Claudio Altafini

Consensus Problems on Networks With Antagonistic Interactions

In a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents, expresses by a communication graph with nonnegative weights. The question we ask in this paper is the following: is it possible to achieve a form of agreement also in presence of antagonistic interactions, modeled as negative weights on the communication graph? The answer to this question is affirmative: on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign. Necessary and sufficient conditions are obtained to describe cases in which this is possible. These conditions have strong analogies with the theory of monotone systems. Linear and nonlinear Laplacian feedback designs are proposed.

Computing global structural balance in large-scale signed social networks

Structural balance theory affirms that signed social networks (i.e., graphs whose signed edges represent friendly/hostile interactions among individuals) tend to be organized so as to avoid conflictual situations, corresponding to cycles of negative parity. Using an algorithm for ground-state calculation in large-scale Ising spin glasses, in this paper we compute the global level of balance of very large online social networks and verify that currently available networks are indeed extremely balanced. This property is explainable in terms of the high degree of skewness of the sign distributions on the nodes of the graph. In particular, individuals linked by a large majority of negative edges create mostly “apparent disorder,” rather than true “frustration.”

Controllability of quantum mechanical systems by root space decomposition of su(N)

The controllability property of the unitary propagator of an N-level quantum mechanical system subject to a single control field is described using the structure theory of semisimple Lie algebras. Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few degenerate cases.

Modeling and Control of Quantum Systems: An Introduction

The scope of this work is to provide a self-contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems, as well as a brief survey on the main approaches to control synthesis. While part of the existing theory, especially in the open-loop setting, stems directly from classical control theory (most notably geometric control and optimal control), a number of tools specifically tailored for quantum systems have been developed since the 1980s, in order to take into account their distinctive features: the probabilistic nature of atomic-scale physical systems, the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback-design problems. The relevant dynamical models for both closed and open quantum systems are presented, along with the main results on their controllability and stability. A brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies.

Controllability properties for finite dimensional quantum Markovian master equations

Various notions from geometric control theory are used to characterize the behavior of the Markovian master equation for N-level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states. It is shown that the system can be accessible but neither small-time controllable nor controllable in finite time. In particular, if the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time. The two level case is treated in detail.

Dynamics of Opinion Forming in Structurally Balanced Social Networks

The aim of this paper is to shed light on how the social relationships between individuals influence their opinions in the case of structurally balanced social networks. If we represent a social network as a signed graph in which individuals are the nodes and the signs of the edges represent friendly or hostile relationships, then the property of structural balance corresponds to the social community being splittable into two antagonistic factions, each containing only friends. A classical example of this situation is a two-party political system. The paper studies the process of opinion forming on such a social community, starting from the observation that the property of structural balance is formally analogous to the monotonicity property of dynamical systems. The paper shows that under the assumption that individuals are positively influenced by their friends and negatively influenced by their enemies, monotone dynamical systems, due to their order-preserving solutions, are natural candidates to describe the highly predictable process of opinion forming on structurally balanced networks.

Some properties of the general n-trailer

If some of the trailers of the so-called n-trailer system are connected via a kingpin hitch, the kinematic model of the system is more complicated that in the standard case. However, something can still be said about the structural properties of the system, like controllability. The more complicated equations can be interpreted in terms of virtual steering wheels placed on the off-axle joints with steering angle which is a non-linear feedback from the original configuration state. Quite remarkably, the extra singularities of the system have an explanation in terms of these virtual steering wheels. This is also sufficient to assert that the general n-trailer problem can be embedded into the corresponding multisteering n-trailer system. The multi-chained form available for this last system can be recovered also for the general n-trailer if we replace the extra steering inputs with the aforementioned feedback loops.

Feedback Stabilization of Isospectral Control Systems on Complex Flag Manifolds: Application to Quantum Ensembles

The convex set of density operators of an N-level quantum mechanical system foliated as a complex flag manifold, where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix. For an isospectral bilinear control system evolving on such an orbit, the state feedback stabilization problem admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ldquoroot-spacerdquo-like structure of the cone of density operators. The converging conditions are time independent but depend on the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing topological obstructions to global stabilizability.

A Path-Tracking Criterion for an LHD Articulated Vehicle:

A path-tracking criterion for the so-called LHD (load-haul-dump) truck used in underground mining is proposed in this paper. It exploits the particular configuration of this vehicle, composed of two units connected by an actuated articulation. The task is to follow the path represented by the middle of the tunnel, maintaining the whole vehicle at a reduced distance from the path itself, to decrease the risk of crashes against the walls of the tunnel. This is accomplished via feedback through the synthesis of an appropriate path-tracking criterion. The criterion is based on monitoring the distances of the midpoints of both axles of the vehicle from their orthogonal projections on the path, using two different moving frames simultaneously. Local asymptotic stability to paths of constant curvature is achieved by means of linear-state feedback.

Following a path of varying curvature as an output regulation problem

Given a path of nonconstant curvature, local asymptotic stability can be proven for the general n trailer whenever the curvature can be considered as the output of an exogenous dynamical system. The controllers that provide convergence to zero of the tracking error chosen for the path-following problem are composed of a prefeedback that input-output linearizes the system, plus a linear controller.