Generic uniqueness and conjugate points for optimal control problems
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2025
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866929681771528192 |
|---|---|
| author | Bressan, Alberto Mazzola, Marco Nguyen, Khai T. |
| author_facet | Bressan, Alberto Mazzola, Marco Nguyen, Khai T. |
| contents | The paper is concerned with an optimal control problem on $\mathbb{R}^n$, where the dynamics is linear w.r.t.~the control functions. For a terminal cost $ψ$ in a $mathcal{G}_δ$ set of $\mathcal{C}^4(\mathbb{R}^n)$ (i.e., in a countable intersection of open dense subsets), two main results are proved.Namely: the set $Γ_ψ\subset\mathbb{R}^n$ of conjugate points is closed, with locally bounded $(n-2)$-dimensional Hausdorff measure. Moreover, the set of initial points $y\in \mathbb{R}^n\setminusΓ_ψ$, which admit two or more globally optimal trajectories, is contained in the union of a locally finite family of embedded manifolds. In particular, the value function is continuously differentiable on an open, dense subset of $\mathbb{R}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10572 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generic uniqueness and conjugate points for optimal control problems Bressan, Alberto Mazzola, Marco Nguyen, Khai T. Optimization and Control 49K05, 49L12 The paper is concerned with an optimal control problem on $\mathbb{R}^n$, where the dynamics is linear w.r.t.~the control functions. For a terminal cost $ψ$ in a $mathcal{G}_δ$ set of $\mathcal{C}^4(\mathbb{R}^n)$ (i.e., in a countable intersection of open dense subsets), two main results are proved.Namely: the set $Γ_ψ\subset\mathbb{R}^n$ of conjugate points is closed, with locally bounded $(n-2)$-dimensional Hausdorff measure. Moreover, the set of initial points $y\in \mathbb{R}^n\setminusΓ_ψ$, which admit two or more globally optimal trajectories, is contained in the union of a locally finite family of embedded manifolds. In particular, the value function is continuously differentiable on an open, dense subset of $\mathbb{R}^n$. |
| title | Generic uniqueness and conjugate points for optimal control problems |
| topic | Optimization and Control 49K05, 49L12 |
| url | https://arxiv.org/abs/2501.10572 |