arXiv subjects

Franco Rampazzo

Publications and source records attributed to Franco Rampazzo.

At least 19 recordsLinked to original sources

arXiv2026

Minimizers that are not Impulsive Minimizers and Higher Order Abnormality

This paper addresses two related problems in optimal control. The first investigation consists of compatibility issues between two classical approaches to deriving necessary conditions for optimal control problems with a final target: the set-separation approach and penalization techniques. These methods generally lead to non-equivalent conditions, mainly due to their reliance on different notions of tangency at the target. We address this issue by considering Quasi Differential Quotient (QDQ) approximating cones (which are fit for the set-separation approach) and identifying conditions under which the Clarke tangent cone (which is a typical tool within penalization techniques) is also a QDQ approximating cone. In particular, we show that this property holds under suitable local invariance assumptions or when the target coincides locally with an $r$-prox regular set. In the second part of the paper we apply this compatibility result to the study of infimum-gap phenomena in optimal control problems with unbounded controls and impulsive extensions. In particular, we establish a connection between the occurrence of infimum gaps for strict-sense minimizers and abnormality in a higher-order Maximum Principle involving Lie brackets. While the abnormality-gap correspondence beyond first-order conditions has been already established for extended-sense --i.e. impulsive-- minimizers, a topological argument involving the former and the utilization of the above compatibility issues allow us to extend this correspondence to strict-sense minimizers.

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arXiv2026

Higher-Order Normality and No-Gap Conditions in Impulsive Control with $L^1$-Control Topology

In optimal control, extending the class of admissible controls is a common strategy to guarantee the existence of optimal solutions. However, such extensions may introduce a gap between the infimum of the original problem and the minimum of the extended one, especially in the presence of endpoint constraints. Since Warga's seminal work, normality of first-order necessary conditions for extended minimizers has been recognized as a sufficient condition to avoid this phenomenon, though it is far from being necessary. In this paper, we consider impulsive extensions of control-affine systems with unbounded controls. We establish that a notion of \textit{higher-order normality}, based on iterated Lie brackets of the systems vector fields, suffices to prevent an infimum gap. The key novelty of this manuscript consists in showing that this holds under a local topology defined by the $L^1$-distance between controls, rather than the more common $L^\infty$-distance between trajectories. Among the reasons that motivate the interest in this issue, let us mention that a counterexample by R. B. Vinter shows that for a different extension -- based on convexification of the velocity set -- a local extended minimizer that is normal with respect to the $L^1$-norm of the controls may still exhibit a gap. Our method relies on set-separation techniques. Such an approach makes it possible to derive higher-order conditions and to exploit the corresponding notion of higher-order normality.

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arXiv2025

An $L^\infty$ Rashevskii-Chow Theorem

Consider a finite family $\{f_1,\dots,f_\nu\}$ of $C^\infty$ vector fields on a $n$-dimensional ($n\in\mathbb{N}$), smooth manifold $\mathcal{M}$. The celebrated Rashevskii-Chow theorem states that, provided the vector fields $\{f_1,\dots,f_\nu\}$, together with their iterated Lie brackets, span the whole tangent space at some $x_*\in\mathcal{M}$, then any $x$ in a neighborhood of $x_*$ can be connected to $x_*$ by means of a finite concatenation of integral curves of $\{\pm f_1,\dots,\pm f_\nu\}$. This result finds applications in a number of areas, e.g., in control theory, in Sub-Riemannian geometry, and the theory of degenerate elliptic and parabolic partial differential equations, to mention a few. Here we extend this basic result to families of vector fields, which are considerably less regular, in particular, by allowing iterated Lie brackets to be just bounded measurable. This is technically made possible by the utilization of set-valued Lie brackets, which have already proven to be useful in extending commutativity type results, Frobenius' theorem, and also higher-order necessary conditions for optimal control problems, to the setting of non-smooth vector fields.

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arXiv2025

Higher order necessary conditions for optimal controls not ranging in the interior

Goh's and Legendre-Clebsch necessary conditions for optimal controls of affine-control systems are usually established under the hypothesis that the minimizing control lies in the interior of the control set $U$. In this paper we investigate the possibility of establishing Goh's and Legendre-Clebsch necessary conditions without this assumption, so that even control sets with empty interiors or optimal controls touching the boundary of $U$ can be taken into consideration.

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arXiv2023

An Abstract Maximum Principle for constrained minimum problems

This article makes no claim to originality, other than, perhaps, the simple statement here called the {\it Abstract Maximum Principle}. Actually, the whole contents are strongly based on some H. Sussmann's and coauthors' papers, in which, in a much more general context, the set-separation approach is regarded as foundational for necessary conditions for minima. So, rather than being the exposition of original material, this paper has mainly a pedagogical purpose. From the Abstract Maximum Principle it is possible to deduce several necessary conditions for both finite dimensional minimum problems and for optimal control problems. More in general, this Principle seems apt to capture some consequences of the geometric and topological idea of (possibly vector-valued) minimization in a parametrized problem.

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arXiv2023

Goh and Legendre-Clebsh conditions for nonsmooth control systems

Higher order necessary conditions for a minimizer of an optimal control problem are generally obtained for systems whose dynamics is at least continuously differentiable in the state variable. Here, by making use of the notion of set-valued Lie bracket introduced in "Set-valued differentials and a nonsmooth version of Chow-Rashevski's theorem" by F. Rampazzo and H.J.Sussmann and extended in "Iterated Lie brackets for nonsmooth vector fields" by E. Feleqi and F.Rampazzo , we obtain Goh and Legendre-Clebsh type conditions for a control affine system with Lipschitz continuous dynamics. In order to manage the simultaneous lack of smoothness of the adjoint equation and of the Lie bracket-like variations, we will exploit the notion of Quasi Differential Quotient, introduced in "A geometrically based criterion to avoid infimum-gaps in Optimal Control" by M. Palladino and F.Rampazzo. We finally exhibit an example where the established higher order condition is capable to rule out the optimality of a control verifying a first order Maximum Principle.

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arXiv2023

A Lie-bracket-based notion of stabilizing feedback in optimal control

For a control system two major issues can be considered: the stabilizability with respect to a given target, and the minimization of an integral functional (while the trajectories reach this target). Here we consider a problem where stabilizability or controllability are investigated together with the further aim of a "cost regulation", namely a state-dependent upper bounding of the functional. This paper is devoted to a crucial step in the program of establishing a chain of equivalences among degree-k stabilizability with regulated cost, asymptotic controllability with regulated cost, and the existence of a degree-k Minimum Restraint Function (which is a special kind of Control Lyapunov Function). Besides the presence of a cost we allow the stabilizing "feedback" to give rise to directions that range in the union of original directions and the family of iterated Lie bracket of length less or equal to $k$. In the main result asymptotic controllability [resp. with regulated cost] is proved to be necessary for degree-k stabilizability [resp. with regulated cost]. Further steps of the above-mentioned logical chain are proved in companion papers, so that also a Lyapunov-type inverse theorem -- i.e. the possibility of deriving existence of a Minimum Restraint Function from stabilizability -- appears as quite likely.

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arXiv2023

HJ inequalities involving Lie brackets and feedback stabilizability with cost regulation

With reference to an optimal control problem where the state has to approach asymptotically a closed target while paying a non-negative integral cost, we propose a generalization of the classical dissipative relation that defines a Control Lyapunov Function to a weaker differential inequality. The latter involves both the cost and the iterated Lie brackets of the vector fields in the dynamics up to a certain degree k greater than or equal to 1, and we call any of its (suitably defined) solutions a degree-k Minimum Restraint Function. We prove that the existence of a degree-k Minimum Restraint Function allows us to build a Lie-bracket-based feedback which sample stabilizes the system to the target while regulating (i.e., uniformly bounding) the cost.

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arXiv2023

Integral representation for bracket-generating multi-flows

If $f_1,f_2$ are smooth vector fields on an open subset of an Euclidean space and $[f_1,f_2]$ is their Lie bracket, the asymptotic formula $$Ψ_{[f_1,f_2]}(t_1,t_2)(x) - x =t_1t_2 [f_1,f_2](x) +o(t_1t_2),$$ where we have set $ Ψ_{[f_1,f_2]}(t_1,t_2)(x) := \exp(-t_2f_2)\circ\exp(-t_1f_1)\circ\exp(t_2f_2)\circ\exp(t_1f_1)(x)$, is valid for all $t_1,t_2$ small enough. In fact, the integral, exact formula \begin{equation}\label{abstractform} Ψ_{[f_1,f_2]}(t_1,t_2)(x) - x = \int_0^{t_1}\int_0^{t_2}[f_1,f_2]^{(s_2,s_1)} (Ψ(t_1,s_2)(x))ds_1\,ds_2 , \end{equation} where $ [f_1,f_2]^{(s_2,s_1)}(y) := D\Big(\exp(s_1f_1)\circ \exp(s_2f_2{)}\Big)^{-1}\cdot [f_1,f_2](\exp(s_1f_1)\circ \exp(s_2f_2){(y)}), $ with ${y = Ψ(t_1,s_2)(x)}$ has also been proven. Of course the integral formula can be regarded as an improvement of the asymptotic formula. In this paper we show that an integral representation holds true for any iterated bracket made from elements of a family of vector fields ${f_1,\dots,f_{k}}$. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving nonsmooth vector fields.

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arXiv2022

Goh conditions for minima of nonsmooth problems with unbounded controls

Higher order necessary conditions for a minimizer of an optimal control problem are generally obtained for systems whose dynamics is continuously differentiable in the state variable. Here, by making use of the notion of set-valued Lie bracket we obtain a Goh-type condition for a control affine system with Lipschitz continuous dynamics and unbounded controls. In order to manage the simultaneous lack of smoothness of the adjoint equation and of the Lie bracket-like variations we make use of the notion of Quasi Differential Quotient. We conclude the paper with a worked out example where the established higher order condition is capable to rule out the optimality of a control verifying the standard maximum principle.

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arXiv2021

Rotational controls and uniqueness of constrained viscosity solutions of Hamilton-Jacobi PDE

The classical inward pointing condition (IPC) for a control system whose state $x$ is constrained in the closure $C:=\barΩ$ of an open set $Ω$ prescribes that at each point of the boundary $x\in \partial Ω$ the intersection between the dynamics and the interior of the tangent space of $\bar Ω$ at $x$ is nonempty. Under this hypothesis, for every system trajectory $x(.)$ on a time-interval $[0,T]$, possibly violating the constraint, one can construct a new system trajectory $\hat x(.)$ that satisfies the constraint and whose distance from $x(.)$ is bounded by a quantity proportional to the maximal deviation $d:=\mathrm{dist}(Ω,x([0,T]))$. When (IPC) is violated, the construction of such a constrained trajectory is not possible in general. However, for a control system of the form $\dot{x}=f_1(x)u_1+f_2(x)u_2$, we prove in this paper that a "higher order" inward pointing condition involving Lie brackets of the dynamics' vector fields allows for a novel construction of a constrained trajectory $\hat x(.)$ whose distance from the reference trajectory $x(.)$ is bounded by a quantity proportional to $\sqrt{d}$. Our method requires a further assumption of non-positiveness of a sort of curvature and is based on the implementation of a suitable "rotating" control strategy. As an application, we establish the continuity up to the boundary of the value function $V$ of a classical optimal control problem, a continuity that allows to regard $V$ as the unique constrained viscosity solution of the corresponding Bellman equation.

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arXiv2021

Quasi Differential Quotients

We explore basic properties and some applications of Quasi Differential Quotients ($QDQ$s) and the related $QDQ$-approximating multi-cones. A $QDQ$, which is a special kind of H.Sussmann's Approximate Generalized Differential Quotient ($AGDQ$), consists in a notion of generalized differentiation for set-valued maps. $QDQ$s have the advantage over $AGDQ$s of allowing a genuine, non-punctured, Open Mapping result, so implying stronger set-separation theorems. They have already proved quite useful in the investigation of some connections occurring between infimum gap phenomena and the normality of minima. Moreover, $QDQ$-approximating multi-cones are fit in optimal control to deduce Maximum Principles that involve (set-valued) Lie brackets of nonsmooth vector fields.

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arXiv2019

A geometrically based criterion to avoid infimum-gaps in Optimal Control

In optimal control theory the expression infimum gap means a strictly negative difference between the infimum value of a given minimum problem and the infimum value of a new problem obtained by the former by extending the original family V of controls to a larger family W. Now, for some classes of domain-extensions -- like convex relaxation or impulsive embedding of unbounded control problems -- the normality of an extended minimizer has been shown to be sufficient for the avoidance of an infimum gaps. A natural issue is then the search of a general hypothesis under which the criterium 'normality implies no gap' holds true. We prove that, far from being a peculiarity of those specific extensions and from requiring the convexity of the extended dynamics, this criterium is valid provided the original family V of controls is abundant in the extended family W. Abundance, which is stronger than the mere C^0-density of the original trajectories in the set of extended trajectories, is a dynamical-topological notion introduced by J. Warga, and is here utilized in a 'non-convex' version which, moreover, is adapted to differential manifolds. To get the main result, which is based on set separation arguments, we prove an open mapping result valid for Quasi-Differential-Quotient (QDQ) approximating cones, a notion of 'tangent cone' resulted as a peculiar specification of H. Sussmann's Approximate-Generalized-Differential-Quotients (AGDQ) approximating cone.

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arXiv2019

A Higher-order Maximum Principle for Impulsive Optimal Control Problems

We consider a nonlinear system, affine with respect to an unbounded control $u$ which is allowed to range in a closed cone. To this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to $u$. This lack of coercivity gives the problem an {\it impulsive} character, meaning that minimizing sequences of trajectories happen to converge towards discontinuous paths. As is known, a distributional approach does not make sense in such a nonlinear setting, where, instead, a suitable embedding in the graph-space is needed. We provide higher order necessary optimality conditions for properly defined impulsive minima, in the form of equalities and inequalities involving iterated Lie brackets of the dynamical vector fields. These conditions are derived under very weak regularity assumptions and without any constant rank conditions.

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arXiv2019

Necessary conditions involving Lie brackets for impulsive optimal control problems

We obtain higher order necessary conditions for a minimum of a Mayer optimal control problem connected with a nonlinear, control-affine system, where the controls range on an m-dimensional Euclidean space. Since the allowed velocities are unbounded and the absence of coercivity assumptions makes big speeds quite likely, minimizing sequences happen to converge toward "impulsive", namely discontinuous, trajectories. As is known, a distributional approach does not make sense in such a nonlinear setting, where instead a suitable embedding in the graph space is needed. We will illustrate how the chance of using impulse perturbations makes it possible to derive a Higher Order Maximum Principle which includes both the usual needle variations (in space-time) and conditions involving iterated Lie brackets. An example, where a third order necessary condition rules out the optimality of a given extremal, concludes the paper.

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arXiv2018

Normality and Gap Phenomena in Optimal Unbounded Control

Optimal unbounded control problems with affine control dependence may fail to have minimizers in the class of absolutely continuous state trajectories. For this reason, extended impulsive versions --which cannot be of measure-theoretical type-- have been investigated, in which the domain is enlarged to include discontinuous state trajectories of bounded variation, and for which existence of minimizers is guaranteed. It is of interest to know whether the passage from the original optimal control problem to its extension introduces an infimum gap. This paper provides sufficient conditions for the absence of an infimum gap based on normality of extremals. In certain cases, the normality conditions reduce to simple verifiable criteria, which improve on earlier, directly-derived sufficient conditions for no infimum gap.

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arXiv2016

Lyapunov-like functions involving Lie brackets

For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k>=1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a smooth degree-k control Lyapunov function does exist, for some k>1. The extension is performed under very weak regularity assumptions on the system, to the point that, for instance, (set valued) Lie brackets of locally Lipschitz vector fields are considered as well.

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arXiv2016

GAC, savings, and unbounded inputs

Let a control system and a target be given on an open subset of an Euclidean space. The existence of a Control Lyapunov Function - namely a positive definite, semiconcave, solution of the Hamilton-Jacobi inequality corresponding to the control vector field -- guarantees Global Asymptotic Controllability (GAC). In this case, however, minimization is not an issue. Instead, if a Lagrangean with non-negative values is considered as well, an optimal control problem can be defined in relation to the corresponding integral functional. In the first part of the present paper we show that the existence of a Minimum Restraint Function -- a solution of a strict Hamilton-Jacobi inequality involving the Lagrangian and a non-negative "savings multiplier" -- provides not only global asymptotic controllability but also savings, namely a state-dependent upper bound for the infima. This extends a former result, where the control set was assumed to be compact. Here we allow unbounded controls and replace inputs' values' compactness with a quite mild hypothesis concerning the dependence of the data on inputs: such condition is met, for instance, by control vector fields that are compositions of Lipschitz maps with polynomials and exponentials of the control variable. In the second part of the paper we focus on the case when the dynamics is a polynomial in the control variable. Through some analysis of convexity properties of vector-valued polynomials' ranges, we prove some simplified versions of the main result, in terms of either affine representability or reduction to weak subsystems for the original dynamics.

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