# A sufficient optimality condition for delayed state-linear optimal control problems

## TL;DR

This paper establishes a sufficient optimality condition for delayed state-linear control problems by transforming them into equivalent non-delayed problems, enabling the application of existing theorems.

## Contribution

It introduces a novel method to handle delays in state-linear optimal control problems by converting them into non-delayed problems for analysis.

## Key findings

- Proved a sufficient optimality condition for delayed state-linear control problems.
- Demonstrated the method with an illustrative example.
- Extended the applicability of known theorems to delayed systems.

## Abstract

We give answer to an open question by proving a sufficient optimality condition for state-linear optimal control problems with time delays in state and control variables. In the proof of our main result, we transform a delayed state-linear optimal control problem to an equivalent non-delayed problem. This allows us to use a well-known theorem that ensures a sufficient optimality condition for non-delayed state-linear optimal control problems. An example is given in order to illustrate the obtained result.