arXiv subjects

Marco Morandotti

Publications and source records attributed to Marco Morandotti.

At least 37 records · Page 2Linked to original sources

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

Characterizing BV- and BD-ellipticity for a class of positively 1-homogeneous surface energy densities

Lower semicontinuity of surface energies in integral form is known to be equivalent to BV-ellipticity of the surface density. In this paper, we prove that BV-ellipticity coincides with the simpler notion of biconvexity for a class of densities that depend only on the jump height and jump normal, and are positively 1-homogeneous in the first argument. The second main result is the analogous statement in the setting of bounded deformations, where we show that BD-ellipticity reduces to symmetric biconvexity. Our techniques are primarily inspired by constructions from the analysis of structured deformations and the general theory of free discontinuity problems.

math.AP↗

arXiv2024

Measure structured deformations

Measure structured deformations are introduced to present a unified theory of deformations of continua. The energy associated with a measure structured deformation is defined via relaxation departing either from energies associated with classical deformations or from energies associated with structured deformations. A concise integral representation of the energy functional is provided both in the unconstrained case and under Dirichlet conditions on a part of the boundary.

math.AP↗

arXiv2024

Direct minimization of the Canham--Helfrich energy on generalized Gauss graphs

The existence of minimizers of the Canham--Helfrich functional in the setting of generalized Gauss graphs is proved. As a first step, the Canham--Helfrich functional, usually defined on regular surfaces, is extended to generalized Gauss graphs, then lower semicontinuity and compactness are proved under a suitable condition on the bending constants ensuring coerciveness; the minimization follows by the direct methods of the Calculus of Variations. Remarks on the regularity of minimizers and on the behavior of the functional in case there is lack of coerciveness are presented.

math.OC↗

arXiv2023

$Γ$-convergence of discrete energies modeling self-aggregation of stochastic particles

In this work, we demonstrate that a functional modeling the self-aggregation of stochastically distributed lipid molecules can be obtained as the $Γ$-limit of a family of discrete energies driven by a sequence of independent and identically distributed random variables. These random variables are intended to describe the asymptotic behavior of lipid molecules that satisfy an incompressibility condition. The discrete energy keeps into account the interactions between particles. We resort to transportation maps to compare functionals defined on discrete and continuous domains, and we prove that, under suitable conditions on the scaling of these maps as the number of random variables increases, the limit functional features an interfacial term with a Wasserstein-type penalization.

math.PR↗

arXiv2023

Optimal control problems in transport dynamics with additive noise

Motivated by the applications, a class of optimal control problems is investigated, where the goal is to influence the behavior of a given population through another controlled one interacting with the first. Diffusive terms accounting for randomness in the evolution are taken into account. A well-posedness theory under very low regularity of the control vector fields is developed, as well as a rigorous derivation from stochastic particle systems.

math.OC↗

arXiv2022

Mean-field limits for entropic multi-population dynamical systems

The well-posedness of a multi-population dynamical system with an entropy regularization and its convergence to a suitable mean-field approximation are proved, under a general set of assumptions. Under further assumptions on the evolution of the labels, the case of different time scales between the agents' locations and labels dynamics is considered. The limit system couples a mean-field-type evolution in the space of positions and an instantaneous optimization of the payoff functional in the space of labels.

math.AP↗

arXiv2022

The variational modeling of hierarchical structured deformations

Hierarchical (first-order) structured deformations are studied from the variational point of view. The main contributions of the present research are the first steps, at the theoretical level, to establish a variational framework to minimize mechanically relevant energies defined on hierarchical structured deformations. Two results are obtained here: (i) an approximation theorem and (ii) the assignment of an energy to a hierarchical structured deformation by means of an iterative procedure. This has the effect of validating the proposal made in [Deseri & Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations at multiple submacroscopic levels. J.~Elast., 135 (2019), 149--182] to study deformations admitting slips and separations at multiple submacroscopic levels. An explicit example is provided to illustrate the behavior of the proposed iterative procedure and relevant directions for future research are highlighted.

math.OC↗

arXiv2022

Periodic homogenization in the context of structured deformations

An energy for first-order structured deformations in the context of periodic homogenization is obtained. This energy, defined in principle by relaxation of an initial energy of integral type featuring contributions of bulk and interfacial terms, is proved to possess an integral representation in terms of relaxed bulk and interfacial energy densities. These energy densities, in turn, are obtained via asymptotic cell formulae defined by suitably averaging, over larger and larger cubes, the bulk and surface contributions of the initial energy. The integral representation theorem, the main result of this paper, is obtained by mixing blow-up techniques, typical in the context of structured deformations, with the averaging process proper of the theory of homogenization.

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 selective optimal control via transient leadership

A mean-field selective optimal control problem of multipopulation dynamics via transient leadership is considered. The agents in the system are described by their spatial position and their probability of belonging to a certain population. The dynamics in the control problem is characterized by the presence of an activation function which tunes the control on each agent according to the membership to a population, which, in turn, evolves according to a Markov-type jump process. This way, a hypothetical policy maker can select a restricted pool of agents to act upon based, for instance, on their time-dependent influence on the rest of the population. A finite-particle control problem is studied and its mean-field limit is identified via $Γ$-convergence, ensuring convergence of optimal controls. The dynamics of the mean-field optimal control is governed by a continuity-type equation without diffusion. Specific applications in the context of opinion dynamics are discussed with some numerical experiments.

math.OC↗

arXiv2020

Upscaling and spatial localization of non-local energies with applications to crystal plasticity

We describe multiscale geometrical changes via structured deformations $(g,G)$ and the non-local energetic response at a point $x$ via a function $Ψ$ of the weighted averages of the jumps $[u_{n}](y)$ of microlevel deformations $u_{n}$ at points $y$ within a distance $r$ of $x$. The deformations $u_{n}$ are chosen so that $\lim_{n\to \infty }u_{n}=g$ and $\lim_{n\to \infty }\nabla u_{n}=$ $G$. We provide conditions on $Ψ$ under which the upscaling "$n\to \infty$" results in a macroscale energy that depends through $Ψ$ on (1) the jumps $[g]$ of $g$ and the "disarrangment field" $\nabla g-G$, (2) the "horizon" $r$, and (3) the weighting function $α_{r}$ for microlevel averaging of $[u_{n}](y)$. We also study the upscaling "$n\to \infty$" followed by spatial localization "$r\to 0$" and show that this succession of processes results in a purely local macroscale energy $I(g,G)$ that depends through $Ψ$ upon the jumps $[g]$ of $g$ and the "disarrangment field" $\nabla g-G$, alone. In special settings, such macroscale energies $I(g,G)$ have been shown to support the phenomena of yielding and hysteresis, and our results provide a broader setting for studying such yielding and hysteresis. As an illustration, we apply our results in the context of the plasticity of single crystals.

math.OC↗

arXiv2020

An alternate Lagrangian scheme for spatially inhomogeneous evolutionary games

An alternate Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics and is alternate, as it consists of the following two steps: first the distribution of strategies or labels is updated according to a best performance criterion and then this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

math.AP↗

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

Mean-field analysis of multi-population dynamics with label switching

The mean-field analysis of a multi-population agent-based model is performed. The model couples a particle dynamics driven by a nonlocal velocity with a Markow-type jump process on the probability that each agent has of belonging to a given population. A general functional analytic framework for the well-posedness of the problem is established, and some concrete applications are presented, both in the case of discrete and continuous set of labels. In the particular case of a leader-follower dynamics, the existence and approximation results recently obtained in [2] are recovered and generalized as a byproduct of the abstract approach proposed.

math.AP↗

arXiv2018

Discrete-to-continuum limits of particles with an annihilation rule

In the recent trend of extending discrete-to-continuum limit passages for gradient flows of single-species particle systems with singular and nonlocal interactions to particles of opposite sign, any annihilation effect of particles with opposite sign has been side-stepped. We present the first rigorous discrete-to-continuum limit passage which includes annihilation. This result paves the way to applications such as vortices, charged particles, and dislocations. In more detail, the discrete setting of our discrete-to-continuum limit passage is given by particles on the real line. Particles of the same type interact by a singular interaction kernel; those of opposite sign interact by a regular one. If two particles of opposite sign collide, they annihilate, i.e., they are taken out of the system. The challenge for proving a discrete-to-continuum limit is that annihilation is an intrinsically discrete effect where particles vanish instantaneously in time, while on the continuum scale the mass of the particle density decays continuously in time. The proof contains two novelties: (i) the observation that empirical measures of the discrete dynamics (with annihilation rule) satisfy the continuum evolution equation that only implicitly encodes annihilation, and (ii) the fact that, by imposing a relatively mild separation assumption on the initial data, we can identify the limiting particle density as a solution to the same continuum evolution equation.

math.AP↗

arXiv2018

Confinement of dislocations inside a crystal with a prescribed external strain

A system of $n$ screw dislocations in an isotropic crystal undergoing antiplane shear is studied in the framework of linear elasticity. Imposing a suitable boundary condition for the strain, namely requesting the non-vanishing of its boundary integral, results in a confinement effect. More precisely, in the presence of an external strain with circulation equal to n times the lattice spacing, it is energetically convenient to have n distinct dislocations lying inside the crystal. The result is obtained by formulating the problem via the core radius approach and by studying the asymptotics as the core size vanishes. An iterative scheme is devised to prove the main result. This work sets the basis for studying the upscaling problem, i.e., the limit as $n\to\infty$, which is treated in [17].

math.AP↗

arXiv2018

Analysis of a perturbed Cahn-Hilliard model for Langmuir-Blodgett films

An advective Cahn-Hilliard model motivated by thin film formation is studied in this paper. The one-dimensional evolution equation under consideration includes a transport term, whose presence prevents from identifying a gradient flow structure. Existence and uniqueness of solutions, together with continuous dependence on the initial data and an energy equality are proved by combining a minimizing movement scheme with a fixed point argument. Finally, it is shown that, when the contribution of the transport term is small, the equation possesses a global attractor and converges, as the transport term tends to zero, to a purely diffusive Cahn-Hilliard equation.

math.AP↗