Generic uniqueness and conjugate points for optimal control problems

2025/01/17 by Bressan, Alberto, Mazzola, Marco, Nguyen, Khai T. · 1 citation
#49K05 #49L12 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2501.10572

Abstract

The paper is concerned with an optimal control problem on ℝⁿ, where the dynamics is linear w.r.t.~the control functions. For a terminal cost ψ in a mathcalG_δ set of C⁴(ℝⁿ) (i.e., in a countable intersection of open dense subsets), two main results are proved.Namely: the set Γ_ψ⊂ℝⁿ of conjugate points is closed, with locally bounded (n-2)-dimensional Hausdorff measure. Moreover, the set of initial points y∈ ℝⁿ∖Γ_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 ℝⁿ.

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