To the Boundary Value Problem of Ordinary Differential Equations

A method for solving of a boundary value problem for ordinary differential equations with boundary conditions at phase and integral constraints is proposed. The base of the method is an immersion principle based on the general solution of the first order Fredholm integral equation which allows to reduce the original boundary value problem to the special problem of the optimal equation. 1 Problem statement We consider the following boundary value problem ˙ x = A(t)x + B(t) f (x, t) + µ(t), t ∈ I = [t 0 , t 1 ] (1.1) with boundary conditions (x(t 0) = x 0 , x(t 1) = x 1) ∈ S ⊂ R 2n , (1.2) with phase constraints x(t) ∈ G(t) : G(t) = {x ∈ R n | γ(t) ≤ F(x, t) ≤ δ(t), t ∈ I}, (1.3) and integral constraints 3g j (x) ≤ c j , g j (x) = t 1 t 0 f 0j (x(t), t) dt, j = 1, m 2 ; (1.6)

The following problems are stated.Problem 1.To find necessary and sufficient conditions for the existence of solutions of boundary value problem (1.1)- (1.6).
As it follows from the problem statement, it is necessary to prove the existence of the pair (x 0 , x 1 ) ∈ S such that the solution of (1.1) proceeded from the point x 0 at the time t 0 passes through the point x 1 at the time t 1 , along with the solution of the system (1.1) for each time the phase constraint is satisfied (1.3), and integrals (1.6) satisfy (1.4), (1.5).In particular, the set S is defined by the relation where H j (x 0 , x 1 ), j=1, p are convex functions in the variables (x 0 , x 1 ), s are given vectors and numbers, •, • is the scalar product.
In many cases, in practice the process under study is described by the equation of the form (1.1) in the phase space of the system defined by the phase constraint of the form (1.3).Outside this domain the process is described by completely different equations or the process under investigation does not exist.In particular, such phenomena take place in the research of dynamics of nuclear and chemical reactors (outside the domain (1.3) reactors do not exist.)Integral constraints of the form (1.4) characterize the total load experienced by the elements and nodes in the system (for example, total overload of cosmonauts), which should not exceed the specified values and equations of the form (1.5) correspond to the total limits for the system (for example, fuel consumption is equal to a predetermined value).
The essence of the method consists in the fact that at the first stage of research by transformation and introducing a fictitious control the initial problem is immersed in the control problem.Further, the existence of solutions of the original problem and the construction of its solution is carried out by solving the problem of optimal control of a special kind.With this approach, the necessary and sufficient conditions for the existence of the solution of the boundary value problem (1.1)-(1.6)can be obtained from the condition to achieve the lower bound of the functional on a given set, and the solution of the original boundary problem is the limit points of minimizing sequences.

Integral equation
The basis of the proposed method of solving problems 1 and 2 are the following theorems about the properties of solutions of the first order Fredholm integral equation: where n × n order is positive definite, where "*" is a sign of transposition.
Theorem 3.2.Let the matrix C(t 0 , t 1 ) be positive definite.Then the general solution of the integral equation (3.1) has the form where v(•) ∈ L 2 (I, R m ) is an arbitrary function, a ∈ R n is an arbitrary vector.

Immersion principle
Along with the differential equation (2.2) with boundary conditions (2.3) we consider the linear control system where µ 2 (t) = B 3 µ(t), t ∈ I.
Let the matrix B(t) = (B 1 (t), B 2 (t)) of (n + m 2 ) × (m 2 + m) order, and the vector-function It is easy to see that the control w(•) ∈ L 2 (I, R m+m 2 ) which transfers the trajectory of system (4.1) from any initial state ξ 0 to any desired state ξ 1 is a solution of the integral equation where is the fundamental matrix of solutions of the linear homoge- As follows from (3.1), (4.4), the matrix K(t 0 , t) = (t 0 , t)B(t).We introduce the following notations where function z(t) = z(t, v), t ∈ I is a solution of the differential equation Solution of the differential equation (4.1) corresponding to the control w(t)∈W is defined by the formula ) The lemma is proved.
We consider the following optimization problem: minimize the functional where We denote Theorem 4.3.Let the matrix W(t 0 , t 1 ) > 0, X * = ∅.In order for the boundary value problem (1.1)-(1.6) to have a solution, it is necessary and sufficient that the value J(θ * ) = 0 = J * , where ∈ X is optimal control for the problem (4.12)-(4.15).
Sufficiency.Let J(θ * ) = 0.This is possible if and only if w * . Sufficiency is proved.The theorem is proved.
The transition from the boundary value problem (1.1)-(1.6) to the problem (4.12)-(4.15) is called the principle of immersion.

Optimization problem
We consider the solution of the optimization problem (4.12)-(4.15).Note, that the function where Theorem 5.1.Let the matrix be W(t 0 , t 1 ) > 0, the function F 0 (t, q) is defined and continuously differentiable in q = (θ, z, z), and the following conditions hold: Then the functional (4.12) at conditions (4.13)-(4.15) is continuous and differentiable by Fréchet in any point θ ∈ X, and , and function ψ(t), t ∈ I is solution of the adjoint system (5.2) In addition, the gradient J (θ), θ ∈ X satisfies to Lipschitz condition where K > 0 is Lipschitz constant.
Theorem 5.3.Let the conditions of Theorem 5.1 hold, the set Λ 0 be bounded, J(θ), θ ∈ X be convex functional.Then the following statements hold.
4) The following convergence rate is satisfied Proof.The first assertion follows from the fact that Λ 0 is bounded closed convex set of a reflexive Banach space X, as well as from the weak lower semi-continuity of functional J(θ) on weakly bicompact set Λ 0 .The second assertion follows from estimation J Hence from the convexity of functional J(θ n ) at Λ 0 follows, that {θ n } is minimizing.The third assertion follows from weak bicompactness of set Λ 0 .Estimation of convergence rate follows from inequality J(θ n ) − J(θ * * ) ≤ c 1 θ n − θ n+1 .The last statement follows from Theorem 4.3.The theorem is proved.
We note, that if f (x, t), f 0j (x, t), j = 1, m 2 , F(x, t) are linear functions with respect to x, then the functional J(θ) is convex.

Example
The equation of motion of the system is where the phase constraint is given as Integral constraint is defined by (5.9) Denoting ϕ = x 1 , φ = ẋ1 = x 2 , the equation (5.7) can be written in vector form integral constraint (5.9) can be written as π 0 x 1 (t) dt ≤ 1. (5.12)

Immersion principle
Linear controlled system (see (4.1)-(4.3))has the form (5.16) where the matrix B 1 and linear homogeneous system ω = A 1 ω are The fundamental matrix of solution of the linear homogeneous system ω = A 1 ω is determined by the formula Here As follows from Theorem 4.1, the control (5.17) where z(t, v), t ∈ I = [0, π] is solution of the differential equation ).

Optimization problem
As for this example f = y 1 , F = y 1 , then optimization problem (4.12)-(4.15)can be written as: minimize the functional at conditions (5.18), where The partial derivatives of F 0 read as follows:

5 )
The boundary value problem (1.1)-(1.6)has a solution if and only if 1, m is known matrix of n × m order with piecewise continuous elements in t at fixed t 0 , u(•) ∈ L 2 [I, R m ] is the source function, I = [t 0 , t 1 ], a ∈ R n is given n-dimensional vector.
2) is an arbitrary function, y(t), t ∈ I is determined by the formula (4.7).