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Cristina Pignotti

Publications and source records attributed to Cristina Pignotti.

At least 19 recordsLinked to original sources
arXiv2026

Well-posedness and exponential stability for abstract evolution equations with delay in the nonlinear source: frictional and viscoelastic cases

In this paper, we establish the well-posedness and exponential stability of a class of semilinear neutral abstract evolution equations with constant time delay. Two damping mechanisms are considered: frictional damping of the form $CC^*u_t$ and viscoelastic damping with memory. Under suitable assumptions, we prove global well-posedness and exponential decay of solutions for sufficiently small initial data. This result is of considerable importance, as it provides a unified framework for the analysis of a wide class of neutral systems arising in structural mechanics, seismic isolation, and control theory. The abstract results are illustrated by some concrete examples. The analysis relies on a combination of semigroup theory, energy estimates, Duhamel's formula, and Gronwall-type arguments.

math.AP↗
arXiv2026

Exponential Consensus and Flocking in Multi-Agent Systems with Infinite Fading Memory

In this paper, we study the emergent collective dynamics of multi-agent systems driven by infinite distributed fading memory of Volterra type. We establish a unified theoretical framework covering both first-order opinion consensus dynamics and second-order velocity alignment flocking kinematics. By introducing Dafermos past-history transformations, the governing integro-differential systems are reformulated into dynamical systems on an extended product Hilbert spaces. For first-order dynamics, we prove that fading memory inherently provides a hidden dissipative mechanism, guaranteeing unconditional global exponential consensus with or without instantaneous communication forces. For second-order dynamics, we obtain unconditional exponential flocking for the pure fading memory system and we give a sufficient condition for flocking when an instantaneous interaction is also present. In particular, this condition is always satisfied, namely the flocking occurs unconditionally, when the influence function has a divergent tail.

math.DS↗
arXiv2026

Asymptotic consensus and flocking under decaying persistent excitation on rooted digraphs

In this paper, we investigate first- and second-order alignment models with non-universal interaction, time delays and possible communication failures, extending the results in [17] to interaction digraphs that are only assumed to be rooted and to a weaker Persistence Excitation Condition. In particular, we allow the amount of interaction over time intervals of fixed length to decay polynomially in time. For the first-order Hegselmann-Krause type model, we prove asymptotic convergence to consensus under a suitable condition relating the decay exponent of the communication weights to the maximal distance from the root. For the second-order Cucker-Smale model, we establish asymptotic flocking under an additional assumption on the decay of the influence function. These results show that collective behavior can still emerge under progressively weakening communication and without requiring strong connectivity of the interaction graph.

math.OC↗
arXiv2026

Consensus and flocking with transmission and reaction delays

We investigate consensus formation and flocking behavior in multi-agent systems subject to two distinct types of delays: a transmission delay accounting for information exchange between agents, and a reaction delay representing the processing time before agents adjust their states. For a simplified linear two-agent system, we provide explicit insight into how these delays affect asymptotic stability. We then derive sufficient conditions for asymptotic consensus and flocking in the general multi-agent setting with a nonlinear, globally positive influence function. These conditions require the delays to be sufficiently small relatively to the initial data and the decay rate of the influence function. The analysis is based on a Lyapunov functional approach combined with a Halanay-type inequality. Our results establish rigorous conditions under which collective behavior emerges in delayed multi-agent systems where both communication and reaction lags are non-negligible, with applications to biological, social, and engineered systems.

math.DS↗
arXiv2025

Time-delayed opinion dynamics with leader-follower interactions: consensus, stability, and mean-field limits

We study a time-delayed variant of the Hegselmann-Krause opinion formation model featuring a small group of leaders and a large group of non-leaders. In this model, leaders influence all agents but only interact among themselves. At the same time, non-leaders update their opinions via interactions with their peers and the leaders, with time delays accounting for communication and decision-making lags. We prove the exponential convergence to consensus of the particle system, without imposing smallness assumptions on the delay parameters. Furthermore, we analyze the mean-field limit in two regimes: (i) with a fixed number of leaders and an infinite number of non-leaders, and (ii) with both populations tending to infinity, obtaining existence, uniqueness, and exponential decay estimates for the corresponding macroscopic models.

math.AP↗
arXiv2025

Indirect stabilization of nonlinear coupled wave equations in the presence of time delay or indefinite damping

This paper explores the exponential stability of two nonlinear wave equations coupled through their velocities. The analysis is divided into two main cases. First, we consider a system where one equation is damped, while the other experiences a discrete time delay. By reformulating the problem in an abstract framework, we use semigroup theory and energy methods to establish well-posedness and derive conditions that guarantee exponential energy decay. In the second case, we examine a scenario where a frictional damping term appears in the first equation, while the second equation contains an indefinite damping term, namely with a sign-changing coefficient. Although this setup can be viewed as a special case of the first, we analyze it separately and show that exponential stability still holds under a weaker condition.

math.AP↗
arXiv2024

Semiconcavity for the minimum time problem in presence of time delay effects

In this paper, we deal with a minimum time problem in presence of a time delay $τ.$ The value function of the considered optimal control problem is no longer defined in a subset of $\mathbb{R}^{n}$, as it happens in the undelayed case, but its domain is a subset of the Banach space $C([-τ,0];\mathbb{R}^{n})$. For the undelayed minimum time problem, it is known that the value function associated with it is semiconcave in a subset of the reachable set and is a viscosity solution of a suitable Hamilton-Jacobi-Belmann equation. The Hamilton-Jacobi theory for optimal control problems involving time delays has been developed by several authors. Here, we are rather interested in investigating the regularity properties of the minimum time functional. Extending classical arguments, we are able to prove that the minimum time functional is semiconcave in a suitable subset of the reachable set.

math.OC↗
arXiv2024

Opinion dynamics under common influencer assumption or leadership control

We study Hegselmann-Krause type opinion formation models with non-universal interaction and time-delayed coupling. We assume the presence of a common influencer between two different agents. Moreover, we explore two cases in which such an assumption does not hold but leaders with independent opinion are present. By using careful estimates on the system's trajectories, we are able to prove asymptotic convergence to consensus estimates. Some numerical tests illustrate the theoretical results.

math.OC↗
arXiv2024

First and second-order Cucker-Smale models with non-universal interaction, time delay and communication failures

In this paper, we deal with first and second-order alignment models with non-universal interaction, time delay and possible lack of connection between the agents. More precisely, we analyze the situation in which the system's agents do not transmit information to all the other agents and also agents that are linked to each other can suspend their interaction at certain times. Moreover, we take into account of possible time lags in the interactions. To deal with the considered "non-universal" connection, a graph topology over the structure of the model has to be considered. Under a so-called Persistence Excitation Condition, we establish the exponential convergence to consensus for both models whenever the digraph that describes the interaction between the agents is strongly connected.

math.OC↗
arXiv2024

Asymptotic synchronization of Kuramoto oscillators with time delay and non-universal interaction

We study the emergence of synchronization in the Kuramoto model on a digraph in the presence of time delays. Assuming the digraph is strongly connected, we first establish a uniform bound on the phase diameter and subsequently prove the asymptotic frequency synchronization of the oscillators under suitable assumptions on the initial configurations. In the case of an all-to-all connection, we obtain an exponential synchronization estimate. Additionally, we present numerical simulations, providing further insights into the synchronization and oscillatory behaviors of the oscillator frequencies depending on the network structure and the magnitude of the time delay.

math.OC↗
arXiv2024

Energy decay for semilinear evolution equations with memory and time-dependent time delay feedback

In this paper, we study well-posedness and exponential stability for semilinear second order evolution equations with memory and time-varying delay feedback. The time delay function is assumed to be continuous and bounded. Under a suitable assumption on the delay feedback, we are able to prove that solutions corresponding to small initial data are globally defined and satisfy an exponential decay estimate.

math.AP↗
arXiv2024

Opinion dynamics of two populations with time-delayed coupling

We study a Hegselmann-Krause type opinion formation model for a system of two populations. The two groups interact with each other via subsets of individuals, namely the leaders, and natural time delay effects are considered. By using careful estimates of the system's trajectories, we are able to prove an asymptotic convergence to consensus result. Some numerical tests illustrate the theoretical result and point out some possible applications.

math.OC↗
arXiv2024

Exponential decay of solutions to linear evolution equations with time-dependent time delay

In this note, we analyze an abstract evolution equation with time-dependent time delay and time-dependent delay feedback coefficient. We assume that the operator corresponding to the nondelayed part of the model generates an exponentially stable semigroup. Under an appropriate assumption on the delay feedback, we prove the well-posedness and an exponential stability estimate for our model. Applications of our abstract results to concrete models are also illustrated.

math.OC↗
arXiv2023

Hegselmann-Krause and Cucker-Smale type models with attractive-repulsive interaction

In this paper, we analyze a Hegselmann-Krause opinion formation model and a Cucker-Smale flocking model with attractive-repulsive interaction. To be precise, we investigate the situation in which the individuals involved in an opinion formation or a flocking process attract each other in certain time intervals and repeal each other in other ones. Under quite general assumptions, we prove the convergence to consensus for the Hegselmann-Krause model and the exhibition of asymptotic flocking for the Cucker-Smale model in presence of positive-negative interaction. With some additional conditions, we are able to improve the convergence to consensus for the solutions of the Hegselmann-Krause model, namely we establish an exponential convergence to consensus result.

math.OC↗
arXiv2023

Convergence to consensus results for Hegselmann-Krause type models with attractive-lacking interaction

In this paper, we analyze a Hegselmann-Krause opinion formation model with attractive-lacking interaction. More precisely, we investigate the situation in which the individuals involved in an opinion formation process interact among themselves but can eventually suspend the exchange of information among each other at some times. Under quite general assumptions, we prove the exponential convergence to consensus for the Hegselmann-Krause model in presence of possible lack of interaction. We then extend the analysis to an analogous model in presence of time delays.

math.OC↗
arXiv2023

Exponential decay estimates for semilinear wave-type equations with time-dependent time delay

In this paper, we analyze a semilinear damped second order evolution equation with time-dependent time delay and time-dependent delay feedback coefficient. The nonlinear term satisfies a local Lipschitz continuity assumption. Under appropriate conditions, we prove well-posedness and exponential stability of our model for small initial data. Our arguments combine a Lyapunov functional approach with some continuity arguments. Moreover, as an application of our abstract results, the damped wave equation with a source term and delay feedback is analyzed.

math.AP↗
arXiv2022

Consensus for the Hegselmann-Krause model with time variable time delays

In this paper, we analyze a Hegselmann-Krause opinion formation model with time-variable time delay and prove that, if the influence function is always positive, then there is exponential convergence to consensus without requiring any smallness assumptions on the time delay function. The analysis is then extended to a model with distributed time delay.

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