control-toolbox / control-toolbox/CTSolvers.jl

[Roadmap] Homotopy methods & conjugate point detection

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roadmap
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Description

## Objective

Implement homotopy methods and conjugate point detection, reproducing the capabilities of [hampath](https://gitlab.inria.fr/ct/hampath).

## Context

This requires:
- Creating a new package for homotopy in optimal control
- Integrating [BifurcationKit.jl](https://github.com/bifurcationkit/BifurcationKit.jl)
- Handling the notion of `parameters` in OCP definitions (see [OptimalControl #655](https://github.com/control-toolbox/OptimalControl.jl/issues/655))

## Why

Homotopy methods are essential for solving difficult optimal control problems where a good initial guess is hard to find. Conjugate point detection is important for verifying optimality in indirect methods. hampath demonstrated these capabilities; we need to bring them into the Julia ecosystem.

## References

- [hampath](https://gitlab.inria.fr/ct/hampath)
- [BifurcationKit.jl](https://github.com/bifurcationkit/BifurcationKit.jl)
- [OptimalControl #655 — Add parameters](https://github.com/control-toolbox/OptimalControl.jl/issues/655)
- [Roadmap comment](https://github.com/orgs/control-toolbox/discussions/68#discussioncomment-17584843)

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