Francesca Albertini

Notions of controllability for bilinear multilevel quantum systems

In this note, we define four different notions of controllability of physical interest for multilevel quantum mechanical systems. These notions involve the possibility of driving the evolution operator as well as the state of the system. We establish the connections among these different notions as well as methods to verify controllability.

Continuous control-Lyapunov functions for asymptotically controllable time-varying systems

This paper shows that, for time varying systems, global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control-Lyapunov function with respect to the set.

Discrete-time transitivity and accessibility: analytic systems

A basic open question for discrete-time nonlinear systems is that of determining when, in analogy with the classical continuous-time “positive form of Chow’s Lemma,” accessibility follows from transitivity of a natural group action.This paper studies the problem and establishes the desired implication for analytic systems in several cases: (i) compact state space, (ii) under a Poisson stability condition, and (iii) in a generic sense. In addition, the paper studies accessibility properties of the “controllability sets” recently introduced in the context of dynamical systems studies. Finally, various examples and counterexamples are provided relating the various Lie algebras introduced in past work.

State observability in recurrent neural networks

Abstract We obtain a characterization of observability for a class of nonlinear systems which appear in neural networks research.

The Lie algebra structure and controllability of spin systems

In this paper, we study the controllability properties and the Lie algebra structure of networks of particles with spin immersed in an electro-magnetic field. We relate the Lie algebra structure to the properties of a graph whose nodes represent the particles and an edge connects two nodes if and only if the interaction between the two corresponding particles is active. For networks with different gyromagnetic ratios, we provide a necessary and sufficient condition of controllability in terms of the properties of the above-mentioned graph and describe the Lie algebra structure in every case. For these systems all the controllability notions, including the possibility of driving the evolution operator and/or the state, are equivalent. For general networks (with possibly equal gyromagnetic ratios), we give a sufficient condition of controllability. A general form of interaction among the particles is assumed which includes both Ising and Heisenberg models as special cases. Assuming Heisenberg interaction we provide an analysis of low-dimensional cases (number of particles less than or equal to three) which includes necessary and sufficient controllability conditions as well as a study of their Lie algebra structure. This also provides an example of quantum mechanical systems where controllability of the state is verified while controllability of the evolution operator is not.

Notions of controllability for quantum mechanical systems

In this paper, we define three different notions of controllability for quantum mechanical systems involving the possibility of driving the evolution operator as well as the state of the system. By using general results on transitivity of transformation groups on spheres we establish the connections among these different notions of controllability. Motivated by the physical model of multilevel quantum systems, we also study the relation between the controllability in arbitrary small time of a system varying on a compact transformation Lie group and the corresponding system on the associated homogeneous space. As an application, we prove for the system of two interacting spin 1/2 particles the negative result that not every state transfer can be obtained in arbitrary time.

Observability and Forward–Backward Observability of Discrete-Time Nonlinear Systems

Abstract. In this paper we study the observability properties of nonlinear discrete-time systems. Two types of contributions are given. First we present observability criteria in terms of appropriate codistributions. For particular, but significant, classes of systems we provide criteria that require only a finite number of computations. Then we consider invertible systems (which includes discrete-time models obtained by sampling continuous-time systems) and prove that the weaker notion of forward–backward observability is equivalent to the stronger notion of (forward) observability.

Asymptotic stability of continuous-time systems with saturation nonlinearities

Abstract A new criterion is established for global asymptotic stability of second-order systems modeled by equations of the type dot x = σ(Ax) , where σ is the saturation function. The derivation is based on the Bendixons theorem on limit cycles and a closer study of the trajectories of the systems. Applications to stabilization of more general cascade nonlinear systems are also discussed.

Quantum symmetries and Cartan decompositions in arbitrary dimensions

Decompositions of Lie groups are used in systems and control, both to analyse dynamics and to design control algorithms for systems with state varying on a Lie group. In this paper, we investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition, there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd–even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in [6–8] in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of N qubits to the case where the component subsystems have arbitrary dimensions. We present an example of application of the results to control design for quantum systems.

Forward accessibility for recurrent neural networks

Gives an algebraic characterization of forward accessibility for recurrent neural networks. >