Cristina Pignotti

University of L'Aquila | Università dell'Aquila · Department of Information Engineering, Computer Science and Mathematics

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 sy...

We consider several classes of degenerate hyperbolic equations involving delay terms and suitable nonlinearities. The idea is to rewrite the problems in an abstract way and, using semigroup theory and energy method, we study well posedness and stability. Moreover, some illustrative examples are given. Keywords: fourth order degenerate operator, sec...

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 i...

In this paper, we analyze a Hegselmann-Krause opinion formation model with attractive-lacking interaction. More precisely, we investigate the situation in which the agents 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 assu...

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 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...

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 sys-tem's trajectories, we are able to prove an asymptotic convergence to consensus result...

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 exp...

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 del...

We analyze Hegselmann-Krause opinion formation models with leadership in presence of time delay effects. In particular, we consider a model with a pointwise time variable time delay and a model with a distributed delay. In both cases we show that, when the delays satisfy suitable smallness conditions, then the leader can control the system, leading...

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...

In this paper, we deal with a minimum time problem in presence of a time delay $\tau.$ 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 $C([-\tau,0];\mathbb{R}^{n})$. For the undelayed minimum time pr...

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 arg...

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 de...

We study abstract linear and nonlinear evolutionary systems with single or multiple delay feedbacks, illustrated by several concrete examples. In particular, we assume that the operator associated with the undelayed part of the system generates an exponentially stable semigroup and that the delay damping coefficients are locally integrable in time....

In this paper, we analyze a semilinear abstract damped wave-type equation with time delay. We assume that the delay feedback coefficient is variable in time and belongs to \begin{document}$ L^1_{loc}([0, +\infty)). $\end{document} Under suitable assumptions, we show well-posedness and exponential stability for small initial data. Our strategy combi...

In this paper we study a class of semilinear wave-type equations with viscoelastic damping and delay feedback with time variable coefficient. By combining semigroup arguments, careful energy estimates and an iterative approach we are able to prove, under suitable assumptions, a well-posedness result and an exponential decay estimate for solutions c...

In this paper we analyze a semilinear abstract damped wave-type equation with time delay. We assume that the delay feedback coefficient is variable in time and belonging to $L^1_{loc}([0, +\infty)).$ Under suitable assumptions, we show well-posedness and exponential stability for small initial data. Our strategy combines careful energy estimates an...

In this paper we analyze a coupled system between a transport equation and an ordinary differential equation with time delay (which is a simplified version of a model for kidney blood flow control). Through a careful spectral analysis we characterize the region of stability, namely the set of parameters for which the system is exponentially stable....

We analyze Hegselmann-Krause opinion formation models with leadership in presence of time delay effects. In particular, we consider a model with pointwise time variable time delay and a model with a distributed delay. In both cases we show that, when the delays satisfy suitable smallness conditions, then the leader can control the system, leading t...

In this paper, we study Hegselmann–Krause models with a time‐variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study t...

In this paper we study a class of semilinear wave type equations with viscoelastic damping and delay feedback with time variable coefficient. By combining semigroup arguments, careful energy estimates and an iterative approach we are able to prove, under suitable assumptions, a well-posedness result and an exponential decay estimate for solutions c...

In this paper we analyze a coupled system between a transport equation and an ordinary differential equation with time delay (which is a simplified version of a model for kidney blood flow control). Through a careful spectral analysis we characterize the region of stability, namely the set of parameters for which the system is exponentially stable....

We analyze the stability of Maxwell equations in bounded domains taking into account electric and magnetization effects. Well-posedness of the model is obtained by means of semigroup theory. A passitivity assumption guarantees the boundedness of the associated semigroup. Further the exponential or polynomial decay of the energy is proved under suit...

We consider the KdV–Burgers equation and its linear version in presence of a delay feedback. We prove well-posedness of the models and exponential decay estimates under appropriate conditions on the damping coefficients. Our arguments rely on a Lyapunov functional approach combined with a step by step procedure and semigroup theory.

We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi's counterexample. Here we assume a condition on the support of off-diagonal coefficients that "keeps away" the counterexample and allows us to prove local boundedness of weak solutions.

We discuss the asymptotic frequency synchronization for the non-identical Kuramoto oscillators with time delayed interactions. We provide explicit lower bound on the coupling strength and upper bound on time delay in terms of initial configurations ensuring exponential synchronization. This generalizes not only the frequency synchronization estimat...

We analyze the stability of Maxwell equations in bounded domains taking into account electric and magnetization effects. Well-posedness of the model is obtained by means of semigroup theory. A passitivity assumption guarantees the boundedness of the associated semigroup. Further the exponential or polynomial decay of the energy is proved under suit...

In this paper, we study Hegselmann-Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study t...

We extend the analysis developed in Pignotti and Reche Vallejo (J Math Anal Appl 464:1313–1332, 2018) [34] in order to prove convergence to consensus results for a Cucker–Smale type model with hierarchical leadership and distributed delay. Flocking estimates are obtained for a general interaction potential with divergent tail. We analyze also the m...

We consider the KdV-Burgers equation and its linear version in presence of a delay feedback. We prove well-posedness of the models and exponential decay estimates under appropriate conditions on the damping coefficients. Our arguments rely on a Lyapunov functional approach combined with a step by step procedure and semigroup theory.

We study a Cucker-Smale-type flocking model with distributed time delay where individuals interact with each other through normalized communication weights. Based on a Lyapunov functional approach, we provide sufficient conditions for the velocity alignment behavior. We then show that as the number of individuals N tends to infinity, the N-particle...

We give maximum principles for solutions u:Ω→ℝN to a class of quasilinear elliptic systems whose prototype is [Formula presented]where α∈{1,…,N} is the equation index and Ω is an open, bounded subset of ℝⁿ. We assume that coefficients [Formula presented] are measurable with respect to x, continuous with respect to y∈ℝN, bounded and elliptic. In vec...

We study abstract linear and nonlinear evolutionary systems with single or multiple delay feedbacks, illustrated by several concrete examples. In particular, we assume that the operator associated with the undelayed part of the system generates an exponentially stable semigroup and that the delay damping coefficients are locally integrable in time....

We extend the analysis developed in [33] in order to prove convergence to consensus results for a Cucker-Smale type model with hierarchical leadership and distributed delay. Flocking estimates are obtained for a general interaction potential with divergent tail. We analyze also the model when the ultimate leader can change its velocity. In this cas...

We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the $C_0$-semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a gene...

It is well-known that wave-type equations with memory, under appropriate assumptions on the memory kernel, are uniformly exponentially stable. On the other hand, time delay effects may destroy this behavior. Here, we consider the stabilization problem for second-order evolution equations with memory and intermittent delay feedback. We show that, un...

We consider abstract evolution equations with on–off time delay feedback. Without the time delay term, the model is described by an exponentially stable semigroup. We show that, under appropriate conditions involving the delay term, the system remains asymptotically stable. Under additional assumptions exponential stability results are also obtaine...

We analyze the Cucker-Smale model under hierarchical leadership in presence of a time delay. By using a Lyapunov functional approach and some induction arguments we will prove convergence to consensus for every positive delay $\tau$. These results seem to point out the advantage of a hierarchical structure in order to contrast time delay effects th...

We analyze the Cucker-Smale model under hierarchical leadership in presence of a time delay. By using a Lyapunov functional approach and some induction arguments we will prove convergence to consensus for every positive delay $\tau$. These results seem to point out the advantage of a hierarchical structure in order to contrast time delay effects th...

We consider the celebrated Cucker-Smale model in finite dimension, modelling interacting collective dynamics and their possible evolution to consensus. The objective of this paper is to study the effect of time delays in the general model. By a Lyapunov functional approach, we provide convergence results to consensus for symmetric as well as nonsym...

We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the $C_0$-semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a gene...

We consider abstract evolution equations with on-off time delay feedback. Without the time delay term, the model is described by an exponentially stable semigroup. We show that, under appropriate conditions involving the delay term, the system remains asymptotically stable. Under additional assumptions exponential stability results are also obtaine...

We study the stabilization problem for the wave equation with localized Kelvin-Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an exponential stability result holds. In this sense, this extends the re...

In this paper we study well-posedness and asymptotic stability for a class of nonlinear second-order evolution equations with intermittent delay damping. More precisely, a delay feedback and an undelayed one act alternately in time. We show that, under suitable conditions on the feedback operators, asymptotic stability results are available. Concre...

In this paper we consider a stabilization problem for the abstract-wave equation with delay. We prove an exponential stability result for appropriate damping coefficient. The proof of the main result is based on a frequency-domain approach.

We consider abstract semilinear evolution equations with a time delay feedback. We show that, if the $C_0$-semigroup describing the linear part of the model is exponentially stable, then the whole system retains this good property when a suitable smallness condition on the time delay feedback is satisfied. Some examples illustrating our abstract ap...

We consider a Dirichlet problem for the Allen-Cahn equation in a smooth, bounded or unbounded, domain $\Omega\subset {\bf R}^n.$ Under suitable assumptions, we prove an existence result and a uniform exponential estimate for symmetric solutions. In dimension n=2 an additional asymptotic result is obtained. These results are based on a pointwise est...

We study the asymptotic behaviour of the wave equation with viscoelastic damping in presence of a time-delayed damping. We prove exponential stability if the amplitude of the time delay term is small enough.

We consider second-order evolution equations in an abstract setting with intermittently delayed/ not-delayed damping. We give sufficient conditions for asymptotic and exponential stability, improving and generalising our previous results from [19]. In particular, under suitable conditions, we can consider unbounded damping operators. Some concrete...

We consider second-order evolution equations in an abstract setting with damping and time delay and give sufficient conditions ensuring exponential stability. Our abstract framework is then applied to the wave equation, the elasticity system and the Petrovsky system.

In this paper we consider some stabilization problems for the wave equation with switching time-delay. We prove exponential stability results for appropriate damping coefficients. The proof of the main results is based on D'Alembert formula, observability inequality and some energy estimates. More general problems, like the Petrovsky system, are al...

We consider second-order evolution equations in an abstract setting with intermittently delayed/not-delayed damping and give sufficient conditions ensuring asymptotic and exponential stability results. Our abstract framework is then applied to the wave equation, the elasticity system, and the Petrovsky system. For the Petrovsky system with clam...

In this paper, we consider the wave equation with internal distributed time delay and local damping in a bounded and smooth domain Ω⊂Rn. When the local damping acts on a neighborhood of a suitable part of the boundary of Ω, we show that an exponential stability result holds if the coefficient of the delay term is sufficiently small.

We consider a stabilization problem for abstract second-order evolution equations with dynamic boundary feedback laws with a delay and distributed structural damping. We prove an exponential stability result under a suitable condition between the internal damping and the boundary laws. The proof of the main result is based on an identity with multi...

In this paper we consider some stabilization problems for the wave equation with switching. We prove exponential stability results for appropriate damping coefficients. The proof of the main results is based on D'Alembert formula and some energy estimates.

We consider the semilinear elliptic equation Δu=W ' (u) with Dirichlet boundary condition in a Lipschitz, possibly unbounded, domain Ω⊂ℝ n · Under suitable assumptions on the potential W, we deduce a condition on the size of the domain that implies the existence of a positive solution satisfying a uniform pointwise estimate. Here, uniform means tha...

We study the stabilization problem by interior damping of the wave equation with boundary or internal time-varying delay feedback in a bounded and smooth domain. By introducing suitable Lyapunov functionals exponential stability estimates are obtained if the delay effect is appropriately compensated by the internal damping.

In this paper we consider a boundary stabilization problem for the wave equation with interior delay. We prove an exponential stability result under some Lions geometric condition. The proof of the main result is based on an identity with multipliers that allows us to obtain a uniform decay estimate for a suitable Lyapunov functional.

We consider the wave equation with a time -varying delay term in the boundary condition in a bounded and smooth domain Ω ⊂ IR n . Under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and suitable Lyapounov functionals. Such analysis is also extended to a nonlinear ve...

This paper is devoted to the asymptotic analysis of simple models of fluid-structure interaction, namely a system between the heat and wave equations coupled via some transmission conditions at the interface. The heat part induces the dissipation of the full system. Here we are interested in the behavior of the model when the thickness of the heat...

We consider the wave equation in a bounded region with a smooth boundary with distributed delay on the boundary or into the domain. In both cases, under suitable assumptions, we prove the exponential stability of the solution. These results are obtained by introducing suitable energies and by proving some observability inequalities. For an internal...

We study a class of Schrodinger operators of the form L-epsilon := -epsilon(2) d(2)/ds(2) + V, where V : R -> R is a non-negative function singular at 0, that is V(0) = 0. Under suitable assumptions on the potential V, we derive sharp lower and upper bounds for the fundamental solution h(epsilon). Moreover, we obtain information on the spectrum of...

In this paper we study an optimal control problem with mixed constraints related to a multisector linear model with endogenous growth. The main aim is to establish a set of necessary and a set of sufficient conditions which are the basis for studying the qualitative properties of optimal trajectories. The presence of possibly degenerate mixed const...

We give exponential and polynomial stability results for the wave equation with variable coefficients in a bounded domain of ℝ n , subject to a Dirichlet boundary condition on one part of the boundary and boundary conditions of memory type on the other part of the boundary. Moreover, analogous stability results are given for a system of Maxwell’s...

We consider, in a bounded and smooth domain, Maxwell's equations with a delay term in the boundary or in the internal feedbacks. Under suitable assumptions we obtain exponential stability results. Some instability examples are also given.

We consider the stabilization of Maxwell¿s equations with space variable coefficients in a bounded region with a smooth boundary, subject to dissipative boundary conditions of memory type on the boundary. Under suitable conditions on the domain and on the permeability and permittivity coefficients, we prove the exponential/polynomial decay of the e...

Observability estimates for Maxwell's system with variable coefficients are established using the differential geometry method recently developed for scalar wave equations. The main tool is that Maxwell's system is reducible to a perturbed vectorial wave equation with a decoupled principal part.

In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtaine...

We consider the internal stabilization of Maxwell's equations with Ohm's law with space variable coefficients in a bounded region with a smooth boundary. Our result is mainly based on an observability estimate, obtained in some particular cases by the multiplier method, a duality argument and a weakening of norm argument, and arguments used in inte...

In this paper we give semiconcavity results for the value function of some constrained optimal control problems with infinite horizon in a half-space. In particular, we assume that the control space is the l1-ball or the l∞-ball in Rn.

In this paper we prove a relaxation result for a class of degenerate Hamilton--Jacobi equations; i.e., equations which do not admit a strict subsolution, extending in some directions the results in [6] and [7].

We consider the stabilization of Maxwell's equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver–Müller boundary condition. This is based on some stability estimates that are obtained using the “standard" identity with multiplier and appropriate properties of the feedback....

We consider the value function V of optimal control problems with exit time. Under suitable assumptions, through the study of the conjugate points, we prove that the closure of the singular set of V is rectifiable. Moreover, a sharper Hausdorff estimate is given on the set of the conjugate nonsingular points.

A simple exit time problem with degenerate cos is here considered. Using a new technique for constructing admissible trajectories, a semiconcavity result for the value function ν is obtained. Such a property of ν is then applied to obtain optimality conditions.

In this paper a semiconcavity result is obtained for the value function of an optimal exit time problem. The related state equation is of the general form y ˙(t)=f(y(t),u(t)),y(t)∈ℝ n ,u(t)∈U⊂ℝ m · However, suitable assumptions are needed relating f with the running and exit costs. The semiconcavity property is then applied to obtain necessary opti...

Let Omega⊂R3 be a bounded simply connected obstacle with boundary ∂Omega locally Lipschitz, we consider the scattering of a time harmonic electromagnetic wave that hits Omega when ∂Omega is assumed to be perfectly conducting. The scattered electromagnetic field is the solution of an exterior boundary value problem for the vector Helmholtz equation....

In this paper, we prove some regularity results for the value function v of an optimal control problem with state constraints. In particular, we are interested in studying the semiconcavity of v

We consider a system of Maxwell’s equations in a bounded domain Ω ⊂ IR^3, with smooth boundary Γ. We prove some results of boundary and internal observability and controllability. The observability inequalities are obtained by using multiplier techniques and the controllability results by the Hilbert Uniqueness Method of J. L. Lions.

Let D={(x, y) ∈ ℝ2 | x>0, y ∈ ℝ} and u(x, y, t) be the solution of an initial-boundary value problem for the two-dimensional wave equation in the half plane D. The half plane D carries a velocity stratification given by flat layers parallel to the boundary of the half plane characterized by a thickness and a constant velocity. We consider the follo...

We consider the stabilization of the wave equation with space variable coefficients in a bounded region with a smooth boundary, subject to Dirichlet boundary conditions on one part of the boundary and linear or nonlinear dissipative boundary conditions of memory type on the remainder part of the boundary. Our stabilization results are mainly based...

In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtaine...