Optimal control with time-delays via the penalty method

TL;DR

This paper establishes necessary conditions for optimal control problems with time delays and introduces a convergence method to solve delayed problems via simpler variational problems.

Contribution

It provides the first Euler-Lagrange type optimality conditions for problems with different delays in the function and its derivative, and introduces a convergence theorem for delayed control problems.

Findings

Derived necessary optimality conditions for delayed variational problems.

Proved a convergence theorem linking delayed control problems to simpler variational problems.

Extended the calculus of variations framework to include different delays in functions and derivatives.

Abstract

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.