The integral estimations for ordinary differential equations and its application to the non-smooth optimal control problems

TL;DR

This paper develops integral estimates for solutions of ODEs with non-smooth functionals and applies these results to regularize and analyze non-smooth optimal control problems, providing new theoretical conditions.

Contribution

It introduces a novel approach to regularize non-smooth optimal control problems by replacing initial vectors with random vectors and derives optimality conditions.

Findings

Established L_2-norm estimates for integral functionals of ODE solutions.

Proposed a regularization method using random initial vectors.

Derived necessary and sufficient conditions for optimal control existence.

Abstract

The paper studies integral functionals with non-smooth functions from L_2 defined on solutions of ODEs. Some regularity is obtained in the form of estimates of L_2-norm for these functionals. This result is used for regularization of optimal control problems for ODEs with non-smooth functionals: it is suggested to replace the initial vector by a random vector with density. Necessary conditions of optimality and sufficient conditions of existence of optimal control are obtained.