arXiv subjects

Marta Zoppello

Publications and source records attributed to Marta Zoppello.

At least 19 recordsLinked to original sources

arXiv2026

Odd elasticity in a three-link microswimmer: feedback equivalence, global controllability, and the cost of non-reciprocity

We study a Purcell three-link microswimmer whose joints are \emph{odd-elastic}: the torsional stiffness is non-Hermitian, its antisymmetric part $k_o$ injecting mechanical work so that the internal elastic drift is non-conservative. Our main finding is a sharp separation between geometry and cost --- odd elasticity is invisible to the control geometry of the swimmer and visible only in the energy of a manoeuvre. The mechanism is a single algebraic fact: the drift lies in the span of the two control vector fields, so the system is feedback-equivalent to a driftless one and the odd modulus enters the bracket structure only through the scalar $\det\textbf{K}=k^2+k_o^2$. From this we deduce that the abnormal extremals of the energy problem are unchanged by $k_o$, that the swimmer is globally controllable for every non-reciprocity with no threshold, and that its nilpotent model is the Cartan $(2,3,5)$ sub-Riemannian structure, deformed only by the metric scaling $g_\chi=(1+\chi^2)g_0$. The odd modulus acts solely on the cost: casting the optimal-steering problem in sub-Finsler (Randers) form, we prove that a prescribed reorientation is strictly cheaper for either sign of $k_o$. Full Resistive-Force-Theory simulations confirm the analysis and reveal that at isotropic drag the swimmer becomes a pure rotator, turning without translating.

math-ph↗

arXiv2026

Purcell swimmer near a wall

We study the effects of hydrodynamic interactions between a wall and the Purcell three-link swimmer in the two-dimensional case. After deriving the equations of motion in a low Reynolds number regime using Resistive Force Theory with suitably modified drag coefficients, we show, by means of criteria from Geometric Control Theory, that the system is controllable at configurations that are nearly parallel to the wall. Furthermore, we study configurations that are tilted, and we show net displacement with respect to the initial orientation. Some numerical experiments illustrate the analytical results.

physics.flu-dyn↗

arXiv2025

Controllability and Displacement Analysis of a Three-Link Elastic Microswimmer: A Geometric Control Approach

This study investigates the dynamics and controllability of a Purcell three-link microswimmer equipped with passive elastic torsional coils at its joints. By controlling the spontaneous curvature, we analyse the swimmers motion using both linear and weakly nonlinear approaches. Linear analysis reveals steady harmonic solutions for small-amplitude controls but does not predict any net displacement, whereas weakly nonlinear analysis predicts translation along the orientation of the central link. Using geometric control theory, we prove that the system is small time locally controllable near equilibrium and derive displacement estimates for periodic piecewise constant controls, which are validated through numerical simulations. These findings indicate that oscillatory controls can enable motion in all directions near equilibrium. This work offers foundational insights into the controllability of elastic microswimmers, paving the way for advanced motion planning and control strategies.

math-ph↗

arXiv2024

Flows of vector fields and the Kalman Theorem

We give two proofs of the Kalman Theorem, alternative to the most common ones, which infer such a classical result of Control Theory using just very basic facts on flows of vector fields. These proofs are apt to be generalised in diverse directions -- in fact one of them has been already generalised, yielding new criteria for local controllability of non-linear real analytic controlled systems.

math.OC↗

arXiv2024

Controllability and kinetic limit of spherical particles immersed in a viscous fluid

This paper deals with systems of spherical particles immersed in a viscous fluid. Two aspects are studied, namely the controllability of such systems, with particular attention to the case of one active particle and either one or two passive ones, and the kinetic limit of such systems as the number of particles diverges. The former issue is tackled in the framework of geometric control theory, whereas the latter resorts to Boltzmann-type formulations of the system of interacting particles.

math.AP↗

arXiv2024

Control of Microparticles Through Hydrodynamic Interactions

The controllability of passive microparticles that are advected with the fluid flow generated by an actively controlled one is studied. The particles are assumed to be suspended in a viscous fluid and well separated so that the far-field Stokes flow solutions may be used to describe their interactions. Applying concepts from geometric control theory, explicit moves characterized by a small amplitude parameter $\varepsilon$ are devised to prove that the active particle can control one or two passive particles. The leading-order (in $\varepsilon$) theoretical predictions of the particle displacements are compared with those obtained numerically and it is found that the discrepancy is small even when $\varepsilon\approx 1$. These results demonstrate the potential for a single actuated particle to perform complex micromanipulations of passive particles in a suspension.

physics.flu-dyn↗

arXiv2024

Gait controllability of length-changing slender microswimmers

Controllability results of four models of two-link microscale swimmers that are able to change the length of their links are obtained. The problems are formulated in the framework of Geometric Control Theory, within which the notions of fiber, total, and gait controllability are presented, together with sufficient conditions for the latter two. The dynamics of a general two-link swimmer is described by resorting to Resistive Force Theory and different mechanisms to produce a length-change in the links, namely, active deformation, a sliding hinge, growth at the tip, and telescopic links. Total controllability is proved via gait controllability in all four cases, and illustrated with the aid of numerical simulations.

math.OC↗

arXiv2024

Proving the Chow-Rashevskii Theorem \`a la Rashevskii

We give a new independent proof of a generalised version of the theorem by Rashevskii, which appeared in [Uch. Zapiski Ped. Inst. K. 2 (1938), 83 -- 94] and from which the classical Chow-Rashevskii Theorem follows as a corollary. The proof is structured to allow generalisations to the case of orbits of compositions of flows in absence of group structures, thus appropriate for applications in Control Theory. In fact, the same structure of the proof has been successfully exploited in [C. Giannotti, A. Spiro and M. Zoppello, arXiv 2401.07555 \& 2401.07560 (2024)] to determine new controllability criteria for real analytic non-linear control systems. It also yields a corollary, which can be used to derive results under lower regularity assumptions, as it is illustrated by a simple explicit example.

math.DG↗

arXiv2024

Distributions and controllability problems (I)

We consider a non-linear real analytic control system of first order $\dot q^i = f^i(t, q, w)$, with controls $w = (w^\alpha)$ in a connected open set $\mathcal{K} \subset \mathbb{R}^m$ and configurations $q = (q^i)$ in $\mathcal{Q} := \mathbb{R}^n$. The set of points in the extended space-time $\mathcal{M} = \mathbb{R} \times \mathcal{Q} \times \mathcal{K}$, which can be reached from a triple $x_o = (t_o , q_o, w_o) \in \mathcal{M}$ through a continuous graph completion $\gamma(s) = \big(t(s), q(t(s)), w(t(s))\big)$ of the graph of a solution $t \to (q(t), w(t))$, $t \in [t_o ,t_o + T]$, with piecewise real analytic controls, is called the {\it $\mathcal{M}$-attainable set of $x_o$ in time $T$}. We prove that if $y_o$ is an $\mathcal{M}$-attainable point of $x_o$, a large set of other nearby $\mathcal{M}$-attainable points of $x_o$ can be determined starting directly from $y_o$ and applying an appropriate ordered composition of flows of vector fields in a distinguished distribution $\mathcal{D}^{II} \subset T \mathcal{M}$, canonically associated with the control system. We then determine sufficient conditions for such neighbouring points to constitute an orbit of the pseudogroup of local diffeomorphisms generated by the vector fields in $\mathcal{D}^{II}$. If such conditions are satisfied and if the tangent spaces of these orbits have maximal rank projections onto $\mathcal{Q}$, the control system is locally accessible and has the small time local controllability property near the state points of equilibrium. These results lead to new proofs of classical local controllability criterions and yield new methods to establish the accessibility and the small time local controllability of non-linear control systems.

math.OC↗

arXiv2024

Distributions and controllability problems (II)

In [C. Giannotti, A. Spiro, M. Zoppello, {\it Distributions and controllability problems (I)}, preprint posted on ArXiv (2024)], we introduced a new approach to the real analytic non-linear control systems of the form $\dot q^i = f^i(t, q, w)$, with controls $w = (w^\alpha)$ running in a connected open set $\mathcal{K}$ of $ \mathbb{R}^m$ and states represented by points $q = (q^i)$ in a configuration space $\mathcal{Q} := \mathbb{R}^n$. The new approach consists of a differential-geometric study of (a) the oriented piecewise regular curves in the {\it extended space-time} $\mathcal{M} = \mathbb{R} \times \mathcal{Q} \times \mathcal{K}$, which are the (completed) graphs of the piecewise real analytic solutions $t \mapsto (q(t), w(t))$ of the control system, and (b) the local structure of the sets of points of $\mathcal{M}$ that are reachable from an initial point $x_o = (t_o, q_o, w_o) \in \mathcal{M}$ through such (completed) graphs. The main results of that paper are two new criterions which can be used to establish the small time local controllability near stable points of real analytic non-linear systems. The goal of this paper is to offer a friendly user's guide to those criterions, illustrating them by several examples. In particular, we analyse certain non-linear control systems, for which the new criterions show that they are small time locally controllable at their stable points, while, at the best of our knowledge, all other previous criterions are either inconclusive or not applicable.

math.OC↗

arXiv2021

Controlling non-controllable scallops

Any swimmer embedded on a inertialess fluid must perform a non-reciprocal motion to swim forward. The archetypal demonstration of this unique motion-constraint was introduced by Purcell with the so-called "scallop theorem". Scallop here is a minimal mathematical model of a swimmer composed by two arms connected via a hinge whose periodic motion (of opening and closing its arms) is not sufficient to achieve net displacement. Any source of incongruence on the motion or in the forces/torques experienced by such time-reversible scallop will break the symmetry imposed by the Stokes linearity and lead to subsequent propulsion of the scallop. However, little is known about the controllability of time-reversible scalloping systems. Here, we consider two individually non-controllable scallops swimming together. Under a suitable geometric assumption on the configuration of the system, it is proved that non-zero net displacement can be achieved as a consequence of their hydrodynamic interaction. A detailed analysis of the control system of equations is carried out analytically by means of geometric control theory. We obtain an analytic expression for the the displacement after a prescribed sequence of controls in function of the phase difference of the two scallops. Numerical validation of the theoretical results is presented with model predictions in further agreement with the literature.

math.OC↗

arXiv2021

Mean-field and kinetic limit of affine control systems: optimal control through leaders

This paper studies a multi-agent system starting from a single agent dynamics which is a nonlinear affine control system. It analyze what happens when the number of agents goes to infinity using two different approaches, a granular mean-field one, trying to control only a finite number of agents while all the others goes to infinity, and a kinetic one, when in the limit the percentage of controlled agents is preserved. In both cases the optimal control problem starting from the solution for the finite dimensional one is stated.

math.OC↗

arXiv2020

Hysteresis and controllability of affine driftless systems: some case studie

We investigate the controllability of some kinds of driftless affine systems where hysteresis effects are taken into account, both in the realization of the control and in the state evolution. In particular we consider two cases: the one when hysteresis is represented by the so-called play operator, and the one when it is represented by a so-called delayed relay. In the first case we prove that, under some hypotheses, whenever the corresponding non-hysteretic system is controllable, then we can also, at least approximately, control the hysteretic one. This is obtained by some suitably constructed approximations for the inputs in the hysteresis operator. In the second case we prove controllability for a generic hysteretic delayed switching system. Finally, we investigate some possible connections between the two cases.

math.DS↗

arXiv2020

A hybrid differential game with switching thermostatic-type dynamics and cost

In this paper we consider an infinite horizon zero-sum differential game where the dynamics of each player and the running cost are also depending on the evolution of some discrete (switching) variables. In particular, such switching variables evolve according to the switching law of a so-called thermostatic delayed relay, applied to the players' states. We first address the problem of the continuity of both lower and upper value function. Then, by a suitable representation of the problem as a coupling of several exit-time differential games, we characterize those value functions as, respectively, the unique solution of a coupling of several Dirichlet problems for Hamilton-Jacobi-Isaacs equations. The concept of viscosity solutions and a suitable definition of boundary conditions in the viscosity sense is used in the paper. Finally, we give some sufficient conditions for the existence of an equilibrium.

math.OC↗

arXiv2019

The $N$-link swimmer in three dimensions: controllability and optimality results

The controllability of a fully three-dimensional $N$-link swimmer is studied. After deriving the equations of motion in a low Reynolds number fluid by means of Resistive Force Theory, the controllability of the minimal $2$-link swimmer is tackled using techniques from Geometric Control Theory. The shape of the $2$-link swimmer is described by two angle parameters. It is shown that the associated vector fields that govern the dynamics generate, via taking their Lie brackets, all six linearly independent directions in the configuration space; every direction and orientation can be achieved by operating on the two shape variables. The result is subsequently extended to the $N$-link swimmer. Finally, the minimal time optimal control problem and the minimisation of the power expended are addressed and a qualitative description of the optimal strategies is provided.

math.OC↗

arXiv2019

A differential game with exit costs

We study a differential game where two players separately control their own dynamics, pay a running cost, and moreover pay an exit cost (quitting the game) when they leave a fixed domain. In particular, each player has its own domain and the exit cost consists of three different exit costs, depending whether either the first player only leaves its domain, or the second player only leaves its domain, or they both simultaneously leave their own domain. We prove that, under suitable hypotheses, the lower and upper value are continuous and are, respectively, the unique viscosity solution of a suitable Dirichlet problem for a Hamilton-Jacobi-Isaacs equation. The continuity of the values relies on the existence of suitable non-anticipating strategies respecting the domain-constraint. This problem is also treated in this work.

math.OC↗

arXiv2019

Optimal motion of a scallop: some case studies

In this paper we focus on a two-link swimmer called scallop which moves changing dynamics between two fluids regimes. We address and solve explicitly two optimal control problems, the minimum time one and the minimum quadratic cost needed to move the swimmer between two fixed positions using a periodic control. Considering only one switching in the dynamics and exploiting the structure of the equation of motion we are able to split the problem into simpler ones. We solve explicitly each sub-problem obtaining a discontinuous global solution. Then we approximate it through a suitable sequence of continuous functions. Finally, we show numerical simulations suggesting that to switch less times is the best strategy for both costs.

math.OC↗

arXiv2017

Can Magnetic Multilayers Propel Artificial Microswimmers Mimicking Sperm Cells?

We formulate and solve the equations governing the dynamics of a microscopic artificial swimmer composed of a head and of a tail made of a thin film of permanent magnetic material. This is a variant of the model swimmer proposed by Dreyfus et al. in 2005, whose tail is a filament obtained from the assembly of super-paramagnetic beads. The swimmer is actuated by an oscillating magnetic field, and its geometry is inspired by that of sperm cells. Using values for the geometric and material parameters that are realistic for a magnetic multilayer, we show that the model swimmer can reach swimming speeds exceeding one body length per second, under reasonable values of the driving magnetic field. This provides a proof of principle for the viability of the concept. In addition, we discuss the possibility to steer the system along curved paths. Finally, we compare the propulsion mechanism (swimming `gait') of our swimmer with that of sperm cells. The main difference between the two is that, contrary to its biological template, our artificial system does not rely on the propagation of bending waves along the tail, at least for the range of material and geometric parameters explored in this article.

cond-mat.soft↗