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Necessary conditions for a minimum in classical calculus of variation problems in the presence of various degenerations

In the paper, we offer a method for studying an extremal in the classical calculus of variation in the presence of various degenerations. This method is based on introduction of Weierstrass type variations characterized by a numerical parameter. To obtain more effective results, introduced variations are used in two forms: in the form of variations on the right with respect to the given point, and in the form of variations on the left with respect to the same point. The research is conducted under the assumption that along the considered extremal the Weierstrass condition and the Legendre condition degenerate, i.e. they are fulfilled as equalities at separate points or on some intervals. Two types of new necessary conditions are obtained: of equality type and of inequality type conditions for a strong and also a weak local minimum. Given specific examples and counterexample show that some of the necessary minimum conditions obtained in this article are strengthening and refining of the corresponding known results in this direction.


Introduction and Problem Statement
We consider the following vector problem of the classical calculus of variations: J (x (·)) = t1 t0 L (t, x (t) ,ẋ (t)) dt → min (1.1) where R n is n -dimensional Euclidean space, L (·) is a given function, x 0 , x 1 , t 0 , t 1 are the given points. The function L (t, x, y) : (a, b) × R n × R n → R := (−∞, +∞), called an integrant, is assumed to be continuously differentiable by a totality of variables, where (a, b) is an interval, and [t 0 , t 1 ] ⊂ (a, b). The sought-for function x (·) : [t 0 , t 1 ] =: I → R n is a piecewise-smooth vector-function, i.e. is continuous, and its derivative is continuous everywhere on I, except for a finitely many points τ i ∈ (t 0 , t 1 ) , i = 1, m, and at the points τ i the derivative functionẋ (·) has first kind discontinuities (at the points t 0 and t 1 the values of the derivative functionẋ (·) are finite on the right and left, respectively). Herewith, as a rule, the points τ i , i = 1, m are called angular points for the functions x (·). We denote the set of all piecewise-smooth functions on [t 0 , t 1 ] by KC 1 (I, R n ). We call the functions x (·) ∈ KC 1 (I, R n ) satisfying the boundary condition (1.2), admissible functions. Obviously, ifx (·) is a fixed admissible function, then for every θ ∈ [t 0 , t 1 ) (θ ∈ (t 0 , t 1 ] ) there exists a number α > 0 such thatx (·) is continuously differentiable on the semi-interval [θ, θ + α) ⊂ I ((θ − α, θ] ⊂ I). We call this statement the property P (θ+;x (·) , α) (P (θ−;x (·) , α)) for the functionx (·), and we will use it in the future.
Following [3,21,22], we give a local modification of necessary condition for a minimum (1.5). Letx (·) be a weak local minimum in problem (1.1), (1.2). Then there exists a number δ > 0 at which the following inequalities are fulfilled: E L (t, ξ) ≥ 0, ∀t ∈ (t 0 , t 1 ) \ {τ } , E L (t±, ξ) ≥ 0, ∀t ∈ {τ } , E L (t 0 +, ξ) ≥ 0, E L (t 1 −, ξ) ≥ 0, ∀ξ ∈ B δ (0) . (1.8) Here and in what follows, the symbol B δ (0) is a closed ball of radius δ centered at the point 0 ∈ R n . It is clear that the Legendre condition follows from necessary condition (1.8) as a corollary. We formulate this condition. Let the admissible functionx (·) be a weak local minimum in problem (1.1), (1.2), and in addition suppose that the integrant L (t, x,ẋ) is twice differentiable with respect to the variableẋ at the points of the set {(t,x (t) ,ẋ (t)) : t ∈ I}. Then for all ξ ∈ R n the following inequalities hold: (1.9) The admissible function that satisfies the Euler equation, i.e. equality (1.3) is called an extremal in problem (1.1), (1.2). In the classical calculus of variation (see, i.e., [4,15]), as the main goal, the extremal for a minimum was studied, and various necessary and also sufficient conditions were obtained. Recall that (see [4, 15, 33 and etc.]) in the classical calculus of variations a number of known necessary and also sufficient conditions for a minimum remain powerless in the case when at some point t = θ ∈ I for the vector ξ = η ∈ R n \ {0} at least one inequality from (1.5), (1.8) and (1.9) is fulfilled as an equality, i.e. corresponding necessary condition degenerates at the point θ or at the point θ on the right or left. Therefore, the study of problem (1.1), (1.2) in such degenerated cases is of theoretical and practical interest today.
It should be noted that similar problems in theory of optimal control, starting with the work of Kelly [13], in terms of singular controls were thoroughly studied by many authors and a hundreds of papers and monographs containing a number of important results were published [2, 7-10, 13, 14, 18-21, 24-28, 32, and etc.] . Application of the obtained results on singular controls either are ineffective or some their justified modifications are required in degenerated cases when solving problem (1.1), (1.2). Although problem (1.1), (1.2) is a special case of a terminal optimal control problem with equality type phase constraints, its study as an independent problem, allows to get more effective results not being corollaries of theorems proved in the theory of optimal control. Confirmation of the last sentence was shown for example in [6, p.33; 22, 23] and in the present paper in the presence of various degenerations.
In this paper we offer a method for studying an extremum in problem (1.1), (1.2) involving degenerations that are based on the introduction of various forms of Weierstrass-type variations characterized by a numerical parameter. Different necessary conditions for a strong and weak local minimum were obtained. It should be noted that some results of this paper are sharpening and refinement of corresponding statements of the paper [3].
The structure of this paper is set out as follows. In the second section we obtain increment formulas of the functional for problem (1.1), (1.2) under various assumptions on the smoothness of the integrant L (·) and the extremal studied for a minimum (see (2.15), (2.16), (2.31), (2.32), (2.35), (2.36)).
In the third and fourth sections, based on the increment formula of the functional obtained in the second section, we introduce necessary conditions for a strong and a weak local minimum in the presence of various degenerations at separate points and on the interval.
In the last section, by means of special examples we discuss the results obtained in the third and fourth sections.
In this section, by means of special variations we obtain increment formulas of functional under various assumptions on the smoothness of the integrant L (·) and on the considered extremal of problem (1.1), (1.2). Note that these formulas have independent meaning and are the basis for the proof of the main results of this work.
Proof. At first we prove part (i) of the proposition, i.e. the validity of the equality (2.13).
Similar to (2.13) we give the proof of part (ii) of proposition 2.1.
Based on the technique for proving Proposition 2.  .27), under weak assumptions on the smoothness of functions L (·) , Lẋ (·) andx (·) as a corollary of Proposition 2.1, it is easy to arrive at the following statement.
Prove part (ii) of Theorem 3.2, i.e. the validity of inequality (3.14). Considering the assumption of part (ii) of Theorem 3.2, it is easy to get the validity of equality (3.9), that was obtained when proving part (ii) of Theorem 3.1. By virtue of notations (2.11), (2.12) and (2.15), equality (3.9) takes a new form W L (θ,λ, η) = 0. Consequently, we confirm that all assumptions of part (i) of Theorem 3.2 are fulfilled, and the functionx (·) is triple differentiable at the point θ. Therefore, the proof of part (ii) of Theorem 3.2 follows from the statement of part (i) of Theorem 3.2. Theorem 3.2 is completely proved.

Necessary conditions for a minimum in the presence of various degenerations on the interval.
In frequent cases, the Weierstrass condition and also the Legendre condition degenerate on some interval. Such a situation as an independent problem is studied in this section. It is important to note that the research of such cases allows to get analogues of statements (3.6), (3.14) and ( We prove the following theorems. Theorem 4.1. Let the integrant L (·) be continuously differentiable in totality of variables, furthermore, the admissible functionx (·) be an extremal in problem (1.1), (1.2) and along it for the vectors η = 0 and λ − 1 −1λ η, whereλ ∈ (0, 1), the Weierstrass condition degenerates at any point of the interval (t 0 ,t 1 ) ⊂ [t 0 , t 1 ], i.e. the following equalities hold: where E L (·) is determined from (1.6) and the interval (t 0 ,t 1 ) does not contain any angular point of function x (·). Then: (i) if the extremalx (·) is a strong local minimum in problem (1.1), (1.2), then the following equality is fulfilled: where ∆L x (·) is determined by (2.10); (ii) if the extremalx (·) is a weak local minimum in problem (1.1), (1.2), then there exists a number δ > 0 such that for every point η, λ − 1 −1λ η,λ ∈ B δ (0) × B δ (0) × (0, 1) satisfying condition (4.1), equality (4.2) is fulfilled.
The proof of part (ii) of Theorem 4.2, allowing for estimations (4.8), (4.9) and definition of a weak local minimum of the functionx (·), follows from part (i) of Theorem 4.2. By the same token, Theorem 4.2 is completely proved.
Further, unlike the necessary condition for a minimum (4.10), the Kelly condition is obtained only by means of the second variation of the functional when solving a singular optimal control problem.
So, based on what has been said, allowing for below given Example 5.1 we can say that necessary condition for a minimum (4.10) is one reinforced variant of the Kelly condition in problem (1.1), (1.2).
Carrying out similar reasonings, we can confirm that in problem (1.1), (1.2) the minimum condition (4.2), and also minimum condition (4.14) are reinforced variant of the known equality type necessary condition obtained in [8].
We also note that, following the below given Example 5.1, we come to the conclusion that necessary condition (4.10) is not a corollary of the result of the paper [7], and it is more constructive compared to [7], more exactly, no need to solve the matrix differential equation. The similar conclusion refers also to necessary conditions obtained in Section 3 when comparing with appropriate results of the monography [8]. Simple comparison and below given Examples 5.2 and 5.3 show that Theorem 3.3, 3.6 and 3.7 are sharpening and refinement of appropriate results of the paper [3].
It is important to note that degeneration of the form (3.21), generally speaking does not follow from degeneration of the form (3.5), and vice versa (see Example 5.2 and 5.4). Therefore, based on this proposition and formulation of the proved theorems in Section 3 and 4, it is easy to confirm that every condition for a minimum obtained by us has its own application area.
Naturally, it is important to compare the necessary condition for a minimum (4.10) with the known necessary conditions for optimality of singular controls, for example, with the Kelly condition and the optimality condition in the paper [7].
For that we consider the following problem equivalent to problem (5.3):