# Solving Optimal Control Problems for Delayed Control-Affine Systems with Quadratic Cost by Numerical Continuation

## TL;DR

This paper presents a novel numerical homotopy method for efficiently solving fixed-delay optimal control problems in control-affine systems with quadratic cost, starting from the non-delayed solution.

## Contribution

It introduces a homotopy-based approach to handle delays in optimal control problems, overcoming complexities of indirect methods and providing convergence guarantees.

## Key findings

- Method successfully solves delay problems starting from non-delayed solutions.
- Numerical efficiency demonstrated through a practical example.
- Convergence results support the robustness of the approach.

## Abstract

- In this paper we introduce a new method to solve fixed-delay optimal control problems which exploits numerical homotopy procedures. It is known that solving this kind of problems via indirect methods is complex and computationally demanding because their implementation is faced with two difficulties: the extremal equations are of mixed type, and besides, the shooting method has to be carefully initialized. Here, starting from the solution of the non-delayed version of the optimal control problem, the delay is introduced by numerical homotopy methods. Convergence results, which ensure the effectiveness of the whole procedure, are provided. The numerical efficiency is illustrated on an example.