Lie Bracket Research Articles

Stability Analysis of DC-DC Converters Applying Lie Algebra

ABSTRACT Stability analysis of switched DC-DC converters is necessary because instability enhances current ripple and causes converter operation at unwanted voltage/current level. In the present study, stability analysis of switched converters using Lie Algebra has been performed considering the switched linear model of DC-DC converters. The switched linear system is asymptotically stable if the Lie bracket formed by the state matrices of the system is commutative. The Lie Algebra is generated and its derived series is also computed. The Lie Algebra is found to be solvable, consequently the switched system is exponentially stable under arbitrary switching. The effect of duty cycle variation and load resistance of DC-DC converters, working in continuous conduction mode (CCM) on the decay constant and decay coefficient, has also been studied. The decay coefficient and decay rate of this exponentially stable system are observed to increase with an increase in the duty cycle for a specific load resistance.

On 3-dimensional complex Hom-Lie algebras

We study and classify the 3-dimensional Hom-Lie algebras over C. We provide first a complete set of representatives for the isomorphism classes of skew-symmetric bilinear products defined on a 3-dimensional complex vector space g. The well known Lie brackets for the 3-dimensional Lie algebras are included into appropriate isomorphism classes of such products representatives. For each product representative, we provide a complete set of canonical forms for the linear maps g→g that turn g into a Hom-Lie algebra, thus characterizing the corresponding isomorphism classes. As by-products, Hom-Lie algebras for which the linear maps g→g are not homomorphisms for their products are exhibited. Examples also arise of non-isomorphic families of Hom-Lie algebras which share, however, a fixed Lie-algebra product on g. In particular, this is the case for the complex simple Lie algebra sl2(C). Similarly, there are isomorphism classes for which their skew-symmetric bilinear products can never be Lie algebra brackets on g.

Acceleration of automatic differentiation of solutions to parabolic partial differential equations: a higher order discretization

The paper proposes a new automatic/algorithmic differentiation for the solutions to partial differential equations of parabolic type. In particular, we provide a higher order discretization scheme which is a natural extension of the standard automatic differentiation. A Brownian polynomial approach is introduced to avoid the Levy area simulation. The Lie brackets of vector fields associated with stochastic differential equation play an important role in the proposed scheme. The case that the test function is non-smooth but has Gateaux derivative is considered. Numerical examples are shown to confirm the effectiveness

Chain level loop bracket and pseudo‐holomorphic disks

Let L be a Lagrangian submanifold in a symplectic vector space which is closed, oriented and spin. Using virtual fundamental chains of moduli spaces of nonconstant pseudo-holomorphic disks with boundaries on L, one can define a Maurer–Cartan element of a Lie bracket operation in string topology (the loop bracket) defined at chain level. This observation is due to Fukaya, who also pointed out its important consequences in symplectic topology. The goal of this paper is to work out details of this observation. Our argument is based on a string topology chain model previously introduced by the author, and the theory of Kuranishi structures on moduli spaces of pseudo-holomorphic disks, which has been developed by Fukaya–Oh–Ohta–Ono.

Regularity of the Minimum Time and of Viscosity Solutions of Degenerate Eikonal Equations via Generalized Lie Brackets

In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of set-valued iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric, nonlinear systems, which have as a consequence the Holder regularity of the minimum time function in optimal control. We then apply such result to prove Holder continuity of solutions of the Dirichlet boundary value problem for the eikonal equation with low regularity of the coefficients. We also prove that the sufficient conditions for the Holder regularity are essentially necessary, at least for smooth vector fields and target.

Multi-component generalizations of mKdV equation and nonassociative algebraic structures

Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg–de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. If this operation is a Lie bracket, then we arrive at the Lie–Jordan algebras [Speciality of Lie–Jordan algebras, J. Algebra 237 (2001) 621–636].

A matrix method to determine infinitesimally mobile linkages with only first-order infinitesimal mobility

Immobile linkages admitting only (possibly higher-order) infinitesimal mobility are shaky structures. In the past, determination of the order of mobility or shakiness was usually approached in a purely kinematic way namely by the higher order kinematic constraint analysis, involving solutions of higher-order kinematic constraints. In this paper, in terms of screw theory and an appropriate representation of kinematic topology, a matrix method is provided to test whether a multi-loop linkage is immobile and only possesses first-order mobility, without the need to solve the second-order constraint equations. The corresponding linkages are called first-order infinitesimal linkages. To this end, the first- and second-order kinematic constraints of multi-loop linkages are firstly formulated explicitly in matrix form, in terms of a Jacobian matrix and Hessian matrix, respectively, and are combined to a quadratic form. The definitiveness of this quadratic form then provides a sufficient condition for being a first-order infinitesimal linkage. This is related to the concept of prestress-stability. The method is applied to several immobile closed-loop linkages with only infinitesimal mobility. A special example is the 3-UU mechanism, which is a first-order infinitesimal linkage but not prestress-stable. Since higher-order derivatives of screws can be obtained explicitly with Lie brackets, a matrix method may be established, in which higher-order kinematic constraints may be analyzed in a more qualitative way. This paper is a first step towards a matrix method for determination of higher-order infinitesimal linkages.

Gerstenhaber algebra structure on the cohomology of a hom-associative algebra

A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. A direct verification shows that this cup product together with the degree $$-1$$ graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology makes it a Gerstenhaber algebra.

PreHamiltonian and Hamiltonian operators for differential-difference equations

In this paper we are developing a theory of rational (pseudo) difference Hamiltonian operators, focusing in particular on its algebraic aspects. We show that a pseudo-difference Hamiltonian operator can be represented as a ratio AB−1 of two difference operators with coefficients from a difference field , where A is preHamiltonian. A difference operator A is called preHamiltonian if its image is a Lie subalgebra with respect to the Lie bracket of evolutionary vector fields on . The definition of a rational Hamiltonian operator can be reformulated in terms of its factors which simplifies the theory and makes it useful for applications. In particular we show that for a given rational Hamiltonian operator H in order to find a second Hamiltonian operator K compatible with H one only needs to find a preHamiltonian pair A and B such that K = AB−1H is skew-symmetric. We apply our theory to study multi-Hamiltonian structures of Narita–Itoh–Bogayavlensky and Adler–Postnikov equations.

Analyzing Controllability of Bilinear Systems on Symmetric Groups: Mapping Lie Brackets to Permutations

Bilinear systems emerge in a wide variety of fields as natural models for dynamical systems ranging from robotics to quantum dots. Analyzing controllability of such systems is of fundamental and practical importance, for example, for the design of optimal control laws, stabilization of unstable systems, and minimal realization of input-output relations. Tools from Lie theory have been adopted to establish controllability conditions for bilinear systems, and the most notable development was the Lie algebra rank condition (LARC). However, the application of the LARC may be computationally expensive for high-dimensional systems. In this paper, we present an alternative and effective algebraic approach to investigate controllability of bilinear systems. The central idea is to map Lie bracket operations of the vector fields governing the system dynamics to permutation multiplications on a symmetric group, so that controllability and controllable submanifolds can be characterized by permutation cycles. The method is further applicable to characterize controllability of systems defined on undirected graphs, such as multi-agent systems with controlled couplings between agents and Markov chains with tunable transition rates between states, which in turn reveals a graph representation of controllability through the graph connectivity.

The Universal Lie $\infty$-Algebroid of a Singular Foliation

We consider singular foliations \mathcal{F} as locally finitely generated \mathscr{O} -submodules of \mathscr{O} -derivations closed under the Lie bracket, where \mathscr{O} is the ring of smooth, holomorphic, or real analytic functions on a correspondingly chosen manifold. We first collect and/or prove several results about the existence of resolutions of such an \mathcal{F} in terms of sections of vector bundles. For example, these exist always on a compact smooth manifold M if \mathcal{F} admits real analytic generators. We show that every complex of vector bundles (E_\bullet,\mathrm{d}) over M providing a resolution of a given singular foliation \mathcal{F} in the above sense admits the definition of brackets on its sections such that it extends these data into a Lie \infty -algebroid. This Lie \infty -algebroid, including the chosen underlying resolution, is unique up to homotopy and, moreover, every other Lie \infty -algebroid inducing the given \mathcal{F} or any of its sub-foliations factors through it in an up-to-homotopy unique manner. We therefore call it the universal Lie \infty -algebroid of \mathcal{F} . It encodes several aspects of the geometry of the leaves of \mathcal{F} . In particular, it permits us to recover the holonomy groupoid of Androulidakis and Skandalis. Moreover, each leaf carries an isotropy Lie \infty -algebra structure that is unique up to isomorphism. It extends a minimal isotropy Lie algebra, that can be associated to each leaf, by higher brackets, which give rise to additional invariants of the foliation. As a byproduct, we construct an example of a foliation \mathcal{F} generated by r vector fields for which we show by these techniques that, even locally, it cannot result from a Lie algebroid of the minimal rank r .

A Higher-Order Maximum Principle for Impulsive Optimal Control Problems

We consider a nonlinear system, affine with respect to an unbounded control $u$ which is allowed to range in a closed cone. With this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to $u$. This lack of coercivity gives the problem an impulsive character, meaning that minimizing sequences of trajectories happen to converge towards discontinuous paths. As is known, a distributional approach does not make sense in such a nonlinear setting, where, instead, a suitable embedding in the graph space is needed. We provide higher-order necessary optimality conditions for properly defined impulsive minima in the form of equalities and inequalities involving iterated Lie brackets of the dynamical vector fields. These conditions are derived under very weak regularity assumptions and without any constant rank conditions.

Symmetry Analysis of the Fokker Planck Equation

In this work, the infinitesimal criterion of invariance for determining symmetries of partial differential equations is applied to the Fokker Planck equation. The maximum rang condition being satisfied, we determine the Lie point symmetries of this equation. Due to the nature of infinitesimal generators of these symmetries and the stability of Lie brackets, we obtain an infinite number of solutions from which we find examples of solutions for the Fokker Planck equation: other solutions are generated given a particular solution of the equation. Then, the Fokker Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation. We show that this system admits six and an infinite number of infinitesimal generators of point symmetries giving rise to two potential symmetries of the Fokker Planck equation. We then use those potential symmetries to determine solutions of the associated system and therefore provide other solutions of the Fokker Planck equation. Note that these are essentially obtained on the basis of the invariant surface conditions. With respect to these conditions and from the potential symmetries that we have found, we finally show that in particular, some solutions of the considered Fokker Planck equation reduced to the trivial solution (solutions that are zero).

Deformed graded Poisson structures, generalized geometry and supergravity

In recent years, a close connection between supergravity, string effective actions and generalized geometry has been discovered that typically involves a doubling of geometric structures. We investigate this relation from the point of view of graded geometry, introducing an approach based on deformations of graded Poisson structures and derive the corresponding gravity actions. We consider in particular natural deformations of the 2-graded symplectic manifold T*[2]T[1]M that are based on a metric g, a closed Neveu-Schwarz 3-form H (locally expressed in terms of a Kalb-Ramond 2-form B) and a scalar dilaton ϕ. The derived bracket formalism relates this structure to the generalized differential geometry of a Courant algebroid, which has the appropriate stringy symmetries, and yields a connection with non-trivial curvature and torsion on the generalized “doubled” tangent bundle E ≅ TM ⊕ T*M. Projecting onto TM with the help of a natural non-isotropic splitting of E, we obtain a connection and curvature invariants that reproduce the NS-NS sector of supergravity in 10 dimensions. Further results include a fully generalized Dorfman bracket, a generalized Lie bracket and new formulas for torsion and curvature tensors associated to generalized tangent bundles. A byproduct is a unique Koszul-type formula for the torsionful connection naturally associated to a non-symmetric metric, which resolves ambiguity problems and inconsistencies of traditional approaches to non-symmetric gravity theories.

Bounds on time-optimal concatenations of arcs for two-input driftless 3D systems

We study a driftless system on a three-dimensional manifold driven by two scalar controls. We assume that each scalar control has an independent bound on its modulus and we prove that, locally around every point where the controlled vector fields satisfy some suitable nondegeneracy Lie bracket condition, every time-optimal trajectory has at most five bang or singular arcs. The result is obtained using first- and second-order necessary conditions for optimality.

Finite determination of accessibility and singular points of nonlinear systems: An algebraic approach

Exploiting tools from algebraic geometry, the problem of determination of accessibility/strong accessibility is investigated for polynomial systems and also for analytic systems that are immersible into polynomial systems. The results are constructive, and algorithms are given to find the maximum depth of Lie brackets necessary for deciding accessibility/strong accessibility of the system at any point, called here accessibility/strong accessibility index of the system, and known as the degree of non-holonomy in the literature. Alternatively, upper bounds on the accessibility/strong accessibility index are obtained, which can be computed easier. In each approach, the entire set of accessibility/strong accessibility singular points are obtained, as a limiting algebraic set of a strictly increasing chain of ideals, that stabilizes in finite time. Several examples demonstrate the applicability of the results using computer algebra tools.

Deformations of linear Lie brackets

A VB-algebroid is a vector bundle object in the category of Lie algebroids. We attach to every VB-algebroid a differential graded Lie algebra and we show that it controls deformations of the VB-algebroid structure. Several examples and applications are discussed. This is the first in a series of papers devoted to deformations of vector bundles and related structures over differentiable stacks.

A Lie bracket approximation approach to distributed optimization over directed graphs

We consider a group of computation units trying to cooperatively solve a distributed optimization problem with shared linear equality and inequality constraints. Assuming that the computation units are communicating over a network whose topology is described by a time-invariant directed graph, by combining saddle-point dynamics with Lie bracket approximation techniques we derive a methodology that allows to design distributed continuous-time optimization algorithms that solve this problem under minimal assumptions on the graph topology as well as on the structure of the constraints. We discuss several extensions as well as special cases in which the proposed procedure becomes particularly simple.

Kähler metrics via Lorentzian Geometry in dimension four

AbstractGiven a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.

Nijenhuis operators on pre-Lie algebras

First we use a new approach to define a graded Lie algebra whose Maurer–Cartan elements characterize pre-Lie algebra structures. Then using this graded Lie bracket, we define the notion of a Nijenhuis operator on a pre-Lie algebra which generates a trivial deformation of this pre-Lie algebra. There are close relationships between [Formula: see text]-operators, Rota–Baxter operators and Nijenhuis operators on a pre-Lie algebra. In particular, a Nijenhuis operator “connects” two [Formula: see text]-operators on a pre-Lie algebra whose any linear combination is still an [Formula: see text]-operator in certain sense and hence compatible [Formula: see text]-dendriform algebras appear naturally as the induced algebraic structures. For the case of the dual representation of the regular representation of a pre-Lie algebra, there is a geometric interpretation by introducing the notion of a pseudo-Hessian–Nijenhuis structure which gives rise to a sequence of pseudo-Hessian and pseudo-Hessian–Nijenhuis structures. Another application of Nijenhuis operators on pre-Lie algebras in geometry is illustrated by introducing the notion of a para-complex structure on a pre-Lie algebra and then studying para-complex quadratic pre-Lie algebras and para-complex pseudo-Hessian pre-Lie algebras in detail. Finally, we give some examples of Nijenhuis operators on pre-Lie algebras.