathcal{X} \left( t \right) H\_{\text{uv}} \left( t + h \right) \Big] \tilde{u} \le 0, \quad \forall t \in I, \ \forall \tilde{u} \in R'; \tag{2.8}
$$

**c.** <sup>1</sup> = 0, <sup>1</sup> − ℎ + <sup>1</sup> ≤ 0, for all ∈ , if optimal control 0 <sup>⋅</sup> is continuous at the points =1 − ℎ, = 1, 2. Here, <sup>⋅</sup> is the characteristic function of the set 0, 1 − ℎ .

It should be noted that the optimality condition (c) is the corollary of conditions (a) and (b).

**Definition 2.1**. An admissible control 0 , ∈ , satisfying conditions (2.7) and (2.8), is called singular (in classical sense) if

$$\operatorname{rang}\left[H\_{\text{uv}}\left(t\right) + \mathcal{X}\left(t\right)H\_{\text{uv}}\left(t+h\right)\right] = r\_{\text{l}} < r, \ \forall t \in I.$$

In this case, the set is called a singular plot for an admissible control 0 <sup>⋅</sup> . The main goal of this chapter is to study such singular controls.

Let = , , = , , where , ∈ 0, , ∈ 1 , 0 + 1 = . Without loss of generality [[27], p. 138], we assume that the singularity to the control 0 <sup>⋅</sup> is delivered by a vector component ∈ 0, that is,

$$H\_{\rho\rho}(t) + \mathcal{X}(t)H\_{\rho\mathfrak{k}}(t+h) = 0, \ t \in I. \tag{2.9}$$

Note that the general inequality (2.8) implies the equality-type optimality condition for a singular (in classical sense) control 0 <sup>⋅</sup> :

$$H\_{\mu\eta}(t) + \mathcal{X}(t)H\_{\mu\overline{\eta}}(t+h) = 0, \ t \in I. \tag{2.10}$$

**Proposition 2.1**. Let assumptions (A1) and (A2) be fulfilled, the admissible control 0 <sup>⋅</sup> <sup>=</sup> <sup>⋅</sup> , <sup>⋅</sup> be singular (in the classical sense) and condition (2.9) be fulfilled along it. Let also the variations <sup>=</sup> 0 , ∈ + <sup>1</sup>, be non-zero only on , + , where <sup>0</sup>, 1 and 0, 0 , with the number 0 0, ℎ be such that (1) if <sup>0</sup>, 1 − ℎ , then 0 < 1 <sup>−</sup> <sup>ℎ</sup> and (2) if <sup>1</sup> − ℎ, 1 , then 0 < 1 <sup>−</sup> . Then, (a) the variational system (2.4) becomes

$$\begin{cases} \delta \dot{\mathbf{x}}(t) = f\_x(t) \delta \mathbf{x}(t) + f\_p\left(t\right) \delta\_o p\left(t\right) + f\_q\left(t\right) \delta q\left(t\right) + f\_{\tilde{\rho}}\left(t\right) \delta\_o p\left(t - h\right) \\ \qquad + f\_q\left(t\right) \delta q\left(t - h\right), \quad t \in \left[\theta, t\_l\right], \\ \delta \mathbf{x}\left(t\right) = 0, \ t \in \left[t\_0, \theta\right], \ \delta\_o p\left(t\right) = 0, \ \delta q\left(t\right) = 0, \quad t \in \left[t\_0 - h, \theta\right); \end{cases} \tag{2.11}$$

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

(2.12)

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