Paolo Gidoni: publications

Preprints

A. Cagnetta and P. Gidoni, Existence of a periodic solution for superquadratic Hamiltonian systems with possible finite-time blow-up, submitted. [preprint]

Abstract
We prove a sufficient condition for the existence of a T-periodic solution for the planar system ż = F(t,z), characterized by the growth to infinity of the rotations made in one period by solutions starting at increasingly large initial values. Our result applies in particular to superquadratic Hamiltonian systems satisfying the Ambrosetti–Rabinowitz condition. The key novelty of the paper is that we do not require any growth condition on the flow to ensure global existence of solutions, allowing finite-time blow-up. Our method is based on a fixed-point theorem which exploits the rotational properties of the dynamics. To conclude, we discuss a family of examples of Hamiltonian systems showing finite-time blow-up.
P. Gidoni, A topological degree theory for rotating solutions of planar systems, submitted. [preprint]

Abstract
We present a generalized notion of degree for rotating solutions of planar systems. We prove a formula for the relation of such degree with the classical use of Brouwer's degree and obtain a twist theorem for the existence of periodic solutions, which is complementary to the Poincaré–Birkhoff Theorem. Some applications to asymptotically linear and superlinear differential equations are discussed.

Research papers

P. Bettiol, G. Colombo and P. Gidoni, Penalization and necessary optimality conditions for a class of nonsmooth sweeping processes, Set-Valued and Variational Analysis, 33, art. 33 (2025), doi: 10.1007/s11228-025-00770-6 [paper]

Abstract
In this paper we derive necessary optimality conditions for a Mayer problem involving a controlled sweeping process characterized by a moving set which is merely locally prox-regular (not necessarily smooth) and satisfies a constraint qualification condition formulated exclusively in terms of its normal vectors. We employ a penalization method based on the distance from the moving constraint, which allows convergence estimates that are uniform with respect to the control and, moreover, the strong W^1,2-convergence of the approximating solutions. An example of nonsmooth mechanics with finite degrees of freedom is presented.
P. Gidoni and A. Margheri, A Massera-type Theorem on relative-periodic solutions for a second-order model of rectilinear locomotion, SIAM Journal on Applied Dynamical Systems 24, 2180–2204 (2025), doi: 10.1137/24M1706955 [preprint] [paper]

Abstract
We study the existence of a global periodic attractor for the reduced dynamics of a discrete toy model for rectilinear crawling locomotion, corresponding to a limit cycle in the shape and velocity variables. The body of the crawler consists of a chain of point masses, joined by active elastic links and subject to smooth friction forces, so that the dynamics is described by a system of second order differential equations. Our main result is of Massera-type, namely we show that the existence of a bounded solution implies the existence of the global periodic attractor for the reduced dynamics. In establishing this result, a contractive property of the dynamics of our model plays a central role. We then prove sufficient conditions on the friction forces for the existence of a bounded solution, and therefore of the attractor. We also provide an example showing that, if we consider more general friction forces, such as smooth approximations of dry friction, bounded solutions may not exist.
P. Gidoni, M. Morandotti and M. Zoppello, Gait controllability of length-changing slender microswimmers, Mathematics and Mechanics of Complex Systems 12, 471–505 (2024), doi: 10.2140/memocs.2024.12.471. [preprint] [paper]

Abstract
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.
J. Kropáček, C. Maslen, P. Gidoni, P. Cigler, F. Štěpánek and I. Řehoř, Light-Responsive Hydrogel Microcrawlers, Powered and Steered with Spatially Homogeneous Illumination, Soft Robotics 11, 531–538 (2024), doi: 10.1089/soro.2023.0074. [preprint] [paper]

Abstract
Sub-millimeter untethered locomoting robots hold promise to radically change multiple areas of human activity such as microfabrication/assembly or health care. To overcome the associated hurdles of such a degree of robot miniaturization, radically new approaches are being adopted, often relying on soft actuating polymeric materials. Here, we present light-driven, crawling microrobots that locomote by a single degree of freedom actuation of their light-responsive tail section. The direction of locomotion is dictated by the robot body design and independent of the spatial modulation of the light stimuli, allowing simultaneous multidirectional motion of multiple robots. Moreover, we present a method for steering such robots by reversibly deforming their front section, using ultraviolet (UV) light as a trigger. The deformation dictates the robot locomotion, performing right- or left-hand turning when the UV is turned on or off respectively. The robots' motion and navigation are not coupled to the position of the light sources, which enables simultaneous locomotion of multiple robots, steering of robots and brings about flexibility with the methods to deliver the light to the place of robot operation.
P. Gidoni, A. Margheri and C. Rebelo, Limit cycles for dynamic crawling locomotors with periodic prescribed shape, Zeitschrift für angewandte Mathematik und Physik 74, art. 46 (2023), doi: 10.1007/s00033-023-01941-x. [preprint] [paper]

Abstract
We study the asymptotic evolution of a family of dynamic models of crawling locomotion, with the aim to introduce a well-posed characterization of a gait as a limit behaviour. The locomotors, which might have a discrete or continuous body, move on a line with a periodic prescribed shape change, and might possibly be subject to external forcing (e.g. crawling on a slope). We discuss how their behaviour is affected by different types of friction forces, including also set-valued ones such as dry friction. We show that, under mild natural assumptions, the dynamics always converge to a relative periodic solution. The asymptotic average velocity of the crawler yet might still depend on its initial state, so we provide additional assumption for its uniqueness. In particular, we show that the asymptotic average velocity is unique both for strictly monotone friction forces, and also for dry friction, provided in the latter case that the actuation is sufficiently smooth (for discrete models) or that the friction coefficients are always nonzero (for continuous models). We present several examples and counterexamples illustrating the necessity of our assumptions.
P. Gidoni, Existence of a periodic solution for superlinear second order ODEs, Journal of Differential Equations 345, 401–417 (2023), doi: 10.1016/j.jde.2022.11.054 . [preprint] [paper]

Abstract
We prove a necessary and sufficient condition for the existence of a T-periodic solution for the time-periodic second order differential equation x'' + f(t,x) + p(t,x,x') = 0, where f grows superlinearly in x uniformly in time, while p is bounded. Our method is based on a fixed-point theorem which uses the rotational properties of the dynamics.
G. Colombo, P. Gidoni and E. Vilches, Stabilization of periodic sweeping processes and asymptotic average velocity for soft locomotors with dry friction, Discrete & Continuous Dynamical Systems – A, 42, 737–757 (2022), doi: 10.3934/dcds.2021135. [preprint] [paper]

Abstract
We study the asymptotic stability of periodic solutions for sweeping processes defined by a polyhedron with translationally moving faces. Previous results are improved by obtaining a stronger W^{1,2} convergence. Then we present an application to a model of crawling locomotion. Our stronger convergence allows us to prove the stabilization of the system to a running-periodic (or derivo-periodic, or relative-periodic) solution and the well-posedness of an average asymptotic velocity depending only on the gait adopted by the crawler. Finally, we discuss some examples of finite-time versus asymptotic-only convergence.
P. Gidoni and F. Riva, A vanishing inertia analysis for finite dimensional rate-independent systems with nonautonomous dissipation and an application to soft crawlers, Calculus of Variations and Partial Differential Equations 60, art. 191 (2021), doi: 10.1007/s00526-021-02067-6. [preprint] [paper]

Abstract
We study the approximation of quasistatic evolutions, formulated as abstract finite-dimensional rate-independent systems, via a vanishing-inertia asymptotic analysis of dynamic evolutions. We prove the uniform convergence of dynamical solutions to the quasistatic one, employing the concept of energetic solution. Motivated by applications in soft locomotion, we allow time-dependence of the dissipation potential, and translation invariance of the potential energy.
G. Colombo and P. Gidoni, On the optimal control of rate-independent soft crawlers, Journal de Mathématiques Pures et Appliquées 146, 127–157 (2021), doi: 10.1016/j.matpur.2020.11.005. [preprint] [paper]

Abstract
Existence of optimal solutions and necessary optimality conditions for a controlled version of Moreau's sweeping process are derived. The control is a measurable ingredient of the dynamics and the constraint set is a polyhedron. The novelty consists in considering time periodic trajectories, adding the requirement that the control have zero average, and considering an integral functional that lacks weak semicontinuity. A model coming from the locomotion of a soft-robotic crawler, that motivated our setting, is analysed in detail. In obtaining necessary conditions, an improvement of the method of discrete approximations is used.
A. Fonda and P. Gidoni, Coupling linearity and twist: an extension of the Poincaré–Birkhoff Theorem for Hamiltonian systems, Nonlinear Differential Equations and Applications NoDEA 27, art. 55 (2020), doi: 10.1007/s00030-020-00653-9. [preprint] [paper]

Abstract
We provide an extension of the Poincaré–Birkhoff Theorem for systems coupling linear components with twisting components. Applications are given both to weakly coupled Hamiltonian systems where, e.g., a superlinear or sublinear behaviour is assumed in the nonlinear part of the coupling in order to recover the needed twist conditions, and to local perturbations of superintegrable systems, showing the survival of a number of periodic solutions from a lower-dimensional torus.
G. Feltrin and P. Gidoni, Multiplicity of clines for systems of indefinite differential equations arising from a multilocus population genetics model, Nonlinear Analysis: Real World Applications 54, 103108 (2020), doi: 10.1016/j.nonrwa.2020.103108. [preprint] [paper]

Abstract
We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at multiple loci. In mathematical terms, this corresponds to the study of a Neumann boundary value problem with sign-changing coupling-weights and superlinear nonlinearities. Using a topological degree approach, we prove existence of 2^N positive fully nontrivial solutions when the real positive parameters are sufficiently large.
P. Gidoni, G.B. Maggiani and R. Scala, Existence and regularity of solutions for an evolution model of perfectly plastic plates, Communications on Pure and Applied Analysis 18, 1783–1826 (2019), doi: 10.3934/cpaa.2019084. [preprint] [paper]

Abstract
We continue the study of a dynamic evolution model for perfectly plastic plates, recently derived from three-dimensional Prandtl-Reuss plasticity. We extend the previous existence result by introducing non-zero external forces in the model, and we discuss the regularity of the solutions thus obtained. In particular, we show that the first derivatives with respect to space of the stress tensor are locally square integrable.
P. Gidoni, A. Margheri, Lower bound on the number of periodic solutions for asymptotically linear planar Hamiltonian systems, Discrete & Continuous Dynamical Systems – A 39, 585–606 (2019), doi: 10.3934/dcds.2019024. [preprint] [paper]

Abstract
In this work we prove the lower bound for the number of T-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, T-periodic in time, with T-Maslov indices i_0, i_∞ at the origin and at infinity, has at least |i_∞ - i_0| periodic solutions, and an additional one if i_0 is even. Our argument combines the Poincaré–Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.
P. Gidoni, Rate–independent soft crawlers, Quarterly Journal of Mechanics and Applied Mathematics 71, 369–409 (2018), doi: 10.1093/qjmam/hby010. [preprint] [paper]

Abstract
This paper applies the theory of rate-independent systems to model the locomotion of bio-mimetic soft crawlers. We prove the well-posedness of the approach and illustrate how the various strategies adopted by crawlers to achieve locomotion, such as friction anisotropy, complex shape changes and control on the friction coefficients, can be effectively described in terms of stasis domains. Compared to other rate-independent systems, locomotion models do not present any Dirichlet boundary condition, so that all rigid translations are admissible displacements, resulting in a non-coercivity of the energy term. We prove that existence and uniqueness of solution are guaranteed under suitable assumptions on the dissipation potential. Such results are then extended to the case of time-dependent dissipation.
P. Gidoni and A. DeSimone, On the genesis of directional friction through bristle-like mediating elements, ESAIM: Control, Optimisation and Calculus of Variations 23, 1023–1046 (2017), doi: 10.1051/cocv/2017030. [preprint] [paper]

Abstract
We propose an explanation of the genesis of directional dry friction, as emergent property of the oscillations produced in a bristle-like mediating element by the interaction with microscale fluctuations on the surface. Mathematically, we extend a convergence result by Mielke, for Prandtl–Tomlinson-like systems, considering also non-homothetic scalings of a wiggly potential. This allows us to apply the result to some simple mechanical models, that exemplify the interaction of a bristle with a surface having small fluctuations. We find that the resulting friction is the product of two factors: a geometric one, depending on the bristle angle and on the fluctuation profile, and a energetic one, proportional to the normal force exchanged between the bristle-like element and the surface. Finally, we apply our result to discuss the with the nap/against the nap asymmetry.
A. Fonda and P. Gidoni, An avoiding cones condition for the Poincaré–Birkhoff Theorem, Journal of Differential Equations 262, 1064–1084 (2017), doi: 10.1016/j.jde.2016.10.002. [preprint] [paper]

Abstract
We provide a geometric assumption which unifies and generalizes the conditions proposed in [11], [12], so to obtain a higher dimensional version of the Poincaré–Birkhoff fixed point Theorem for Poincaré maps of Hamiltonian systems.
P. Gidoni and A. DeSimone, Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler, Meccanica 52, 587–601 (2017), doi: 10.1007/s11012-016-0408-0. [preprint] [paper]

Abstract
We formulate and solve the locomotion problem for a bio-inspired crawler consisting of two active elastic segments (i.e., capable of changing their rest lengths), resting on three supports providing directional frictional interactions. The problem consists in finding the motion produced by a given, slow actuation history. By focusing on the tensions in the elastic segments, we show that the evolution laws for the system are entirely analogous to the flow rules of elasto-plasticity. In particular, sliding of the supports and hence motion cannot occur when the tensions are in the interior of certain convex regions (stasis domains), while support sliding (and hence motion) can only take place when the tensions are on the boundary of such regions (slip surfaces). We solve the locomotion problem explicitly in a few interesting examples. In particular, we show that, for a suitable range of the friction parameters, specific choices of the actuation strategy can lead to net displacements also in the direction of higher friction.
A. Fonda, M. Garrione and P. Gidoni, Periodic perturbations of Hamiltonian systems, Advances in Nonlinear Analysis 5, 367–382 (2016), doi: 10.1515/anona-2015-0122. [preprint] [paper]

Abstract
We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.
A. Fonda and P. Gidoni, Generalizing the Poincaré–Miranda Theorem: the avoiding cones condition, Annali di Matematica Pura ed Applicata 195, 1347–1371 (2016), doi: 10.1007/s10231-015-0519-6. [preprint] [paper]

Abstract
After proposing a variant of the Poincaré–Bohl theorem, we extend the Poincaré–Miranda theorem in several directions, by introducing an avoiding cones condition. We are thus able to deal with functions defined on various types of convex domains, and situations where the topological degree may be different from ±1. An illustrative application is provided for the study of functionals having degenerate multi-saddle points.
A. DeSimone, P. Gidoni and G. Noselli, Liquid crystal elastomer strips as soft crawlers, Journal of the Mechanics and Physics of Solids 84, 254–272 (2015), doi: 10.1016/j.jmps.2015.07.017. [preprint] [paper]

Abstract
In this paper, we speculate on a possible application of Liquid Crystal Elastomers to the field of soft robotics. In particular, we study a concept for limbless locomotion that is amenable to miniaturisation. For this purpose, we formulate and solve the evolution equations for a strip of nematic elastomer, subject to directional frictional interactions with a flat solid substrate, and cyclically actuated by a spatially uniform, time-periodic stimulus (e.g., temperature change). The presence of frictional forces that are sensitive to the direction of sliding transforms reciprocal, 'breathing-like' deformations into directed forward motion. We derive formulas quantifying this motion in the case of distributed friction, by solving a differential inclusion for the displacement field. The simpler case of concentrated frictional interactions at the two ends of the strip is also solved, in order to provide a benchmark to compare the continuously distributed case with a finite-dimensional benchmark. We also provide explicit formulas for the axial force along the crawler body.
A. Fonda and P. Gidoni, A permanence theorem for local dynamical systems, Nonlinear Analysis, Theory, Methods and Applications 121, 73–81 (2015), doi: 10.1016/j.na.2014.10.011. [preprint] [paper]

Abstract
We provide a necessary and sufficient condition for permanence related to a local dynamical system on a suitable topological space. We then present an illustrative application to a Lotka–Volterra predator–prey model with intraspecific competition.
P. Gidoni, G. Noselli and A. DeSimone, Crawling on directional surfaces, International Journal of Non-Linear Mechanics 61, 65–73 (2014), doi: 10.1016/j.ijnonlinmec.2014.01.012. [preprint] [paper]

Abstract
In this paper we study crawling locomotion based on directional frictional interactions, namely, frictional forces that are sensitive to the sign of the sliding velocity. Surface interactions of this type are common in biology, where they arise from the presence of inclined hairs or scales at the crawler/substrate interface, leading to low resistance when sliding 'along the grain', and high resistance when sliding 'against the grain'. This asymmetry can be exploited for locomotion, in a way analogous to what is done in cross-country skiing (classic style, diagonal stride). We focus on a model system, namely, a continuous one-dimensional crawler and provide a detailed study of the motion resulting from several strategies of shape change. In particular, we provide explicit formulae for the displacements attainable with reciprocal extensions and contractions (breathing), or through the propagation of extension or contraction waves.

Other writings

P. Gidoni, Two explorations in Dynamical Systems and Mechanics. Avoiding cones conditions and higher dimensional twist – Directional friction in bio-inspired locomotion, PhD dissertation. [pdf]

Paolo Gidoni's website

Publications

Preprints

Research papers

Other writings