**6. Conclusion**

As is seen, systems (1.2) and (1.3) are not the most general among all the systems with retarded control. We have chosen it only for definiteness, just to demonstrate the essentials of our method. Nevertheless, the optimality conditions (5.2)–(5.5) can be generalized to the case for more general systems with retarded control.

It should be noted that (1) optimality conditions (5.4) and (5.5), for = 0, are actually the analogs of the equality-type conditions and the Kelly [12] condition, while optimality condition (5.3) is the analog of the Gabasov [11] condition for the considered problem (1.1)–(1.3); (2) optimality condition (5.5), for = 1 is the analog of the Koppa-Mayer [33] condition. Conditions (5.3)–(5.5) were obtained in [10] only for singular controls with complete degree of degeneracy, that is, for the case when 1 = 0 (see Definition 2.1).

We also note that (1) the analog of the Kelly condition and equality-type condition was obtained in [24] by another method for systems with retarded state; (2) optimality-type conditions (5.2)– (5.5) for system with retarded state were obtained in [[31, 32], p. 119]; (3) optimality conditions of type (5.4), (5.5) for systems without retardation were obtained in the papers [[23, 26, 27], p. 145, [29, 30, 33, 34, 39–41], etc.].

The proof of Theorem 5.1 shows that the optimality conditions (5.3)–(5.5) are independent. Also, it is clear that, unlike (5.2), (5.3), and (5.5), the optimality condition (5.4) for 1 <sup>=</sup> <sup>1</sup> (see Definition 2.1) becomes ineffective, though it is effective in the general case for 1 <sup>&</sup>lt; <sup>1</sup>. To illustrate the rich content of condition (5.4), we consider a concrete example:

$$\text{Example.}\,\dot{x}\_1(t) = u\_2(t) + u\_1^2(t-1) - u\_3(t-1),\,\dot{x}\_2(t) = u\_1(t) - u\_2(t)\,\dot{x}\_3$$

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

[ ]() () [ ]( ) \*\* 0, , 0,1,... *L p t tL p t h t I i i i* + + = "Î =

{ [ ]( ) () [ ]( )

% %

 q

**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

**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

0,

As is seen, systems (1.2) and (1.3) are not the most general among all the systems with retarded control. We have chosen it only for definiteness, just to demonstrate the essentials of our method. Nevertheless, the optimality conditions (5.2)–(5.5) can be generalized to the case for

It should be noted that (1) optimality conditions (5.4) and (5.5), for = 0, are actually the analogs of the equality-type conditions and the Kelly [12] condition, while optimality condition (5.3) is the analog of the Gabasov [11] condition for the considered problem (1.1)–(1.3); (2) optimality condition (5.5), for = 1 is the analog of the Koppa-Mayer [33] condition. Conditions (5.3)–(5.5) were obtained in [10] only for singular controls with complete degree of

1.

+ += =

 c q

*M pp M pp h M pp h h i*

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

+ += =

 q

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

 x

+ +

0, 0,1,...

 qq

%

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

[ ]() () [ ]( )

*P pq P pq h i*

+ ++ £=

%

*i i*

*Qp Qp h i*

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

 q q

 qq

% %

c%

*i i*

q cq

*i i i*

0.

The proof of the corollary follows immediately from Theorem 5.1.

= 0 it is sufficient that assumptions (A1), (A3), and (A4) be fulfilled.

q cq

*T*

cq

x

222 Nonlinear Systems - Design, Analysis, Estimation and Control

be fulfilled for all \* \*,

are valid for all , <sup>∩</sup> \* \* and

more general systems with retarded control.

degeneracy, that is, for the case when 1 = 0 (see Definition 2.1).

**6. Conclusion**

equalities hold:

$$\begin{aligned} \dot{x}\_3(t) &= \left(u\_1(t) + u\_2(t)\right) \ge x\_2(t) + u\_3^2(t) + u\_3^2(t-1), \ t \in I: = [0, 2], \\\ u\_i \Big|\_{t} &< 2, \ i = 1, 2, 3, \\\ u\_i \Big|\_{t} &< 2, \ i = 1, 2, 3, \\\ &\quad \end{aligned} \\ \begin{aligned} \dot{x}\_i(0) &= 0, \ x\_i(0) = 0, \ u\_i(t) = 0, \ u\_i(t) = 0, \ t \in [-1, 0), \\\ &\quad \quad \quad \quad t \in [-1, 0), \\\ &\quad \quad \quad \quad \quad \quad \end{aligned}$$

Check for optimality of the control 0 <sup>=</sup> 0, 0, 0 , 1, 2 . In this control according to (2.7), (2.8), (3.9), (3.10), (4.6), (4.9), (4.15), (4.21), and (4.24), we have

 <sup>0</sup> = 0, = 1, 2, 3, <sup>0</sup> = 0, = 1, 2, 3 <sup>0</sup> <sup>=</sup> 1, , 0 , , , , <sup>=</sup> 1 + 2 2 3 <sup>2</sup> 3 <sup>2</sup>, : = ℎ , , where ℎ = 0, , 1, 2, 3 , , <sup>≠</sup> 3, 3 , ℎ33 <sup>=</sup> 2; + 1 <sup>=</sup> ℎ , 0, 1 , where ℎ = 0, , 1, 2, 3 , , <sup>≠</sup> 3, 3 , ℎ33 <sup>=</sup> <sup>2</sup>; + 1 = 0, 1, 2 ; 0 <sup>=</sup> 0 10 <sup>1</sup> 1 0 , , 0 = 0, , where : = 1, 2 , : = 1, 2 ; 1 = 1 = 0, , 0 <sup>=</sup> <sup>0</sup> <sup>2</sup> 2 0 , , 0 + 1 = 0, , 1 = 1 + 1 = 0, , 0 , = 0 , = 0, 1 , = 1 , = 0, 0 , , <sup>=</sup> 1 <sup>1</sup> 1 0 , , 0 , <sup>⋅</sup> = 0, 0 , <sup>⋅</sup> = 0, + + 1 <sup>=</sup> 4, 0, 1 , 2, 1, 2 , where =3, = 3.

Hence, we have the following: (1) admissible control 0 <sup>=</sup> 0, 0, 0 , 1, 2 is singular (in the sense of Definition 2.1) and singularity to it is delivered by the vector component = 1, 2 , that is, equality (5.1) is fulfilled only = 0; (2) optimality conditions (5.2), (5.3), (5.5), and the results of the papers [1–3, 6, 9, 10] cannot say that whether the control 0 <sup>⋅</sup> is an optimal or not. However, optimality condition (5.4) for = 0 is not fulfilled (0 + 0 + 1 <sup>=</sup> 0 −2 2 0 = 0, ), that is, by condition (5.4) (for = 0) we conclude that the control 0 <sup>=</sup> 0, 0, 0 , −1, 2 cannot be optimal.

## **Author details**

Misir J. Mardanov1\* and Telman K. Melikov2

\*Address all correspondence to: misirmardanov@imm.az

1 Institute of Mathematics and Mechanics of ANAS, Baku, Azerbaijan

2 Institute of Mathematics and Mechanics and Institute of Control Systems of ANAS, Baku, Azerbaijan
