Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem wa... more Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem was presented. The model considers a diffusion equation enforcing the balance of the transported masses with a timevarying conductivity that evolves proportionally to the transported flux. In this paper we present an extension of this model that considers a time derivative of the conductivity that grows as a power law of the transport flux with exponent β > 0. A sub-linear growth (0 < β < 1) penalizes the flux intensity and promotes distributed transport, with equilibrium solutions that are reminiscent of Congested Transport Problems. On the contrary, a super-linear growth (β > 1) favors flux intensity and promotes concentrated transport, leading to the emergence of steady-state "singular" and "fractal-like" configurations that resemble those of Branched Transport Problems. We derive a numerical discretization of the proposed model that is accurate, efficient, and robust for a wide range of scenarios. For β > 1 the numerical model is able to reproduce highly irregular and fractal-like formations without any a-priory structural assumption.
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.
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.
Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem wa... more Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem was presented. The model considers a diffusion equation enforcing the balance of the transported masses with a timevarying conductivity that evolves proportionally to the transported flux. In this paper we present an extension of this model that considers a time derivative of the conductivity that grows as a power law of the transport flux with exponent β > 0. A sub-linear growth (0 < β < 1) penalizes the flux intensity and promotes distributed transport, with equilibrium solutions that are reminiscent of Congested Transport Problems. On the contrary, a super-linear growth (β > 1) favors flux intensity and promotes concentrated transport, leading to the emergence of steady-state "singular" and "fractal-like" configurations that resemble those of Branched Transport Problems. We derive a numerical discretization of the proposed model that is accurate, efficient, and robust for a wide range of scenarios. For β > 1 the numerical model is able to reproduce highly irregular and fractal-like formations without any a-priory structural assumption.
In this work we study and expand a model describing the dynamics of a unicellular slime mold, Phy... more In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. We propose an extension of this model that abandons the graph structure and moves to a continuous domain. The new model, that couples an elliptic equation enforcing PP density balance with an ODE governing the dynamics of the flow of information along the PP body, is analyzed by recasting it into an infinite-dimensional dynamical system. We are able to show well-posedness of the proposed model for sufficiently small times under the hypothesis of Hölder continuous diffusion coefficients and essentially bounded forcing functions, which play the role of food sources. Numerical evidence, shows that the model is capable of describing the slime mold dynamics also for large times, accurately reproducing the PP behavior. A notable result related to the original model is that it is equivalent to an optimal transportation problem over the graph as time tends to infinity. In our case, we can only conjecture that our extension presents a time-asymptotic equilibrium. This equilibrium point is precisely the solution of the Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal transportation problems. Numerical results obtained with our approach, which combines P 1 Finite Elements with forward Euler time stepping, show that the approximate solution converges at large times to an equilibrium configuration that well compares with the numerical solution of the MK-equations.
We consider the class of minimising cost problems for dynamical systems with differential constra... more We consider the class of minimising cost problems for dynamical systems with differential constraints of higher order 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, with uptq curve in a compact set K Ă R m of control parameters. For these systems we establish a perfect analogue of the usual Pontryagin Maximum Principle (PMP) for control problems with first order constraints. The proof is a consequence of our previous results on control problems with constraints of variational type. For k " 1, it gives a new independent proof of the classical PMP.
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.
A version of the Arnol’d conjecture, first studied by Conley, Zehnder, giving a generalization of... more A version of the Arnol’d conjecture, first studied by Conley, Zehnder, giving a generalization of the Poincaré-Birkhoff last geometrical theorem, is here proved inside Viterbo’s framework of the generating functions quadratic at infinity. By making this goal, we give short recalls on some tools here exploited, often utilized in symplectic topology.
We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fiel... more We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fields, converging to a fixed near the identity $$C^1$$ C 1 symplectic diffeomorphism $$\psi $$ ψ . This convergence is proved to be uniformly exponentially fast, in a non analytic symplectic topology framework.
Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem wa... more Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem was presented. The model considers a diffusion equation enforcing the balance of the transported masses with a timevarying conductivity that evolves proportionally to the transported flux. In this paper we present an extension of this model that considers a time derivative of the conductivity that grows as a power law of the transport flux with exponent β > 0. A sub-linear growth (0 < β < 1) penalizes the flux intensity and promotes distributed transport, with equilibrium solutions that are reminiscent of Congested Transport Problems. On the contrary, a super-linear growth (β > 1) favors flux intensity and promotes concentrated transport, leading to the emergence of steady-state "singular" and "fractal-like" configurations that resemble those of Branched Transport Problems. We derive a numerical discretization of the proposed model that is accurate, efficient, and robust for a wide range of scenarios. For β > 1 the numerical model is able to reproduce highly irregular and fractal-like formations without any a-priory structural assumption.
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.
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.
Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem wa... more Recently a Dynamic-Monge-Kantorovich formulation of the PDEbased L 1-optimal transport problem was presented. The model considers a diffusion equation enforcing the balance of the transported masses with a timevarying conductivity that evolves proportionally to the transported flux. In this paper we present an extension of this model that considers a time derivative of the conductivity that grows as a power law of the transport flux with exponent β > 0. A sub-linear growth (0 < β < 1) penalizes the flux intensity and promotes distributed transport, with equilibrium solutions that are reminiscent of Congested Transport Problems. On the contrary, a super-linear growth (β > 1) favors flux intensity and promotes concentrated transport, leading to the emergence of steady-state "singular" and "fractal-like" configurations that resemble those of Branched Transport Problems. We derive a numerical discretization of the proposed model that is accurate, efficient, and robust for a wide range of scenarios. For β > 1 the numerical model is able to reproduce highly irregular and fractal-like formations without any a-priory structural assumption.
In this work we study and expand a model describing the dynamics of a unicellular slime mold, Phy... more In this work we study and expand a model describing the dynamics of a unicellular slime mold, Physarum Polycephalum (PP), which was proposed to simulate the ability of PP to find the shortest path connecting two food sources in a maze. The original model describes the dynamics of the slime mold on a finite dimensional planar graph using a pipe-flow analogy whereby mass transfer occurs because of pressure differences with a conductivity coefficient that varies with the flow intensity. We propose an extension of this model that abandons the graph structure and moves to a continuous domain. The new model, that couples an elliptic equation enforcing PP density balance with an ODE governing the dynamics of the flow of information along the PP body, is analyzed by recasting it into an infinite-dimensional dynamical system. We are able to show well-posedness of the proposed model for sufficiently small times under the hypothesis of Hölder continuous diffusion coefficients and essentially bounded forcing functions, which play the role of food sources. Numerical evidence, shows that the model is capable of describing the slime mold dynamics also for large times, accurately reproducing the PP behavior. A notable result related to the original model is that it is equivalent to an optimal transportation problem over the graph as time tends to infinity. In our case, we can only conjecture that our extension presents a time-asymptotic equilibrium. This equilibrium point is precisely the solution of the Monge-Kantorovich (MK) equations at the basis of the PDE formulation of optimal transportation problems. Numerical results obtained with our approach, which combines P 1 Finite Elements with forward Euler time stepping, show that the approximate solution converges at large times to an equilibrium configuration that well compares with the numerical solution of the MK-equations.
We consider the class of minimising cost problems for dynamical systems with differential constra... more We consider the class of minimising cost problems for dynamical systems with differential constraints of higher order 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, with uptq curve in a compact set K Ă R m of control parameters. For these systems we establish a perfect analogue of the usual Pontryagin Maximum Principle (PMP) for control problems with first order constraints. The proof is a consequence of our previous results on control problems with constraints of variational type. For k " 1, it gives a new independent proof of the classical PMP.
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.
A version of the Arnol’d conjecture, first studied by Conley, Zehnder, giving a generalization of... more A version of the Arnol’d conjecture, first studied by Conley, Zehnder, giving a generalization of the Poincaré-Birkhoff last geometrical theorem, is here proved inside Viterbo’s framework of the generating functions quadratic at infinity. By making this goal, we give short recalls on some tools here exploited, often utilized in symplectic topology.
We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fiel... more We propose the construction of a sequence of time one flows of autonomous Hamiltonian vector fields, converging to a fixed near the identity $$C^1$$ C 1 symplectic diffeomorphism $$\psi $$ ψ . This convergence is proved to be uniformly exponentially fast, in a non analytic symplectic topology framework.
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Papers by Franco Cardin