**1. Introduction**

As is known, optimal control problems described by the dynamical systems with retarded control are attracting the attention of many specialists, and the results obtained in this field deal mainly with the first-order necessary optimality conditions [1–8, etc.]. However, theory of singular controls for systems with retarded control has not been studied enough yet [9, 10]. One of the main reasons here is that the methods proposed and developed for ordinary systems (for systems without retardation) in [11–18] are not directly applicable to the singular controls in dynamical systems with aftereffect (see [9, 14–19]). Therefore, to study optimal control problems in the systems with retarded control is of special theoretical interest. Besides, such problems have practical significance as well, because mathematical modelling for some problems of organization of the economic plan and production leads to the problems with retarded control (see, e.g., [20]).

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

As is known, the concept of singular control was first introduced to the theory of optimal processes by Rozenoer [22] in 1959. First results on the necessary optimality conditions for singular controls have been obtained by Kelley [12] in the case of open set , and by Gabasov [11] in the case of arbitrary (in particular, closed) set , where *U* is a set of values of admissible controls. Afterward, Kelley and Gabasov's conditions as well as the methods for treating singular controls proposed in [11, 13] have been significantly generalized in [10, 14–19, 23– 41, etc.] to the cases of (1) controls with higher-order degeneration, (2) multidimensional controls, and (3) various classes of control systems. Considering all these cases, the methods in [11, 13] have been generalized in [17, 37] and for optimality of singular controls, necessary conditions in the form of recurrence sequences are obtained for dynamical systems with delayed in state. Similar results for the problem of dynamic systems with retarded control have been obtained in [10] only for singular controls with full degree of degeneration. Below, by considering a larger class of singular controls, proposing a modified version of the variations transform method [13] and matrix impulse method [11], we generalize all results of [10]. While treating the optimality of singular (in the classical sense) controls, we use the Legendre [[42], p. 413] polynomials as variations of control because such an approach is more convenient.

**1. Problem statement**. Consider the following optimal control problem with retarded control:

$$S(\boldsymbol{u}) = \phi\left(\mathbf{x}\left(t\_1\right)\right) \to \min\_{\boldsymbol{u}} \tag{1.1}$$

$$\dot{\mathbf{x}}(t) = f\left(\mathbf{x}(t), u(t), u(t-h), t\right), \ t \in I := \left[t\_0, t\_1\right], \mathbf{x}(t\_0) = \mathbf{x}\_0,\tag{1.2}$$

$$u\left(t\right) = \left.w\left(t\right)\right|,\ t \in I\_0 := \left[t\_0 - h, t\_0\right),\ u\left(t\right) \in U \subset \mathbb{R}',\ t \in I. \tag{1.3}$$

Here, is an open set in -dimensional Euclidean space , 1 = :: = −∞, + <sup>∞</sup> , is an -vector with phase coordinates, is an -vector of control actions, ℎ = const > 0, 0, 0, 1 are fixed points with <sup>1</sup> > 0 + ℎ; : , , , , : × × × , <sup>⋅</sup> + <sup>0</sup> − ℎ, 0 , are the given functions, where + <sup>0</sup> <sup>−</sup> ℎ, <sup>0</sup> , is a class of piecewise continuous (continuous from the right at discontinuity points and continuous from the left at the point <sup>0</sup> ) vector functions : <sup>0</sup> <sup>−</sup> ℎ, <sup>0</sup> .

The function <sup>⋅</sup> is said to be an admissible control if it belongs to + <sup>1</sup>, and satisfies the condition (1.3), where .

Note that if the function <sup>⋅</sup> and its partial derivative <sup>⋅</sup> are continuous on × × × , then, by using the method of successive approximations as in [21] it is easy to show that every admissible control ⋅ generates a unique absolutely continuous solution  ⋅ of the system (1.2), (1.3) where this solution will be assumed as defined everywhere on .

If the admissible control 0 , <sup>1</sup> is a solution of the problem (1.1)–(1.3), we will call it an optimal control, while the corresponding trajectory <sup>0</sup> , of the system (1.2)–(1.3) will be called an optimal trajectory. The pair 0 <sup>⋅</sup> , 0 <sup>⋅</sup> will be called an optimal process.

While studying the problem (1.1)–(1.3), we will also use the following assumptions:

(A1) let the functional : be twice continuously differentiable in the space ;

(A2) let the function <sup>⋅</sup> and its partial derivatives <sup>⋅</sup> , <sup>⋅</sup> be continuous in the space × × × , where = , , ;

(A3) let the function ⋅ be three times continuously differentiable in the totality of its arguments in the space × × × ;

(A4) let the inclusions ˙ ⋅ <sup>0</sup> <sup>−</sup> ℎ, <sup>0</sup> , and ˙ <sup>0</sup> <sup>⋅</sup> 1, hold for the derivatives ˙ ⋅ and ˙ <sup>0</sup> <sup>⋅</sup> , where , , is a class of piecewise continuous (continuous from the right and left at the points *a* and *b*, respectively) vector functions : , ;

(A5) let the function ⋅ be sufficiently smooth in the totality of its arguments in the space × × × ;

(A6) let the initial function <sup>⋅</sup> + <sup>0</sup> − ℎ, 0 , and admissible control 0 <sup>⋅</sup> be sufficiently piecewise smooth, that is, and

As is known, the concept of singular control was first introduced to the theory of optimal processes by Rozenoer [22] in 1959. First results on the necessary optimality conditions for singular controls have been obtained by Kelley [12] in the case of open set , and by Gabasov [11] in the case of arbitrary (in particular, closed) set , where *U* is a set of values of admissible controls. Afterward, Kelley and Gabasov's conditions as well as the methods for treating singular controls proposed in [11, 13] have been significantly generalized in [10, 14–19, 23– 41, etc.] to the cases of (1) controls with higher-order degeneration, (2) multidimensional controls, and (3) various classes of control systems. Considering all these cases, the methods in [11, 13] have been generalized in [17, 37] and for optimality of singular controls, necessary conditions in the form of recurrence sequences are obtained for dynamical systems with delayed in state. Similar results for the problem of dynamic systems with retarded control have been obtained in [10] only for singular controls with full degree of degeneration. Below, by considering a larger class of singular controls, proposing a modified version of the variations transform method [13] and matrix impulse method [11], we generalize all results of [10]. While treating the optimality of singular (in the classical sense) controls, we use the Legendre [[42], p. 413] polynomials as variations of control because such an approach is more convenient.

**1. Problem statement**. Consider the following optimal control problem with retarded control:

an -vector with phase coordinates, is an -vector of control actions, ℎ = const > 0, 0,

piecewise continuous (continuous from the right at discontinuity points and continuous from

Note that if the function <sup>⋅</sup> and its partial derivative <sup>⋅</sup> are continuous on × × × , then, by using the method of successive approximations as in [21] it is easy to show that every admissible control ⋅ generates a unique absolutely continuous solution

<sup>0</sup> <sup>−</sup> ℎ,

<sup>0</sup> .

<sup>0</sup> − ℎ, 0 , are the given functions, where +

<sup>0</sup> ) vector functions :

The function <sup>⋅</sup> is said to be an admissible control if it belongs to +

*xt f xt ut ut h t t I t t xt x* &() () () ( ) = ( , , ,, : , , , - Î= = ) [ ] 01 0 0 ( ) (1.2)

() (), : ,, , . 00 0 [ ) () *<sup>r</sup> u t wt t I t ht u t U R t I* = Î = - ÎÌ Î (1.3)

(1.1)

<sup>1</sup> > 0 + ℎ; : , , , , : × × × ,

<sup>0</sup> <sup>−</sup> ℎ,

, 1 = :: = −∞, + <sup>∞</sup> , is

<sup>0</sup> , is a class of

<sup>1</sup>, and satisfies the

() () ( ) <sup>1</sup> min*<sup>u</sup> Su xt* = ® j

Here, is an open set in -dimensional Euclidean space

0, 1 are fixed points with

196 Nonlinear Systems - Design, Analysis, Estimation and Control

condition (1.3), where .

<sup>⋅</sup> +

the left at the point

Especially note that more precise assumptions on the analytic properties of φ ⋅ , ⋅ , ⋅ , ⋅ will directly follow from the representation of optimality criteria obtained below.
