# Integral equation in the theory of optimal speed of linear systems with constraints

## DOI:

https://doi.org/10.26577/JMMCS.2020.v106.i2.01## Keywords:

boundary value problems, phase constraints, optimization problem, minimizing sequences, integral equation## Abstract

Boundary-value problems with phase constraints for linear ordinary differential equations are considered.

The necessary and sufficient conditions for the existence of a solution to the boundary value problems of linear ordinary differential equations with boundary conditions from given sets in the presence of phase constraints are obtained.

A method is proposed for constructing a solution to a boundary value problem with phase constraints by constructing minimizing sequences in a functional space.

An estimate of the convergence rate of minimizing sequences is obtained.

The basis of the proposed method for solving boundary value problems with phase constraints is the ability to reduce these problems to one class of the Fredholm integral equation of the first kind.

The Fredholm integral equation of the first kind is among the poorly studied problems of mathematics.

Therefore, basic research on integral equations and the solution based on them of boundary value problems of differential equations is the main promising direction in mathematics.

A new method is proposed for solving boundary value problems of linear ordinary differential equations with phase constraints, which has numerous applications in the theory of dynamical systems.

The scientific novelty of the results is:

Formalization of the general problem of dynamical systems and its reduction to boundary value problems of ordinary differential equations with phase constraints;

A new criterion is found for the existence of a solution to boundary value problems in the form of the immersion principle based on the existence theorem and the construction of a solution to the integral equation;

A new method has been created for solving boundary value problems of linear ordinary differential equations by constructing minimizing sequences for a special initial optimal control problem.

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## How to Cite

*Journal of Mathematics, Mechanics and Computer Science*,

*106*(2), 3–17. https://doi.org/10.26577/JMMCS.2020.v106.i2.01