arXiv subjects

Giovanni Russo

Publications and source records attributed to Giovanni Russo.

At least 55 records · Page 3Linked to original sources

arXiv2024

Self-regulated biological transportation structures with general entropy dissipation: 2D case and leaf-shaped domain

In recent years, the study of biological transportation networks has attracted significant interest, focusing on their self-regulating, demand-driven nature. This paper examines a mathematical model for these networks, featuring nonlinear elliptic equations for pressure and an auxiliary variable, and a reaction-diffusion parabolic equation for the conductivity tensor, introduced in \cite{portaro2022emergence}. The model, based on an energy functional with diffusive and metabolic terms, allows for various entropy generating functions, facilitating its application to different biological scenarios. We proved a local well-posedness result for the problem in Hölder spaces employing Schauder and semigroup theory. Then, after a suitable parameter reduction through scaling, we computed the numerical solution for the proposed system using a recently developed ghost nodal finite element method \cite{astuto2024nodal}. An interesting aspect emerges when the solution is very articulated and the branches occupy a wide region of the domain.

math.AP↗

arXiv2024

On Convex Data-Driven Inverse Optimal Control for Nonlinear, Non-stationary and Stochastic Systems

This paper is concerned with a finite-horizon inverse control problem, which has the goal of reconstructing, from observations, the possibly non-convex and non-stationary cost driving the actions of an agent. In this context, we present a result enabling cost reconstruction by solving an optimization problem that is convex even when the agent cost is not and when the underlying dynamics is nonlinear, non-stationary and stochastic. To obtain this result, we also study a finite-horizon forward control problem that has randomized policies as decision variables. We turn our findings into algorithmic procedures and show the effectiveness of our approach via in-silico and hardware validations. All experiments confirm the effectiveness of our approach.

math.OC↗

arXiv2024

High order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries

In this paper a fourth order finite difference ghost point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high order extension of the second ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods are numerically verified on several test problems.

math.NA↗

arXiv2024

On Weakly Contracting Dynamics for Convex Optimization

We analyze the convergence behavior of \emph{globally weakly} and \emph{locally strongly contracting} dynamics. Such dynamics naturally arise in the context of convex optimization problems with a unique minimizer. We show that convergence to the equilibrium is \emph{linear-exponential}, in the sense that the distance between each solution and the equilibrium is upper bounded by a function that first decreases linearly and then exponentially. As we show, the linear-exponential dependency arises naturally in certain dynamics with saturations. Additionally, we provide a sufficient condition for local input-to-state stability. Finally, we illustrate our results on, and propose a conjecture for, continuous-time dynamical systems solving linear programs.

math.OC↗

arXiv2024

Safe haptic teleoperations of admittance controlled robots with virtualization of the force feedback

Haptic teleoperations play a key role in extending human capabilities to perform complex tasks remotely, employing a robotic system. The impact of haptics is far-reaching and can improve the sensory awareness and motor accuracy of the operator. In this context, a key challenge is attaining a natural, stable and safe haptic human-robot interaction. Achieving these conflicting requirements is particularly crucial for complex procedures, e.g. medical ones. To address this challenge, in this work we develop a novel haptic bilateral teleoperation system (HBTS), featuring a virtualized force feedback, based on the motion error generated by an admittance controlled robot. This approach allows decoupling the force rendering system from the control of the interaction: the rendered force is assigned with the desired dynamics, while the admittance control parameters are separately tuned to maximize interaction performance. Furthermore, recognizing the necessity to limit the forces exerted by the robot on the environment, to ensure a safe interaction, we embed a saturation strategy of the motion references provided by the haptic device to admittance control. We validate the different aspects of the proposed HBTS, through a teleoperated blackboard writing experiment, against two other architectures. The results indicate that the proposed HBTS improves the naturalness of teleoperation, as well as safety and accuracy of the interaction.

cs.RO↗

arXiv2024

Modeling and Contractivity of Neural-Synaptic Networks with Hebbian Learning

This paper is concerned with the modeling and analysis of two of the most commonly used recurrent neural network models (i.e., Hopfield neural network and firing-rate neural network) with dynamic recurrent connections undergoing Hebbian learning rules. To capture the synaptic sparsity of neural circuits we propose a low dimensional formulation. We then characterize certain key dynamical properties. First, we give biologically-inspired forward invariance results. Then, we give sufficient conditions for the non-Euclidean contractivity of the models. Our contraction analysis leads to stability and robustness of time-varying trajectories -- for networks with both excitatory and inhibitory synapses governed by both Hebbian and anti-Hebbian rules. For each model, we propose a contractivity test based upon biologically meaningful quantities, e.g., neural and synaptic decay rate, maximum in-degree, and the maximum synaptic strength. Then, we show that the models satisfy Dale's Principle. Finally, we illustrate the effectiveness of our results via a numerical example.

math.OC↗

arXiv2024

Positive Competitive Networks for Sparse Reconstruction

We propose and analyze a continuous-time firing-rate neural network, the positive firing-rate competitive network (\pfcn), to tackle sparse reconstruction problems with non-negativity constraints. These problems, which involve approximating a given input stimulus from a dictionary using a set of sparse (active) neurons, play a key role in a wide range of domains, including for example neuroscience, signal processing, and machine learning. First, by leveraging the theory of proximal operators, we relate the equilibria of a family of continuous-time firing-rate neural networks to the optimal solutions of sparse reconstruction problems. Then, we prove that the \pfcn is a positive system and give rigorous conditions for the convergence to the equilibrium. Specifically, we show that the convergence: (i) only depends on a property of the dictionary; (ii) is linear-exponential, in the sense that initially the convergence rate is at worst linear and then, after a transient, it becomes exponential. We also prove a number of technical results to assess the contractivity properties of the neural dynamics of interest. Our analysis leverages contraction theory to characterize the behavior of a family of firing-rate competitive networks for sparse reconstruction with and without non-negativity constraints. Finally, we validate the effectiveness of our approach via a numerical example.

q-bio.NC↗

arXiv2024

Guaranteeing Control Requirements via Reward Shaping in Reinforcement Learning

In addressing control problems such as regulation and tracking through reinforcement learning, it is often required to guarantee that the acquired policy meets essential performance and stability criteria such as a desired settling time and steady-state error prior to deployment. Motivated by this necessity, we present a set of results and a systematic reward shaping procedure that (i) ensures the optimal policy generates trajectories that align with specified control requirements and (ii) allows to assess whether any given policy satisfies them. We validate our approach through comprehensive numerical experiments conducted in two representative environments from OpenAI Gym: the Inverted Pendulum swing-up problem and the Lunar Lander. Utilizing both tabular and deep reinforcement learning methods, our experiments consistently affirm the efficacy of our proposed framework, highlighting its effectiveness in ensuring policy adherence to the prescribed control requirements.

eess.SY↗

arXiv2024

A Multiplex Approach Against Disturbance Propagation in Nonlinear Networks with Delays

We consider both leaderless and leader-follower, possibly nonlinear, networks affected by time-varying communication delays. For such systems, we give a set of sufficient conditions that guarantee the convergence of the network towards some desired behaviour while simultaneously ensuring the rejection of polynomial disturbances and the non-amplification of other classes of disturbances across the network. To fulfill these desired properties, and prove our main results, we propose the use of a control protocol that implements a multiplex architecture. The use of our results for control protocol design is then illustrated in the context of formation control. The protocols are validated both in-silico and via an experimental set-up with real robots. All experiments confirm the effectiveness of our approach.

eess.SY↗

arXiv2023

In vivo learning-based control of microbial populations density in bioreactors

A key problem toward the use of microorganisms as bio-factories is reaching and maintaining cellular communities at a desired density and composition so that they can efficiently convert their biomass into useful compounds. Promising technological platforms for the real time, scalable control of cellular density are bioreactors. In this work, we developed a learning-based strategy to expand the toolbox of available control algorithms capable of regulating the density of a \textit{single} bacterial population in bioreactors. Specifically, we used a sim-to-real paradigm, where a simple mathematical model, calibrated using a few data, was adopted to generate synthetic data for the training of the controller. The resulting policy was then exhaustively tested in vivo using a low-cost bioreactor known as Chi.Bio, assessing performance and robustness. In addition, we compared the performance with more traditional controllers (namely, a PI and an MPC), confirming that the learning-based controller exhibits similar performance in vivo. Our work showcases the viability of learning-based strategies for the control of cellular density in bioreactors, making a step forward toward their use for the control of the composition of microbial consortia.

eess.SY↗

arXiv2023

On constant higher order mean curvature hypersurfaces in $\mathbb H^n \times \mathbb R$

We classify hypersurfaces with rotational symmetry and positive constant $r$-th mean curvature in $\mathbb H^n \times \mathbb R$. Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros--Rosenberg type theorem in $\mathbb H^n \times \mathbb R$: we show that compact connected hypersurfaces of constant $r$-th mean curvature embedded in $\mathbb H^n \times [0,\infty)$ with boundary in the slice $\mathbb H^n \times \{0\}$ are topological disks under suitable assumptions.

math.DG↗

arXiv2023

Multiscale Modeling with Differential Equations

Many physical systems are governed by ordinary or partial differential equations (see, for example, Chapter ''Differential equations'', ''System of Differential Equations''). Typically the solution of such systems are functions of time or of a single space variable (in the case of ODE's), or they depend on multidimensional space coordinates or on space and time (in the case of PDE's). In some cases, the solutions may depend on several time or space scales. An example governed by ODE's is the damped harmonic oscillator, in the two extreme cases of very small or very large damping, the cardiovascular system, where the thickness of the arteries and veins varies from centimeters to microns, shallow water equations, which are valid when water depth is small compared to typical wavelength of surface waves, and sorption kinetics, in which the range of interaction of a surfactant with an air bubble is much smaller than the size of the bubble itself. In all such cases a detailed simulation of the models which resolves all space or time scales is often inefficient or intractable, and usually even unnecessary to provide a reasonable description of the behavior of the system. In the Chapter ''Multiscale modeling with differential equations'' we present examples of systems described by ODE's and PDE's which are intrinsically multiscale, and illustrate how suitable modeling provide an effective way to capture the essential behavior of the solutions of such systems without resolving the small scales.

math.NA↗

arXiv2023

Time multiscale modeling of sorption kinetics I: uniformly accurate schemes for highly oscillatory advection-diffusion equation

In this paper we propose a numerical method to solve a 2D advection-diffusion equation, in the highly oscillatory regime. We use an efficient and robust integrator which leads to an accurate approximation of the solution without any time step-size restriction. Uniform first and second order numerical approximations in time are obtained with errors, and at a cost, that are independent of the oscillation frequency. {This work is part of a long time project, and the final goal is the resolution of a Stokes-advection-diffusion system, in which the expression for the velocity in the advection term, is the solution of the Stokes equations.} This paper focuses on the time multiscale challenge, coming from the velocity that is an $\varepsilon-$periodic function, whose expression is explicitly known. We also introduce a two--scale formulation, as a first step to the numerical resolution of the complete oscillatory Stokes-advection-diffusion system, that is currently under investigation. This two--scale formulation is also useful to understand the asymptotic behaviour of the solution.

math.NA↗

arXiv2023

On a Probabilistic Approach for Inverse Data-Driven Optimal Control

We consider the problem of estimating the possibly non-convex cost of an agent by observing its interactions with a nonlinear, non-stationary and stochastic environment. For this inverse problem, we give a result that allows to estimate the cost by solving a convex optimization problem. To obtain this result we also tackle a forward problem. This leads to the formulation of a finite-horizon optimal control problem for which we show convexity and find the optimal solution. Our approach leverages certain probabilistic descriptions that can be obtained both from data and/or from first-principles. The effectiveness of our results, which are turned in an algorithm, is illustrated via simulations on the problem of estimating the cost of an agent that is stabilizing the unstable equilibrium of a pendulum.

math.OC↗

arXiv2023

Multiscale Modeling of Sorption Kinetics

In this paper we propose and validate a multiscale model for the description of particle diffusion in presence of trapping boundaries. We start from a drift-diffusion equation in which the drift term describes the effect of bubble traps, and is modeled by a short range potential with an attractive term and a repulsive core. The interaction of the particles attracted by the bubble surface is simulated by the Lennard-Jones potential that simplifies the capture due to the hydrophobic properties of the ions. In our model the effect of the potential is replaced by a suitable boundary condition derived by mass conservation and asymptotic analysis. The potential is assumed to have a range of small size $\varepsilon$. An asymptotic expansion in the $\varepsilon$ is considered, and the boundary conditions are obtained by retaining the lowest order terms in the expansion. Another aspect we investigate is saturation effect coming from high concentrations in the proximity of the bubble surface. The validity of the model is carefully checked with several tests in 1D, 2D and different geometries.

math.NA↗

arXiv2023

Asymmetry and condition number of an elliptic-parabolic system for biological network formation

We present results of numerical simulations of the tensor-valued elliptic-parabolic PDE model for biological network formation. The numerical method is based on a non-linear finite difference scheme on a uniform Cartesian grid in a 2D domain. The focus is on the impact of different discretization methods and choices of regularization parameters on the symmetry of the numerical solution. In particular, we show that using the symmetric alternating-direction implicit (ADI) method for time discretization helps preserve the symmetry of the solution, compared to the (non symmetric) ADI method. Moreover, we study the effect of regularization by isotropic background permeability $r>0$, showing that increased condition number of the elliptic problem due to decreasing value of $r$ leads to loss of symmetry. We show that in this case, neither the use of the symmetric ADI method preserves the symmetry of the solution. Finally, we perform numerical error analysis of our method making use of Wasserstein distance.

math.NA↗

arXiv2023

Optimal Decision-Making for Autonomous Agents via Data Composition

We consider the problem of designing agents able to compute optimal decisions by composing data from multiple sources to tackle tasks involving: (i) tracking a desired behavior while minimizing an agent-specific cost; (ii) satisfying safety constraints. After formulating the control problem, we show that this is convex under a suitable assumption and find the optimal solution. The effectiveness of the results, which are turned in an algorithm, is illustrated on a connected cars application via in-silico and in-vivo experiments with real vehicles and drivers. All the experiments confirm our theoretical predictions and the deployment of the algorithm on a real vehicle shows its suitability for in-car operation.

math.OC↗

arXiv2023

The mathematical theory of Hughes' model: a survey of results

We provide an overview of the results on Hughes' model for pedestrian movements available in the literature. After the first successful approaches to solving a regularised version of the model, researchers focused on the structure of the Riemann problem, which led to local-in-time existence results for Riemann-type data and paved the way for a WFT (Wave-Front Tracking) approach to the solution semigroup. In parallel, a DPA (Deterministic Particles Approximation) approach was developed in the spirit of follow-the-leader approximation results for scalar conservation laws. Beyond having proved to be powerful analytical tools, the WFT and the DPA approaches also led to interesting numerical results. However, only existence theorems on very specific classes of initial data (essentially ruling out non-classical shocks) have been available until very recently. A proper existence result using a DPA approach was proven not long ago in the case of a linear coupling with the density in the eikonal equation. Shortly after, a similar result was proven via a fixed point approach. We provide a detailed statement of the aforementioned results and sketch the main proofs. We also provide a brief overview of results that are related to Hughes' model, such as the derivation of a dynamic version of the model via a mean-field game strategy, an alternative optimal control approach, and a localized version of the model. We also present the main numerical results within the WFT and DPA frameworks.

math.AP↗