Construction of a functional by a given second-order Ito stochastic equation

1 Introduction: Problem statement

The theory of inverse problems of dynamics, which originated in the seminal work of Yerugin [1], has been quite fully developed for ordinary differential equations (ODEs) in [2–10]. In a study by Yerugin [1], a set of ODEs is constructed by a given integral curve. In [2,3], Galiullin proposed a classification of the main types of inverse problems of dynamics in the class of ODEs and developed general methods for their solution. In [11–15], the solvability conditions were obtained for inverse problems of dynamics in the class of stochastic Ito differential equations.

In recent decades, the increased interest in the Helmholtz problem [16] has marked a new stage in the study of inverse problems for differential systems. Inverse problems of dynamics related to the Helmholtz problem are described in a study by Galiullin [17].

The classical Helmholtz problem is the problem of constructing equivalent differential equations in the Lagrangian form by given second-order ODEs. Mayer [18] and Suslov [19] independently showed that the classical Helmholtz conditions are not only necessary but also sufficient conditions for the transition from Newton equations to equivalent Lagrangian equations.

First, the solution of the Helmholtz problem in a certain broader class of differential equations allows us to extend to this class the well-developed mathematical methods of classical mechanics. Second, since the Helmholtz problem is also an inverse problem of the calculus of variations, it is often useful to replace the problem of finding the solution of a differential equation with the equivalent problem of finding the extremals of the functional constructed from this equation using numerical methods. Speaking of the Helmholtz problem, we should mention the two-volume monograph by Santilli [20,21]. This monograph occupies a special place in the study of the Helmholtz problem for the variety of aspects and the completeness of the presentation of this problem. It is devoted to the problem of the representation of second-order ODEs in the form of Lagrange, Hamilton, and Birkhoff equations. In the studies by Budochkina and Savchin [22–24], the methods for solving the Helmholtz problem are extended to the class of partial differential equations (PDEs).

Further research into the Helmholtz problem is presented in by Santilli [20,21] and Filippov et al. [25]. These works contain the authors’ own investigations, mainly in the class of ODEs and PDEs, and give a historical review of the development and generalization of the above problem.

The Helmholtz problem in the class of stochastic equations can be formally divided into two interrelated subproblems.

Note that [26,27] solved mainly Problem 1 (the first part of the stochastic Helmholtz problem). In a study by Tleubergenov and Azhymbaev [26], Problem was is solved in the class of stochastic differential equations equivalent almost surely by the method of additional variables. In a study by Tleubergenov et al. [27], the solvability of Problem 1 was studied in the class of stochastic differential equations equivalent in distribution.

In what follows, we assume that the vector function F ( z , t ) and the matrix σ ( z , t ) , z = ( x T , x ˙ T ) T , satisfy the following conditions:

1. F ( z , t ) and σ ( z , t ) are continuous in t and satisfy the Lipschitz condition in z , i.e.,

   ∥ σ ( z ′ , t ) − σ ( z ″ , t ) ∥ 2 + ∥ F ( z ′ , t ) − F ( z ″ , t ) ∥ 2 ≤ L ( 1 + ∣ z ′ − z ″ ∣ 2 ) for all z ′ , z ″ ∈ R 2 n ;

2. The linear growth condition

   ∥ σ ( z , t ) ∥ 2 + ∥ F ( z , t ) ∥ 2 ≤ L ( 1 + ∣ z ∣ 2 )

   is satisfied for all z ∈ R 2 n .

Let ( Ω , U , P ) be a probability space with flow { U t } . Here, { ξ 1 ( t , ω ) , … , ξ m ( t , ω ) } , ω ∈ Ω , is a system of independent Wiener processes [28].

This article is devoted to the solution of Problem 2 (the second part of the stochastic Helmholtz problem).

One of the methods, widely used in the class of Ito stochastic differential equations is the so-called method of moment functions (or statistical theory of dynamical systems) (see, e.g., [28]). The essence of this method is that the study of stochastic differential equations, and a random process x ( t , ω ) is reduced to the study of a system of ODEs with respect to moments of different orders E x r ( t , ω ) , r = 1 , 2 , … .

2 The Hamilton principle in the presence of random perturbations

Let us apply the method of moment functions to extend the Hamilton principle to the class of second-order Ito stochastic equations.

We consider a mechanical system characterized by the Lagrange function L = L ( q , q ˙ , t ) , where q ν and q ˙ ν ( ν = 1 , n ¯ ) are the generalized coordinates and generalized velocities, respectively. Suppose that the system is subject to generalized random perturbing forces Q ν , so that the equations of motion of the given system are of the form

We assume that the random perturbing forces Q ν admit the representation Q ν = σ ν j ( q , q ˙ , t ) ξ ˙ 1 ∕ 2 j , ( j = 1 , m ¯ ) , where ξ ˙ 1 ∕ 2 j is a white noise in the sense of Stratonovich [29].

Let us apply the operation of mathematical expectation [28] to equation (3). Taking into account the property of a white noise in the sense of Stratonovich, E [ σ ν j ξ ˙ 1 ∕ 2 j ] = 1 2 E ∂ σ ν j ∂ q ˙ i σ i j , we arrive at the equation in E L of the form

We consider the following cases.

The functions σ ν j = σ ν j ( q , t ) are independent of generalized velocities. Hence, ∂ σ ν j ∂ q ˙ i ≡ 0 , and equation (4) takes the form

The functions σ ν , j = σ ν j ( q , q ˙ , t ) depend on generalized velocities and satisfy the following relations:

These conditions are derived from the work of Santilli [20, pp. 194–195]. This work, in particular, presents a study of the problem of reducing a first-order ODE to the equation of the Lagrange form

and derives the Helmholtz-type conditions, which are equivalent to relations (6) with respect to the functions F ν = 1 2 ∂ σ ν j ∂ q ˙ i σ i j .

Conditions in equation (6) ensure, following [20], the existence of a function Θ of the form

Thus, in the case when σ ν j = σ ν j ( q , q ˙ , t ) satisfies equation (6), equation (4) takes the form

or when E L ˜ = E L − Θ , we obtain

If we assume that the Lagrange function is a second-degree function with respect to the generalized velocities, then from the identity ∂ 3 L ∂ q ˙ ν ∂ q ˙ i ∂ q ˙ k ≡ 0 , we have d ∘ ( ⋅ ) d t ≡ d ( ⋅ ) d t . Hence, in this case, equation (11) can be written in the form of equation (5).

We now consider the question: what equation of the Lagrangian structure is satisfied by the second-order initial momentum Γ L = E L 2 , if the original equation in L is given in the form of equation (3)?

Let us expand the expression d d t ∂ L 2 ∂ q ˙ ν − ∂ L 2 ∂ q ν . Due to the fact that the white noise in the original equation (3) is given in the Stratonovich form [29], we apply the “normal” rule for differentiation of composite functions:

Since d L d t = ∂ L ∂ t + ∂ L ∂ q μ q ˙ μ + ∂ L ∂ q ˙ μ q ¨ μ , we obtain

where Φ ν = 2 ∂ L ∂ t + ∂ L ∂ q μ q ˙ μ + ∂ L ∂ q ˙ μ q ¨ μ ∂ L ∂ q ˙ ν .

Let us rewrite the resulting equation in the form

where a ν k = 2 ∂ L ∂ q ˙ ν ∂ L ∂ q ˙ k and b ν = 2 ∂ L ∂ q ˙ ν ∂ L ∂ t + ∂ L ∂ q μ q ˙ μ , and consider the possibility of the following representation:

If we assume that the function L is such that b ν and a ν k satisfy the Helmholtz conditions [20, p. 65]

then there exists a function, Θ 1 = Θ 1 ( q , q ˙ , t ) which admits the representation of equation (13) in the form

where ρ ν j = 2 L σ ν j .

We then apply the operation of mathematical expectation to equation (15), and due to E [ ρ ν j ξ ˙ 1 ∕ 2 j ] = 2 E L ∂ ( L σ ν j ) ∂ q ˙ i σ i j (see [20]), we obtain

Suppose that the right-hand side of equation (16) satisfies the Helmholtz conditions that are equivalent to conditions in equation (6), and for the following first-order equation:

these conditions are equivalent to relations

Then, according to Santilli [20], there exists

and hence, under conditions in equation (17), equation (16) is equivalent to the following equation:

or when Γ L ˜ = Γ L − E Θ 1 − E Θ 2 , we obtain

To derive the equation of the Lagrangian structure with respect to the variance D L = E ( L − E L ) 2 , we use the formula D L = L − ( E L ) 2 :

It follows from equation (15) that

and

Thus, we obtain

where

If we assume that Ψ ν satisfies both conditions in equation (14) and the Helmholtz conditions

imposed on the functions ϕ ν = − 2 d E L d t ⋅ ∂ E L ∂ q ˙ ν = E [ a ν k ′ q ¨ k + b ν ′ ] , so that the representation a ν k ′ q ¨ k + b ν ′ ≡ d d t ∂ Θ 3 ∂ q ˙ ν − ∂ Θ 3 ∂ q ν takes place, then equation (20) is equivalent to the following equation:

where D L ˜ = D L − E Θ 1 − E Θ 3 .

We preliminary note the following:

From the relations

it follows that

(see [28, p. 182]).

Equation (10) solved with respect to the highest derivative has the form

where α k ν = ∂ 2 L ∂ q ˙ k ∂ q ˙ ν . It is assumed that the matrix ( α k ν ) is nonsingular and the inverse matrix has the form ( α k ν − 1 ) . In terms of differentials, we obtain

where

and

The differentials d L and d ∂ L ∂ q ˙ ν are of the forms

and

respectively. Here, A 1 = ∂ L ∂ t + ∂ L ∂ q μ q ˙ μ + ∂ L ∂ q ˙ μ X μ + 1 2 ∂ 2 L ∂ q ˙ k ∂ q ˙ μ σ k j σ μ j , and B 1 j = ∂ L ∂ q ˙ μ Y μ j .

Based on equations (24) and (29), we calculate the expression

Then,

Furthermore, due to equation (28), we obtain

Hence, applying the operation of mathematical expectation and taking into account E [ 2 ∂ L ∂ q ˙ ν B 1 j + 2 L σ ν j ] ξ ˙ 0 j = 0 , we arrive at the following equation:

where R ν = 2 ∂ L ∂ q ˙ ν A 1 + ∂ L ∂ q ˙ μ α l μ − 1 σ l j σ ν j .

Similarly, if we use L ∘ = L − E L , instead of L , we obtain the equation in D L of the following form:

where R ∘ ν = 2 ∂ L ∘ ∂ q ˙ ν A ∘ 1 + ∂ L ∘ ∂ q ˙ μ α l μ − 1 σ l j σ ν j .

Thus, if the original Lagrange equation in the presence of random perturbations is given in the Langevin-Stratonovich form (3), then

1. the averaged Lagrangian E L satisfies the equation (4), and under the additional assumption (6) with respect to σ ν j , there is a function Θ ( q , q ˙ , t ) such that equation (9) holds for E L ˜ = E L − Θ ,

2. it follows from relation (12) that the second-order initial moment L satisfies the following equation:

   (32) d d t ∂ Γ L ∂ q ˙ ν − ∂ Γ L ∂ q ν = E Φ ν + 2 E L ∂ ( L σ ν j ) ∂ q ˙ i σ i j ,

   and under the additional conditions in equations (14) and (17), there exist some functions Θ 1 and Θ 2 such that the following equation holds with respect to Γ L ˜ = Γ L − E Θ 1 − E Θ 2 :

   (33) d d t ∂ Γ L ˜ ∂ q ˙ ν − ∂ Γ L ˜ ∂ q ν = 0 .

3. The variance D L satisfies the equation

   d d t ∂ D L ∂ q ˙ ν − ∂ D L ∂ q ν = E Φ ν − 2 d E L d t ⋅ ∂ E L ∂ q ˙ ν ,

   and under conditions in equations (17) and (22), there exist some functions Θ 1 and Θ 3 such that the generalized variance D L ˜ = D L − E Θ 1 − E Θ 3 satisfies equation (23).

If the original Lagrange equation in the presence of random perturbations is given in the Langevin-Ito form (10), then, by using the Ito differentiation operator d ∘ ( ⋅ ) d t = d ( ⋅ ) d t + 1 2 ∂ 2 ( ⋅ ) ∂ q ˙ i ∂ q ˙ k σ i j σ k j , we obtain the following:

1. The averaged Lagrangian E L satisfies equation (11). If L = O ( q ˙ 2 ) , then d ∘ ( ⋅ ) d t ≡ d ( ⋅ ) d t , and equation (11) is equivalent to equation (9).

2. The second-order initial moment Γ L satisfies equation (30).

3. The variance D L satisfies equation (31).

Preliminarily, similar to the classical case, let us consider the Hamiltonian action W = ∫ t 0 t 1 L ( q , q ˙ , t ) d t and introduce the following notions in the extended configuration space ( q , t ) ∈ R n + 1 :

E ∫ t 0 t 1 L d t is the mathematical expectation of the Hamiltonian action;

∫ t 0 t 1 L d t ≡ E ∫ t 0 t 1 L 2 d t is the second-order initial moment of the Hamiltonian action;

∫ t 0 t 1 D L d t ≡ E ∫ t 0 t 1 ( L − E L ) 2 d t is the variance of the Hamiltonian action.

Let q 0 ν ( ω ) and q 1 ν ( ω ) be Gaussian random variables fixed at moments t 0 and t 1 .