Franco Cardin - Academia.edu (original) (raw)
Papers by Franco Cardin
Some aspects of a geometrical version of the Cauchy problem for the Hamilton-Jacobi equation are ... more Some aspects of a geometrical version of the Cauchy problem for the Hamilton-Jacobi equation are studied in the general framework of symplectic mechanics. The knowledge of a global complete solution allows us to solve explicitly generalized Cauchy problems by global solutions, here represented by Morse families generating Lagrangian manifolds. This leads in a natural way to a general version of Huygens' principle.
Continuum Mechanics and Thermodynamics, 2019
We present a dynamical interpretation of the Monge-Kantorovich theory in a stationary regime. Thi... more We present a dynamical interpretation of the Monge-Kantorovich theory in a stationary regime. This new principle, akin to the Fermat principle of geometric optics, captures the geodesic character of many distribution networks such as plant roots, river basins and the physiological transportation network of metabolites in living systems. Our general continuum framework allows us to map a previously proposed phenomenological principle into a stationary Monge optimization principle in the Kantorovich relaxed format. Keywords Monge-Kantorovich • Metabolic scaling • Fermat principle 1 Introduction There are many instances where one is faced with the problem of transporting, at a certain rate, material from a point-like source to an extended (continuum) region endowed with a density of rates of consumption. For example, this happens in the case of a bakery, a dairy store, a water source or blood supply in our bodyproviding, respectively, bread, milk, water or metabolites. There is also the equally important dual problem of collecting material, from a (continuum) region, produced with given density of rates and transporting it to some collection center. This happens, for example, for the roots of a plant which collect water from the underlying soil and convey it to the base of the trunk or a river basin where the rain is conveyed into streams of increasing order and finally into the main river. In view of the many possible applications, it is quite relevant to solve the underlying optimization problem of what is the most efficient way to design a transportation network given the spatial distribution of sinks or sources. Our aim is to present a dynamical 'stationary' version of the classical 'static' Monge variational principle for optimal transport, and more precisely, by invoking the Kantorovich relaxed version of Monge problem, which is essential in order to discuss the stationary dynamics from a point-like source to the whole distribution region-the dual problem being similar. Communicated by Andreas Öchsner.
Ricerche di Matematica
We propose some crystalline materials showing a strong correspondence with a construction by Ball... more We propose some crystalline materials showing a strong correspondence with a construction by Ball and Murat for elastostatic problems. Such a construction, translated into a space-time setting, is producing a plenty of turbulence self similar solutions.
arXiv (Cornell University), Jul 19, 2023
In this work, we propose a geometric framework for analyzing mechanical manipulation, for instanc... more In this work, we propose a geometric framework for analyzing mechanical manipulation, for instance, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. In the first part of the paper, we review how quasi-static mechanical manipulation tasks can be naturally described via the so-called force-space, i.e. the cotangent bundle of the configuration space, and its Lagrangian submanifolds. Then, via a second order analysis, we derive the control Hessian of total energy. As this is not necessarily positive-definite, from an optimal control perspective, we propose the use of the squared-Hessian, also motivated by insights derived from both mechanics (Gauss' Principle) and biology (Separation Principle). In the second part of the paper, we apply such methods to the problem of an elastically-driven, inverted pendulum. Despite its apparent simplicity, this example is representative of an important class of robotic manipulation problems for which we show how a smooth elastic potential can be derived by regularizing mechanical contact. We then show how graph theory can be used to connect each numerical solution to 'nearby' ones, with weights derived from the very metric introduced in the first part of the paper.
arXiv (Cornell University), Oct 13, 2021
Exploiting our previous results on higher order controlled Lagrangians in [Nonlinear Anal. 207 (2... more Exploiting our previous results on higher order controlled Lagrangians in [Nonlinear Anal. 207 (2021), 112263], we derive here an analogue of the classical first order Pontryagin Maximum Principle (PMP) for cost minimising problems subjected to higher order differential constraints d k x j dt k " f j`t , xptq, dx dt ptq,. .. , d k´1 x dt k´1 ptq, uptq˘, t P r0, T s, where uptq is a control curve in a compact set K Ă R m. This result and its proof can be considered as a detailed illustration of one of the claims of that previous paper, namely that the results of that paper, originally established in a smooth differential geometric framework, yield directly properties holding under much weaker and more common assumptions. In addition, for further clarifying our motivations, in the last section we display a couple of quick indications on how the two-step approach of this paper (i.e., a preliminary easy-to-get differential geometric discussion followed by a refining analysis to weaken the regularity assumptions) might be fruitfully exploited also in the context of control problems governed by partial differential equations or in studies on the dynamics of controlled mechanical systems.
arXiv (Cornell University), May 11, 2023
Several authors have recently highlighted the need for a new dynamical paradigm in the modelling ... more Several authors have recently highlighted the need for a new dynamical paradigm in the modelling of brain workings and evolution. In particular, the models should include the possibility of dynamic synaptic weights T ij in the neuron-neuron interaction matrix, radically overcoming the classical Hopfield setting. Krotov and Hopfield proposed a non constant, still symmetric model, leading to a vector field describing a gradient type dynamics and then including a Lyapunov-like energy function. In this note, we firstly will detail the general condition to produce a Hopfield like vector field of gradient type obtaining, as a particular case, the Krotov-Hopfield condition. Secondly, we abandon the symmetry because of two relevant physiological facts: (1) the actual neural connections have a marked directional character and (2) the gradient structure deriving from the symmetry forces the dynamics always towards stationary points, prescribing every pattern to be recognized. We propose a novel model including a set limited but varying controls |ξ ij | ≤ K used for correcting a starting constant interaction matrix, T ij = A ij + ξ ij. Besides, we introduce a reasonable family of controlled variational functionals to be optimized. This allows us to reproduce the following three possible outcomes when submitting a pattern to the learning system. If (1) the dynamics leads to an already existing stationary point without activating the controls, the system has recognized an existing pattern. If (2) a new stationary point is reached by the activation of controls, then the system has learned a new pattern. If (3) the dynamics is unable to reach neither existing or new stationary points, then the system is unable to recognize or learn the pattern submitted. A further feature (4), appears to model forgetting and restoring memory.
Lecture notes of the Unione Matematica Italiana, Oct 29, 2014
The connection between swimming and control theory is attracting increasing attention in the rece... more The connection between swimming and control theory is attracting increasing attention in the recent literature [7, 1, 3, 4]. Starting from an idea of Alberto Bressan [2] we study the system of a planar body whose position and shape are described by a nite number of parameters, and is immersed in a 2-dimensional ideal and incompressible uid. This special case has an interesting geometric nature and there is an attractive mathematical framework for it, since it can be interpreted in terms of gauge eld on the space of shapes. We focus [9] on a class of deformations near the identity since they are dieomeorphisms whose existence is ensured by the Riemann mapping theorem. They can be represented by converging series of complex numbers of which we keep only a nite number of terms. We show that thanks to the linearity of the Euler equations which govern the motion of the uid, the system’s Lagrangian is the sum of the kinetic energy of the body and of the uid. We focus our attention on a cr...
Communications in Contemporary Mathematics, 2021
We extend to the spatial case a technique of integration of the close encounters formulated by Tu... more We extend to the spatial case a technique of integration of the close encounters formulated by Tullio Levi-Civita for the planar restricted three-body problem. We consider the Hamiltonian introduced in the Kustaanheimo–Stiefel regularization and construct a complete integral of the related Hamilton–Jacobi equation by means of a series convergent in a neighborhood of the collisions with the primary or secondary body.
Some aspects of a geometrical version of the Cauchy problem for the Hamilton-Jacobi equation are ... more Some aspects of a geometrical version of the Cauchy problem for the Hamilton-Jacobi equation are studied in the general framework of symplectic mechanics. The knowledge of a global complete solution allows us to solve explicitly generalized Cauchy problems by global solutions, here represented by Morse families generating Lagrangian manifolds. This leads in a natural way to a general version of Huygens' principle.
Continuum Mechanics and Thermodynamics, 2019
We present a dynamical interpretation of the Monge-Kantorovich theory in a stationary regime. Thi... more We present a dynamical interpretation of the Monge-Kantorovich theory in a stationary regime. This new principle, akin to the Fermat principle of geometric optics, captures the geodesic character of many distribution networks such as plant roots, river basins and the physiological transportation network of metabolites in living systems. Our general continuum framework allows us to map a previously proposed phenomenological principle into a stationary Monge optimization principle in the Kantorovich relaxed format. Keywords Monge-Kantorovich • Metabolic scaling • Fermat principle 1 Introduction There are many instances where one is faced with the problem of transporting, at a certain rate, material from a point-like source to an extended (continuum) region endowed with a density of rates of consumption. For example, this happens in the case of a bakery, a dairy store, a water source or blood supply in our bodyproviding, respectively, bread, milk, water or metabolites. There is also the equally important dual problem of collecting material, from a (continuum) region, produced with given density of rates and transporting it to some collection center. This happens, for example, for the roots of a plant which collect water from the underlying soil and convey it to the base of the trunk or a river basin where the rain is conveyed into streams of increasing order and finally into the main river. In view of the many possible applications, it is quite relevant to solve the underlying optimization problem of what is the most efficient way to design a transportation network given the spatial distribution of sinks or sources. Our aim is to present a dynamical 'stationary' version of the classical 'static' Monge variational principle for optimal transport, and more precisely, by invoking the Kantorovich relaxed version of Monge problem, which is essential in order to discuss the stationary dynamics from a point-like source to the whole distribution region-the dual problem being similar. Communicated by Andreas Öchsner.
Ricerche di Matematica
We propose some crystalline materials showing a strong correspondence with a construction by Ball... more We propose some crystalline materials showing a strong correspondence with a construction by Ball and Murat for elastostatic problems. Such a construction, translated into a space-time setting, is producing a plenty of turbulence self similar solutions.
arXiv (Cornell University), Jul 19, 2023
In this work, we propose a geometric framework for analyzing mechanical manipulation, for instanc... more In this work, we propose a geometric framework for analyzing mechanical manipulation, for instance, by a robotic agent. Under the assumption of conservative forces and quasi-static manipulation, we use energy methods to derive a metric. In the first part of the paper, we review how quasi-static mechanical manipulation tasks can be naturally described via the so-called force-space, i.e. the cotangent bundle of the configuration space, and its Lagrangian submanifolds. Then, via a second order analysis, we derive the control Hessian of total energy. As this is not necessarily positive-definite, from an optimal control perspective, we propose the use of the squared-Hessian, also motivated by insights derived from both mechanics (Gauss' Principle) and biology (Separation Principle). In the second part of the paper, we apply such methods to the problem of an elastically-driven, inverted pendulum. Despite its apparent simplicity, this example is representative of an important class of robotic manipulation problems for which we show how a smooth elastic potential can be derived by regularizing mechanical contact. We then show how graph theory can be used to connect each numerical solution to 'nearby' ones, with weights derived from the very metric introduced in the first part of the paper.
arXiv (Cornell University), Oct 13, 2021
Exploiting our previous results on higher order controlled Lagrangians in [Nonlinear Anal. 207 (2... more Exploiting our previous results on higher order controlled Lagrangians in [Nonlinear Anal. 207 (2021), 112263], we derive here an analogue of the classical first order Pontryagin Maximum Principle (PMP) for cost minimising problems subjected to higher order differential constraints d k x j dt k " f j`t , xptq, dx dt ptq,. .. , d k´1 x dt k´1 ptq, uptq˘, t P r0, T s, where uptq is a control curve in a compact set K Ă R m. This result and its proof can be considered as a detailed illustration of one of the claims of that previous paper, namely that the results of that paper, originally established in a smooth differential geometric framework, yield directly properties holding under much weaker and more common assumptions. In addition, for further clarifying our motivations, in the last section we display a couple of quick indications on how the two-step approach of this paper (i.e., a preliminary easy-to-get differential geometric discussion followed by a refining analysis to weaken the regularity assumptions) might be fruitfully exploited also in the context of control problems governed by partial differential equations or in studies on the dynamics of controlled mechanical systems.
arXiv (Cornell University), May 11, 2023
Several authors have recently highlighted the need for a new dynamical paradigm in the modelling ... more Several authors have recently highlighted the need for a new dynamical paradigm in the modelling of brain workings and evolution. In particular, the models should include the possibility of dynamic synaptic weights T ij in the neuron-neuron interaction matrix, radically overcoming the classical Hopfield setting. Krotov and Hopfield proposed a non constant, still symmetric model, leading to a vector field describing a gradient type dynamics and then including a Lyapunov-like energy function. In this note, we firstly will detail the general condition to produce a Hopfield like vector field of gradient type obtaining, as a particular case, the Krotov-Hopfield condition. Secondly, we abandon the symmetry because of two relevant physiological facts: (1) the actual neural connections have a marked directional character and (2) the gradient structure deriving from the symmetry forces the dynamics always towards stationary points, prescribing every pattern to be recognized. We propose a novel model including a set limited but varying controls |ξ ij | ≤ K used for correcting a starting constant interaction matrix, T ij = A ij + ξ ij. Besides, we introduce a reasonable family of controlled variational functionals to be optimized. This allows us to reproduce the following three possible outcomes when submitting a pattern to the learning system. If (1) the dynamics leads to an already existing stationary point without activating the controls, the system has recognized an existing pattern. If (2) a new stationary point is reached by the activation of controls, then the system has learned a new pattern. If (3) the dynamics is unable to reach neither existing or new stationary points, then the system is unable to recognize or learn the pattern submitted. A further feature (4), appears to model forgetting and restoring memory.
Lecture notes of the Unione Matematica Italiana, Oct 29, 2014
The connection between swimming and control theory is attracting increasing attention in the rece... more The connection between swimming and control theory is attracting increasing attention in the recent literature [7, 1, 3, 4]. Starting from an idea of Alberto Bressan [2] we study the system of a planar body whose position and shape are described by a nite number of parameters, and is immersed in a 2-dimensional ideal and incompressible uid. This special case has an interesting geometric nature and there is an attractive mathematical framework for it, since it can be interpreted in terms of gauge eld on the space of shapes. We focus [9] on a class of deformations near the identity since they are dieomeorphisms whose existence is ensured by the Riemann mapping theorem. They can be represented by converging series of complex numbers of which we keep only a nite number of terms. We show that thanks to the linearity of the Euler equations which govern the motion of the uid, the system’s Lagrangian is the sum of the kinetic energy of the body and of the uid. We focus our attention on a cr...
Communications in Contemporary Mathematics, 2021
We extend to the spatial case a technique of integration of the close encounters formulated by Tu... more We extend to the spatial case a technique of integration of the close encounters formulated by Tullio Levi-Civita for the planar restricted three-body problem. We consider the Hamiltonian introduced in the Kustaanheimo–Stiefel regularization and construct a complete integral of the related Hamilton–Jacobi equation by means of a series convergent in a neighborhood of the collisions with the primary or secondary body.