arXiv subjects

Piernicola Bettiol

Publications and source records attributed to Piernicola Bettiol.

3 recordsLinked to original sources

arXiv2026

Necessary Conditions for Free End-Time Sweeping Process Problems

This paper derives necessary optimality conditions for free end-time optimal control problems governed by sweeping process dynamics. In contrast to classical optimal control problems, the presence of the normal cone term in the state equation introduces discontinuities with respect to the state variable and prevents the standard Lipschitz regularity assumptions typically used in optimal control theory. As a result, both the analysis of trajectories and the formulation of optimality conditions require refined tools. We address problems with general endpoint constraints and time-dependent data, considering both the case in which the dynamics are Lipschitz continuous in time and the more challenging setting of merely measurable time dependence. Special attention is devoted to the additional transversality conditions associated with the free end-time variable. In the Lipschitz case, these conditions are expressed in terms of a function of bounded variation that coincides almost everywhere with the maximized Hamiltonian. In the measurable case, we show that a suitable interpretation can be obtained via the notions of sub- and super-essential values of the maximized Hamiltonian, recently introduced in the literature, leading to strengthened transversality conditions. These conditions also involve an additional term reflecting the interaction between the costate arc and the normal cone to the moving set at the optimal endpoints. Our approach combines suitable perturbation techniques with a penalization method tailored to sweeping processes. The moving set is modelled as a finite intersection of inequality-defined sets satisfying suitable constraint qualifications, including a positive linear independence condition and a diagonal dominance property of the associated Gramian matrix. We also introduce a local version of the constraint qualification condition. Our results apply to a broad class of problems, including those involving unbounded moving sets such as polyhedra. The resulting optimality system extends and sharpens recent results for classical free end-time problems to the non-Lipschitz, discontinuous framework of sweeping processes. The relevance of the transversality conditions obtained in this article is illustrated through two simple examples.

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arXiv2022

Uniform boundedness for the optimal controls of a discontinuous, non-convex Bolza problem

We consider a Bolza type optimal control problem of the form \begin{equation}\min J_{t}(y,u):=\int_t^TΛ(s,y(s), u(s))\,ds+g(y(T))\tag{P$_{t,x}$}\end{equation} Subject to: \begin{equation}\label{tag:admissible}\tag{D}\begin{cases} y\in AC([t,T];\mathbb R^n)\\y'=b(y)u\text{ a.e. } s\in [t,T], \,y(t)=x\\u(s)\in \mathcal U\text{ a.e. } s\in [t,T],\, y(s)\in \mathcal S\,\,\forall s\in [t,T], \end{cases} \end{equation} where $Λ(s,y,u)$ is locally Lipschitz in $s$, just Borel in $(y,u)$, $b$ has at most a linear growth and both the Lagrangian $Λ$ and the running cost function $g$ may take the value $+\infty$. If $b\equiv 1$ and $g\equiv 0$ problem (P$_{t,x}$) is the classical one of the calculus of variations. We suppose the validity a slow growth condition in $u$, introduced by Clarke in 1993, including Lagrangians of the type $Λ(s,y,u)=\sqrt{1+|u|^2}$ and $Λ(s,y,u)=|u|-\sqrt{|u|}$ and the superlinear case. If $Λ$ is real valued, any family of optimal pairs $(y_*, u_*)$ for (P$_{t,x}$) whose energy $J_t(y_*, u_*)$ is equi-bounded as $(t,x)$ vary in a compact set, has $L^\infty$ -- equibounded optimal controls. If $Λ$ is extended valued, the same conclusion holds under an additional lower semicontinuity assumption on $(s,u)\mapstoΛ(s,y,u)$ and on the structure of the effective domain. No convexity, nor local Lipschitz continuity is assumed on the variables $(y,u)$. As an application we obtain the local Lipschitz continuity of the value function under slow growth assumptions.

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arXiv2019

Necessary Optimality Conditions For Average Cost Minimization Problems

Control systems involving unknown parameters appear a natural framework for applications in which the model design has to take into account various uncertainties. In these circumstances the performance criterion can be given in terms of an average cost, providing a paradigm which differs from the more traditional minimax or robust optimization criteria. In this paper, we provide necessary optimality conditions for a nonrestrictive class of optimal control problems in which unknown parameters intervene in the dynamics, the cost function and the right end-point constraint. An important feature of our results is that we allow the unknown parameters belonging to a mere complete separable metric space (not necessarily compact).

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