Conditions for Optimality of Singular Controls in Dynamic Systems with Retarded Control

In this chapter, we consider an optimal control problem with retarded control and study a larger class of singular (in the classical sense) controls. For the optimality of singular controls, the various necessary conditions in the recurrent forms are obtained. These conditions contain also the analogs of Kelly, Koppa-Mayer, Gabasov, and equality-type conditions. While proving the main results, the Legendre polynomials are used as variations of control.


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][12][13][14][15][16][17][18] are not directly applicable to the singular controls in dynamical systems with aftereffect (see [9,[14][15][16][17][18][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]).
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.
The function ⋅ is said to be an admissible control if it belongs to + 1 , and satisfies the condition (1.3), where .
Note that if the function ⋅ and its partial derivative ⋅ 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 , ∈ 1 is a solution of the problem (1.1)-(1.3),we will call it an optimal control, while the corresponding trajectory 0 , ∈ of the system (1.2)-(1.3)will be called an optimal trajectory.The pair 0 ⋅ , 0 ⋅ 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 ⋅ and its partial derivatives ⋅ , ⋅ 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 ˙⋅ ∈ 0 − ℎ, 0 , and ˙0 ⋅ ∈ 1 , hold for the derivatives ˙⋅ and ˙0 ⋅ , 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 ⋅ ∈ + 0 − ℎ, 0 , and admissible control 0 ⋅ be sufficiently piecewise smooth, that is, and Especially note that more precise assumptions on the analytic properties of φ ⋅ , ⋅ , ⋅ , ⋅ will directly follow from the representation of optimality criteria obtained below.
where 1 0 ; ⋅ and 2 0 ; ⋅ are, respectively, the first and the second variations of the functional at the point 0 ⋅ ; , , , ∈ , , ; ⋅ is the variation of the control 0 ⋅ , while ⋅ is the corresponding variation of the trajectory 0 , ∈ , which ⋅ is the solution of the system where : = 0 , 0 , 0 − ℎ , , ∈ and ∈ , , , while the vector function is the solution of the conjugate system Below, we consider that the following conditions are fulfilled: and for (2.6) If 0 ⋅ , 0 ⋅ 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: Here, ⋅ 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).
uu vv

H t t H t h r r t I
In this case, the set is called a singular plot for an admissible control 0 ⋅ .The main goal of this chapter is to study such singular controls.

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 ⋅ the equality (2.9) hold.Use Proposition 2.1.Let the variation = 0 , ∈ + 1 , have the form: \ , , where ∈ 0 , ∈ 0 , 1 , and the number 0 was defined in Proposition 2.1.

t H t x t dt x t H t x t h H t h dt
where , ∈ is the solution of the system (2.11).
By the Cauchy formula, we have where , , , ∈ × is the solution of the system , = 0, > , , = ( is a unit × matrix).
As (A2) and 0 ⋅ ∈ + 1 , are fulfilled, then by (3.1) and (3.4) and for all ∈ 0 , 1 , from q q l q q x q q q e el q q x e q e q d el q q x q c q l q q x q q q e e l q q c q l q q x e q e ì Î ï - where ⋅ is the characteristic function of the set 0 , 1 − ℎ ; / 0, as 0.
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 x q l q l q q c q q l q l q q c q q l q l q q x e é D = ë T x q l q q c q e q l q q l q q x e e x q q c q q l q q q c q q q x e Following [10,14,17], we consider the matrix functions where ⋅ , ⋅ is the solution of the system (3.4).
Proposition 3.1.Let conditions (A1) and (A2) be fulfilled, and the admissible control 0 ⋅ = ⋅ , ⋅ 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:

T S u u M p p M p p h M p p h h o
d d e x q q c q q q c q q q x e e e × = -+ + where the number 0 was defined above (see Proposition 2.1), ⋅ 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).

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 ⋅ 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 ⋅ : , : 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 h t u h t t h u h
The following properties are obvious: (1) \* is a finite set and 1 ∈ *; (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 ˙0 ⋅ , ˙0 ⋅ 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 ⋅ = 0 ⋅ , ⋅ satisfies additionally the following conditions as well: where * = min 0 , , and 0 , were defined above.
Make a passage from the variation = 0 , , ∈ 1 , satisfying (4.2), to a new Obvious, ( ) ( ) Transform the variation of the trajectory as well: in place of , ∈ , consider the function

t g p t p t g p t p t h t I
where 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 x t f t x t g p t p t g p t p t h f t q t f t q t h t t where Now, let us write down the second variation (2.12) in terms of new variables.By (4.4) from (4.5), we have 1 = 1 1 .According to this property and (4.2)-(4.6),for any ∈ 0, * the second variation (2.12), after simple reasoning takes a new form ( ) ( ) where x

T T T xp xp p t g p t H t g p t h H t h p t dt
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

t H t f t H t dt d 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 q e q q e q q e q d d d 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.

P p q t G p t f t g p t H t P p q t H t t I i P p q t G p t f t g p t H t P p q t H t t I
...,  where ⋅ and ⋅ are determined by (3.4) and (3.9), respectively.
Similar to *, we introduce the set * * when assumption (A6) is fulfilled: the admissible control 0 ⋅ is sufficiently smooth or sufficiently smooth from the right at the points θ and − ℎ ∪ ∈ 0 , 1 − ℎ : the admissible control 0 ⋅ is sufficiently smooth or sufficiently smooth

}
from the right at the points and .h q q ± (4.28) The following obvious properties hold: (1) \ * * is a finite set, and 1 ∈ * *, 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 * *.

t L p t L p t h p t dt p t P p q t P p q t h q t p t Q p t Q p t h p t p t P p q t P p q t h q t q t H t H t h q t dt
where ∈ * *, ∈ 0, * * (the number * * was defined above), , ∈ is the solution of the system ... ,

x t g p t p t g p t p t h f t q t f t q t h t t x t t t p t q t t t h k
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 = 0 , , ∈ 1 satisfies the conditions (4.29) and (4.30).
Then by assumption the following representation is valid:

S u p p q p t L p t L p t h p t p t P p q t P p q t h q t p t Q p t Q p t h p t p t P p q t P p q t h q t q t H t H t h q t dt
where ⋅ , , ⋅ , , ⋅ , ⋅ , ⋅ , ∈ , , = 0, 1, ... are defined by (4.23)-(4.26),and − 1 , ∈ is the solution of the system: Nonlinear Systems -Design, Analysis, Estimation and Control ( )

t x t g p t p t g p t p t h f t q t f t q t h x t t t p t q t t t h k
Apply the modified variant of variations transformations method [13] to the system for − 1 , ∈ 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: , . x

t H t x t dt x t H t g p t x t h H t h g p t h p t p t g p t H t g p t g p t h H t h g p t h p t dt
q q e q q e q d d where

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 t g p t H t x t h p t g p t h H t h q t dt
x t H q e q q e q q e q d d d Then, applying the method of integration by parts, we have

t dt d x t h f t h G p t h G p t h p t dt p t g p t G p t g p t h G p t h p t p t G p t f t G p t h f t h q t dt p t
q e q d d d

Optimality conditions
Based on Propositions 3.1, 4.1, and 4.2, we prove the following theorem.
Then for the optimality of the admissible control 0 ⋅ , it is necessary that the relations ..., ,

M p p t M p p h M p p h h i k
x q q c q q c q q q x + + (5.5) be fulfilled for all ∈ * *, ∈ 0 and ∈ 1 .
We first prove the validity of (5.4) for k=0.q q e e e e d q q e q b q q e e e e d q q e ì ae -ö
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 ⋅ , we have e ab q q t t e e ab q q e e e + + -- 0 are the elements of the matrix 0 + 0 + ℎ .
To prove statement (5.5) for = 0, under the conditions (4.2) and (4.3), we write down the vector components of the variation ⋅ = 0 ⋅ , ⋅ in the following form: ( ) ( ) t p t t I s l ds t q t t I q q x q q e e d q q e e e q h q q e e d q q e e e ì ae - According to (4.2), (4.3), (4.7), (4.8), and (5.7), it is easy to prove that ( 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 , ∈

S u u p t L p t t L p t h p t p t P p q t t P p q t h q t q t H t t H t h q t dt o L p L p h P p q P p q h
d e e x q c q q x x q c q q h h q c q q h t t , 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 k , , ≥ 0 (see (5.5) for the case k-1) is valid for all ∈ * *, ∈ 0 and ∈ 1 . Hence, taking into account (5.1), we have From this inequality, we easily get that , + , + ℎ = 0, that is, we get the validity of optimality condition (5.2) for = .

G p t x t h G p t h p t dt
q q e e d q q e e e -ì ae 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 q q e q q e q q q l q q x t t e q q e l q q x t t e q e q d l q q x t t c q l q q x t t e q q e l q q c q l q q x t t e q As , ∈ −1, 1 is the -th Legendre polynomial, then it is easy to get ( ) ( ) ( ) ( ) ( ) q e e t t t t t t t t 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 q q d j d x q l q j l q q c q q l q j l q q c q q l q j l q q x t t e x q l q l q q c q q + + é = ë

t h t g p h g p h h t H t h t g p h dt c d o g p G p g p h G p h g p h G p h c d o
q e q q e q l q l q q c q q l q l q q x t t e x q q c q q l q q q c q q q x t t e (5.17) Substitute (5.15)-(5.17) in (5.9).Then by (3.9), (4.27), and (5.14), we have .
x q q c q q q e c q q q x t t t e q x + + - 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 Choose the variation = 0 , , ∈ 1 in the following way: ..., \ , , , 1, 2,..., , , q q e e d q q e t dt e e ab q q t t t e , and , are the elements of the matrix + + ℎ .
At last, let us prove optimality condition (5.5).Choose the variation = 0 , , ∈ 1 in the following way: Obviously, the variation = 0 , , ∈ 1 defined in (5.22)   + + ℎ = 0, ∈ * * (see (5.4)), from (5.18), we get x q c q q x x q c q q h h q c q q h e q c q q x q q c q q q c q q q x q c q q + + = =

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.