Especially note that more precise assumptions on the analytic properties of will directly follow from the representation of optimality criteria obtained below.
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 be some admissible process. If the process is optimal, then, by using the known technique (see, e.g., [27, p. 51]), it is easy to get
where and are, respectively, the first and the second variations of the functional at the point ; , , ; is the variation of the control , while is the corresponding variation of the trajectory , which is the solution of the system
where , and , while the vector function is the solution of the conjugate system
Below, we consider that the following conditions are fulfilled:
If 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:
for all , if optimal control is continuous at the points . Here, is the characteristic function of the set .
It should be noted that the optimality condition (c) is the corollary of conditions (a) and (b).
Definition 2.1. An admissible control , satisfying conditions (2.7) and (2.8), is called singular (in classical sense) if
In this case, the set is called a singular plot for an admissible control . The main goal of this chapter is to study such singular controls.
Let , where . Without loss of generality [, p. 138], we assume that the singularity to the control is delivered by a vector component , that is,
Note that the general inequality (2.8) implies the equality-type optimality condition for a singular (in classical sense) control :
Proposition 2.1. Let assumptions (A1) and (A2) be fulfilled, the admissible control be singular (in the classical sense) and condition (2.9) be fulfilled along it. Let also the variations be non-zero only on , where and , with the number be such that (1) if , then and (2) if , then . Then, (a) the variational system (2.4) becomes
(b) the following representation is valid for the second variation (2.3):
Proof. To prove (a), it suffices to consider the definition of the variation 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 .
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 the equality (2.9) hold. Use Proposition 2.1. Let the variation have the form:
where , and the number was defined in Proposition 2.1.
Along the singular control satisfying condition (2.9), taking into account (3.1), formula (2.12) takes the form:
where is the solution of the system (2.11).
By the Cauchy formula, we have
where is the solution of the system
(is a unit matrix).
As (A2) and are fulfilled, then by (3.1) and (3.4) and for all , from (3.3) we get
where is the characteristic function of the set ; , as .
By (2.6) and (3.5) and taking into account and for , we calculate separate terms of (3.2). As a result, after simple reasoning, we get
Following [10, 14, 17], we consider the matrix functions
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 , for , we get the validity of the following statement.
Proposition 3.1. Let conditions (A1) and (A2) be fulfilled, and the admissible control be singular (in the classic sense) and the condition (2.9) be fulfilled along it. Then, for each and for all the following expansion is valid:
where the number was defined above (see Proposition 2.1), is the characteristic function of the set and matrix functions , , that are defined by (3.10).
4. Transformation of the second variation of the functional by means of modified variant of variations transformation method
4.1. Expansion of the second variation in Kelley-type variation (first-order transformation)
Let 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 .
Introduce the following set dependent on the admissible control :
The following properties are obvious: (1) is a finite set and ; (2) for every , there exists a sufficiently small number such that ; and (3) by (1.2), (1.3), and (2.5), the derivatives 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 satisfies additionally the following conditions as well:
where , and were defined above.
Make a passage from the variation , satisfying (4.2), to a new variation , where
Transform the variation of the trajectory as well: in place of , consider the function :
As assumptions (A3) and (A4) are fulfilled, then by virtue of property of the set we easily have: the function is continuous and .
By direct differentiation, allowing for (A3), (A4) and (2.11), (4.3), (4.4) from (4.5) we obtain that is the solution of the system
Now, let us write down the second variation (2.12) in terms of new variables. By (4.4) from (4.5), we have . According to this property and (4.2)–(4.6), for any the second variation (2.12), after simple reasoning takes a new form
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 by integration by parts. Then, we have
where is defined by (2.19),
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.
Proposition 4.1. Let assumptions (A1), (A3), (A4), and conditions (2.6) be fulfilled. Also, let the functions be defined by (4.6), (4.9), and (4.15), respectively, and be the solution of the system (4.7) and (4.8). Then along the singular control , satisfying condition (2.9), and on the variations satisfying (4.2), (4.3), the following representation (first-order transformation) is valid:
where was defined above (see (4.2)),
4.2. Higher-order transformation
Let be some process, where is a singular control satisfying condition (2.9), and assumptions (A1), (A5), and (A6) be fulfilled. Introduce the matrix functions calculated along the process and determined by the following recurrent formulas:
Furthermore, similar to (4.15), (4.20), (4.21), and (3.10), consider the functions
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 is sufficiently smooth or sufficiently smooth from the right at the points θ and the admissible control is sufficiently smooth or sufficiently smooth
The following obvious properties hold: (1) is a finite set, and , also ; (2) for every there exists a sufficiently small number , 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 .
Let us consider a variation that in addition satisfies the following conditions as well:
(were defined above).
According to (4.30), we have
The following statement is valid.
Proposition 4.2. Let assumptions (A1), (A5), (A6), and condition (2.6) be fulfilled. Furthermore, let the functions and , where be defined by (4.22)–(4.26), and the set be defined by (4.28). Then along the singular control , satisfying condition (2.9), and on the variations satisfying (4.29) and (4.30), the following representation (-th order transformation, where ) is valid:
where , (the number was defined above), is the solution of the system
Proof. We carry out the proof of Proposition 4.2 by induction. For , 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 inclusively, . We prove the validity of representation (4.32) for the case . Let the variation , satisfies the conditions (4.29) and (4.30). Then by assumption the following representation is valid:
where are defined by (4.23)-(4.26), and is the solution of the system:
Apply the modified variant of variations transformations method  to the system for 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:
According to (4.22), (4.30), (4.31), and (4.39) from (4.40) by direct differentiation, we get the system (4.35) for . Furthermore, as , then by (4.40) we get . Taking into account this equality and by (4.29), (4.30), and (4.40) in (4.37), let us transform the representation (4.36) into new variables . Then,
where is determined by formula (4.38) as well as by (4.22), (4.29), (4.30), (4.35), (2.6) are calculated in the following way:
Taking into account (A5), (A6), (4.29), (4.30), (4.35), and the properties of the set , let us calculate . Then, applying the method of integration by parts, we have
At first, we substitute the last expression in (4.43), and then (4.42)–(4.44) in (4.41). Then by (4.23)–(4.26), (4.33), (4.34), and (4.38), it is easy to get representation (4.32). Consequently, we get the proof for . This completes the proof of Proposition 4.2.
5. Optimality conditions
Based on Propositions 3.1, 4.1, and 4.2, we prove the following theorem.
Theorem 5.1. Let conditions (A1), (A5), and (A6) be fulfilled, and the matrix functions , be defined as in (4.24)–(4.27). Let also the set be defined as in (4.28) and along the singular (in the classical sense) control the following equalities be fulfilled:
where is the characteristic function of the set .
Then for the optimality of the admissible control , it is necessary that the relations
be fulfilled for all and .
Proof. Let be an optimal control. We will prove the theorem by induction. Let , that is, . Then, according to (4.24) and (2.10) we get the proof of optimality condition (5.2) for . The proof of optimality condition (5.3) for directly follows from (3.11) allowing for (2.1) (see Proposition 3.1). Now, based on Proposition 4.1 prove the optimality conditions (5.4) and (5.5) for .
We first prove the validity of (5.4) for k=0.
where are arbitrary fixed points of the set and is the -th coordinate of the vector ; and are arbitrary fixed points, the functions are the Legendre polynomials.
It is clear that the variation , defined by (5.6) satisfies the condition (4.2) and, according to (5.6) the function , defined by (4.3) is of order , and the solution of the system (4.7), (4.8) is of order . Also, according to (4.15) it is easy to see that for every the matrix 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 , we have
where are the elements of the matrix .
Then, we conclude from the arbitrariness of and that the skew-symmetric matrix is also symmetric. Consequently, for every we have . This completes the proof of the optimality condition (5.4) for .
To prove statement (5.5) for , under the conditions (4.2) and (4.3), we write down the vector components of the variation in the following form:
where is a Legendre polynomial, are arbitrary fixed points.
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 :
Hence, taking into account the arbitrariness of , and , we easily get the validity of the optimality condition (5.5) for .
Now suppose that all the statements of Theorem 5.1 are valid for as well. Prove statements (5.2)–(5.5), for . By assumption, the inequalityk (see (5.5) for the case k-1) is valid for all and . Hence, taking into account (5.1), we have
From this inequality, we easily get that , that is, we get the validity of optimality condition (5.2) for .
Now, prove the validity of condition (5.3) for . In formula (4.32), we put
where is the -th Legendre polynomial which the number is defined above (see (4.30)) and .
Obviously, conditions (4.29) and (4.30) are fulfilled for variation (5.8).
As the conditions , , and , , are fulfilled, then by (4.33), (4.34), and (5.8), formula (4.32) takes the form:
Here, by (4.31), (4.35), (5.8), and the Cauchy formula, and are determined as follows:
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
As is the -th Legendre polynomial, then it is easy to get
Taking into account (5.12)–(5.16) and the fact that for we calculate separately each terms of (5.9). As a result, after simple reasoning we get
Substitute (5.15)–(5.17) in (5.9). Then by (3.9), (4.27), and (5.14), we have
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 -th order transformation. As the equalities
taking into account (2.6), we have
where are determined similarly to (4.33) by changing the index by , and is the solution of the system (similar to (4.35))
Choose the variation in the following way:
where , is a Legendre polynomials .
Obviously, by (5.20), the variation defined in (5.20) satisfies conditions (4.29), (4.30) for . Taking into account (5.20), by means of (4.30), (4.31), (4.33), and (5.19), it is easy to calculate
By (5.20) and (5.21), from (5.18) we get
where is determined in (4.25).
Hence, taking into account the skew symmetry of the matrix and the properties of the set , and also by (2.1), (4.30), and (5.20), we have
where , and are the elements of the matrix .
From the last inequality, by arbitrariness of , and it follows that for each , the skew-symmetric matrix is also symmetric. Consequently, , that is, condition (5.4) is proved for i=k.
At last, let us prove optimality condition (5.5). Choose the variation in the following way:
where is the -th Legendre polynomial, , , , .
Obviously, the variation defined in (5.22) satisfies the conditions (4.29) and (4.30) for
By (4.30), (4.31), (5.12), (5.19), (5.22), and (5.23), the following relations hold:
Taking into account (5.23)–(5.25) and validity of the equality (see (5.4)), from (5.18), we get
From this expansion, taking into account (2.1), it follows inequality (5.5).
Therefore, Theorem 5.1 is completely proved.
Corollary 5.1. Let all the conditions of Theorem 5.1 be fulfilled. Let, in addition, the following equalities hold:
Then, for optimality of the singular control , it is necessary that the relations
be fulfilled for all .
The proof of the corollary follows immediately from Theorem 5.1.
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 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 it is sufficient that assumptions (A1), (A3), and (A4) be fulfilled.
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 are valid for all and .
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 , are actually the analogs of the equality-type conditions and the Kelly  condition, while optimality condition (5.3) is the analog of the Gabasov  condition for the considered problem (1.1)–(1.3); (2) optimality condition (5.5), for is the analog of the Koppa-Mayer  condition. Conditions (5.3)–(5.5) were obtained in  only for singular controls with complete degree of degeneracy, that is, for the case when (see Definition 2.1).
We also note that (1) the analog of the Kelly condition and equality-type condition was obtained in  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 (see Definition 2.1) becomes ineffective, though it is effective in the general case for . To illustrate the rich content of condition (5.4), we consider a concrete example:
Example. x˙1(t)=u2(t)+u12(t−1)−u3(t−1), ,
, , , , , , .
Check for optimality of the control . 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
, where , , where , ; ; , where , ; , , , , , , , , , , where .
Hence, we have the following: (1) admissible control is singular (in the sense of Definition 2.1) and singularity to it is delivered by the vector component , that is, equality (5.1) is fulfilled only ; (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 is an optimal or not. However, optimality condition (5.4) for is not fulfilled (, ), that is, by condition (5.4) (for ) we conclude that the control cannot be optimal.
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