# Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach

## TL;DR

This paper proposes a hybrid approach combining the Hamilton-Jacobi-Bellman equation with Pontryagin's Minimum Principle to efficiently initialize control problem solutions, especially in complex scenarios with multiple minima and constraints.

## Contribution

It introduces a novel coupling method that uses HJB to generate initial guesses for PMP, improving convergence to the global minimum over existing techniques.

## Key findings

- The method reduces CPU time to under four minutes for problems up to four dimensions.
- It effectively handles multiple minima, discontinuous controls, and state constraints.
- Numerical tests validate the approach's efficiency and robustness.

## Abstract

The aim of this paper is to investigate from the numerical point of view the possibility of coupling the Hamilton-Jacobi-Bellman (HJB) equation and Pontryagin's Minimum Principle (PMP) to solve some control problems. A rough approximation of the value function computed by the HJB method is used to obtain an initial guess for the PMP method. The advantage of our approach over other initialization techniques (such as continuation or direct methods) is to provide an initial guess close to the global minimum. Numerical tests involving multiple minima, discontinuous control, singular arcs and state constraints are considered. The CPU time for the proposed method is less than four minutes up to dimension four, without code parallelization.