Abstract
This paper addresses infinite horizon control problems under state constraints and some regularity aspects of the associated value function through non-smooth analysis. We focus on the locally bounded variation property of the epigraph of value functions, extending the well-known property of absolute continuity in terms of set-valued maps. Here, we relax common assumptions on the Lagrangian, admitting a larger class of control problems that are not taken into account when uniform lower bounds with respect to an integrable function are considered. We demonstrate that the locally bounded property holds for value functions without assuming any controllability assumptions except minimal viability conditions on the constraint set. Inequalities involving the contingent epiderivative/hypoderivative of the value function are also investigated.
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Notes
Here \(\mathfrak {m}\) stands for the Lebesgue measure.
Let \(f\in L^1_{loc}(\mathbb {R}^n;\mathbb {R})\). A point x is called a Lebesgue point for f if
$$\begin{aligned} \lim _{\begin{array}{c} {\mathfrak {m}}(\mathcal {Q})\rightarrow 0\\ x\in \mathcal {Q} \end{array}} \frac{1}{{\mathfrak {m}}(\mathcal {Q})} \int _{\mathcal {Q}}|f(y)-f(x)| d y=0 \end{aligned}$$where \(\mathcal {Q}\) stands for n-dimensional qubes. It is well known that x is a Lebesgue point for f for a.e. \(x\in \mathbb {R}^n\) and holds
$$\begin{aligned} \lim _{\begin{array}{c} {\mathfrak {m}}(\mathcal {Q})\rightarrow 0\\ x\in \mathcal {Q} \end{array}} \frac{1}{{\mathfrak {m}}(\mathcal {Q})} \int _{\mathcal {Q}} f(y) d y=f(x). \end{aligned}$$
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Basco, V. Locally bounded variations epigraph property of the value function to infinite horizon optimal control problems under state constraints. Boll Unione Mat Ital (2024). https://doi.org/10.1007/s40574-024-00410-1
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DOI: https://doi.org/10.1007/s40574-024-00410-1