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Generic uniqueness and conjugate points for optimal control problems

Alberto Bressan, Marco Mazzola, Khai T. Nguyen

TL;DR

This work analyzes optimal control problems with dynamics affine in the control and a terminal cost ψ belonging to a generic class ${\cal G}_{\delta}$. By bounding solutions to the Pontryagin maximum principle and employing transversality arguments, it shows that for generic ψ the set of conjugate points $\Gamma_ψ$ forms a closed, locally $(n-2)$-dimensional manifold, while the locus of initial conditions with multiple global optima is contained in a finite union of embedded $(n-1)$-manifolds outside $\Gamma_ψ$. These geometric regularities imply that the value function $V$ is $C^1$ on an open dense subset of $\mathbb{R}^n$, and that the generic optimal feedback control has a well-structured, essentially unique behavior on large regions. The results hinge on uniform bounds for globally optimal trajectories, a detailed PMP analysis, and generic transversality arguments, providing a rigorous description of the regularity and structure of optimal controls in this broad class of problems.

Abstract

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 $\mathbb{R}^n$.

Generic uniqueness and conjugate points for optimal control problems

TL;DR

This work analyzes optimal control problems with dynamics affine in the control and a terminal cost ψ belonging to a generic class . By bounding solutions to the Pontryagin maximum principle and employing transversality arguments, it shows that for generic ψ the set of conjugate points forms a closed, locally -dimensional manifold, while the locus of initial conditions with multiple global optima is contained in a finite union of embedded -manifolds outside . These geometric regularities imply that the value function is on an open dense subset of , and that the generic optimal feedback control has a well-structured, essentially unique behavior on large regions. The results hinge on uniform bounds for globally optimal trajectories, a detailed PMP analysis, and generic transversality arguments, providing a rigorous description of the regularity and structure of optimal controls in this broad class of problems.

Abstract

The paper is concerned with an optimal control problem on , where the dynamics is linear w.r.t.~the control functions. For a terminal cost in a set of (i.e., in a countable intersection of open dense subsets), two main results are proved.Namely: the set of conjugate points is closed, with locally bounded -dimensional Hausdorff measure. Moreover, the set of initial points , 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 .
Paper Structure (4 sections, 3 theorems, 116 equations, 1 figure)

This paper contains 4 sections, 3 theorems, 116 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bounds on solutions of the PMP
  3. Generic properties of conjugate points in optimal control problems
  4. Generic structure of the optimal feedback control

Key Result

Lemma 2.1

Assume that the couple $(f,L)$ satisfies (A1)-(A2) and $\psi:{\mathbb R}^n\to {\mathbb R}_+$ is twice continuously differentiable. Then there exist continuous functions $\alpha,\beta,\gamma:{\mathbb R}_+\mapsto {\mathbb R}_+$ such that the following holds. Given any initial point $y\in{\mathbb R}^n Moreover, the adjoint variable $p^*(\cdot)$ satisfies

Figures (1)

  • Figure 1: Trajectories of (\ref{['PMP5']}), with terminal cost given by (\ref{['psidef']}). Notice that some backward trajectories terminate when $x(t)=0$. This happens when the adjoint variable $p(t)$ blows up. For $z>1/2$ backward solutions are still unique, but highly sensitive to the terminal data because the derivative $\psi'(z)= 2e^{z+1}$ has rapid growth.

Theorems & Definitions (7)

  • Definition 1.1
  • Definition 1.2
  • Example 2.1
  • Lemma 2.1
  • Remark 2.1
  • Theorem 3.1
  • Theorem 4.1