lagrangian hamiltonian unified formalism for field theory the rusk skinner formalism was developed in order to give a geometrical unified formalism for describing mechanical systems it incorporates all the characteristics of lagrangian and hamiltonian descriptions of these systems including dynamical equations and solutions constraints legendre map evolution operators equivalence etc in this work we extend this unified framework to first order classical field theories and show how this description comprises the main features of the lagrangian and hamiltonian formalisms both for the regular and singular cases this formulation is a first step toward further applications in optimal control theory for partial differential equations

multivector fields and connections setting lagrangian equations in field theories the integrability of multivector fields in a differentiable manifold is studied then given a jet bundle j1e e m it is shown that integrable multivector fields in e are equivalent to integrable connections in the bundle e m that is integrable jet fields in j1e this result is applied to the particular case of multivector fields in the manifold j1e and connections in the bundle j1e m that is jet fields in the repeated jet bundle j1j1e in order to characterize integrable multivector fields and connections whose integral manifolds are canonical lifting of sections these results allow us to set the lagrangian evolution equations for first order classical field theories in three equivalent geometrical ways in a form similar to that in which the lagrangian dynamical equations of nonautonomous mechanical systems are usually given then using multivector fields we discuss several aspects of these evolution equations both for the regular and singular cases namely the existence and nonuniqueness o

enumeration of the order 14 invariants formed from the riemann tensor the results of a complete enumeration of the scalers formed from the riemann tensor by covariant differentiation multiplication and contraction of order 14 in the derivatives of the metric are presented the corresponding enumeration for the numbers of scalars constructed solely from the weyl tensor is also given

differential geometry palatini gravity and reduction the present article deals with a formulation of the so called vacuum palatini gravity as a general variational principle in order to accomplish this goal some geometrical tools related to the geometry of the bundle of connections of the frame bundle lm are used a generalization of lagrange poincare reduction scheme to these types of variational problems allows us to relate it with the einstein hilbert variational problem relations with some other variational problems for gravity found in the literature are discussed

nilpotent and high order approximations of vector field systems this paper is concerned with the study of structurally stable properties associated with systems described by one or several real analytic vector fields via approximating systems which retain inf

automorphisms of real four dimensional lie algebras and the invariant characterization of homogeneous 4 spaces the automorphisms of all four dimensional real lie algebras are presented in a comprehensive way their action on the space of 4 4 real symmetric and positive definite matrices defines equivalence classes which are used for the invariant characterization of the four dimensional homogeneous spaces which possess an invariant basis

brief state space representation for fractional order controllers this article proposes an infinite dimensional state space realization for linear filters with transfer function c d s c 0 1 s w b 1 s w h d where 0 w b w h and 0

adaptive regulation of high order lower triangular systems an adding a power integrator technique abstract in this paper a new adaptive control technique called adding a power integrator is proposed for a class of linearly parametrized high order nonlinear systems that are of a lower triangular structure but neither feedback linearizable nor affine in the control input using the adaptive adding a power integrator technique as a basic design tool we show how to explicitly construct in a recursive manner a smooth adaptive controller that solves the problem of adaptive regulation with global stability the development of the adding a power integrator technique leads to a significant relaxation on the system nonlinearities such as feedback linearizability and affineness which are known to be two standard hypotheses made in the existing nonlinear adaptive control schemes

numerical values of goldberg s coefficients in the series for log e x e y calcul des coefficients de goldberg dans la serie infinie de log e x e y pour x et y non communitatifs avec des mots de longueur 20

special case closed form of the baker campbell hausdorff formula the baker campbell hausdorff formula is a general result for the quantity where x and y are not necessarily commuting for completely general commutation relations between x and y the free lie algebra the general result is somewhat unwieldy however in specific physics applications the commutator while non zero might often be relatively simple which sometimes leads to explicit closed form results we consider the special case and show that in this case the general result reduces to furthermore we explicitly evaluate the symmetric function demonstrating that and relate this to previously known results for instance this result includes but is considerably more general than results obtained from either the heisenberg commutator or the creation destruction commutator

adding one power integrator a tool for global stabilization of high order lower triangular systems a new feedback design tool called adding a power integrator is introduced and used to solve the problem of global robust stabilization for a significant class of uncertain nonlinear systems that are of a lower triangular form but neither necessarily feedback linearizable fully or partially nor affine in the control input and therefore cannot be dealt with via conventional approaches sufficient conditions are presented under which a globally stabilizing smooth state feedback control law can be explicitly constructed using repeatedly the adding one power integrator technique this novel design technique permits removal of the feedback linearizability and linearity of the control input which have been two common assumptions so far in the literature of nonlinear control

on minimum fuel control of affine nonlinear systems the minimum fuel control problem is investigated for a class of multiinput affine nonlinear systems whose associated lie algebra is nilpotent after preliminary notions and definitions are presented emphasis is on the solution to the optimal control problem of the lie algebra generated by certain system vector fields is nilpotent consequences of the maximum principle are deduced for such systems

nilpotent bases for distributions and control systems abstract let v m be the lie algebra infinite dimensional of real analytic vector fields on the n dimensional manifold m necessary conditions that a real analytic k dimensional distibution on m have a local basis which generates a nilpotent subalgebra of v m are derived two methods for sufficient conditions are given the first depending on the existence of a solution to a system of partial differential equations the second using darboux s theorem to give a computable test for an n 1 dimensional distribution a nonlinear control system in which the control variables appear linearly can be transformed into an orbit equivalent system whose describing vector fields generate a nilpotent algebra if the distribution generated by the original describing vector fields admits a nilpotent basis when this is the case local analysis of the control system is greatly simplified

dynamical systems and nilpotent sub riemannian geometry we consider dynamical systems determined by distributions of real analytic vector fields the equations of motion and the associated gauge transformation are presented in detail a gradation of the associated lie algebras leads to the consideration of polynomial vector fields the minimization of kinetic energy action in the class of horizontal curves associated with the distribution is formulated as a sub riemannian geodesic problem normal geodesics are fully described for the so called gaveau brockett distribution the exponential mapping the unitary spheres and the wave fronts are calculated for particular cases

free nilpotent and nilpotent quadratic lie algebras abstract in this paper we introduce an equivalence between the category of the t nilpotent quadratic lie algebras with d generators and the category of some symmetric invariant bilinear forms over the t nilpotent free lie algebra with d generators taking into account this equivalence t nilpotent quadratic lie algebras with d generators are classified up to isometric isomorphisms and over any field of characteristic zero in the following cases d 2 and t 5 d 3 and t 3

identifying digital and fractional transfer functions from a frequency response in this article an existing exact algorithm for identifying a continuous transfer function from frequency data giving major importance to phase data is further developed the new version allows identifying a digital transfer function or a continuous fractional order transfer function with a fixed order of commensurability i e a ratio of two polynomials in s this is straightforward if the desired transfer function is to have only zeros or only poles if a transfer function with both zeros and poles is sought however a numerical method must be resorted to

on homogeneous finite time control for linear evolution equation in hilbert space based on the notion of generalized homogeneity a new algorithm of feedback control design is developed for a plant modeled by a linear evolution equation in a hilbert space with a possibly unbounded operator the designed control law steers any solution of the closed loop system to zero in a finite time a method of homogeneous extension is presented in order to make the developed control design principles to be applicable for evolution systems with nonhomogeneous operators the design scheme is demonstrated for the heat equation with the control input distributed on the segment 0 1

on the legendre map in higher order field theories the authors show how the construction of a cartan form in higher order field theories defines a legendre map and how the regularity of this map may be described in terms of a sequence of maps between intermediate phase spaces they also show how semi holonomic jets are important in this context they allow the definition of regularity to be applied without ambiguity to the lagrangian itself and they permit the specification of a unique cartan form legendre map and covariant phase space for second order lagrangians

on the analysis of nonlinear nilpotent switched systems using the hall sussmann system abstract we consider an open problem on the stability of nonlinear nilpotent switched systems posed by daniel liberzon partial solutions to this problem were obtained as corollaries of global nice reachability results for nilpotent control systems the global structure is crucial in establishing stability we show that a nice reachability analysis may be reduced to the reachability analysis of a specific canonical system the nilpotent hall sussmann system furthermore local nice reachability properties for this specific system imply global nice reachability for general nilpotent systems we derive several new results revealing the elegant lie algebraic structure of the nilpotent hall sussmann system

first order equivalent to einstein hilbert lagrangian a first order lagrangian l variationally equivalent to the second order einstein hilbert lagrangian is introduced such a lagrangian depends on a symmetric linear connection but the dependence is covariant under diffeomorphisms the variational problem defined by l is proved to be regular and its hamiltonian formulation is studied including its covariant hamiltonian attached to

multisymplectic unified formalism for einstein hilbert gravity we present a covariant multisymplectic formulation for the einstein hilbert model of general relativity as it is described by a second order singular lagrangian this is a gauge field theory with constraints the use of the unified lagrangian hamiltonian formalism is particularly interesting when it is applied to these kinds of theories since it simplifies the treatment of them in particular the implementation of the constraint algorithm the retrieval of the lagrangian description and the construction of the covariant hamiltonian formalism in order to apply this algorithm to the covariant field equations they must be written in a suitable geometrical way which consists of using integrable distributions represented by multivector fields of a certain type we apply all these tools to the einstein hilbert model without and with energy matter sources we obtain and explain the geometrical and physical meaning of the lagrangian constraints and we construct the multimomentum covariant hamiltonian formalisms in both cases as a consequence of the gauge freedom and the constraint algorithm we see how this model is equivalent to a first order regular theory without gauge freedom in the case of the presence of energy matter sources we show how some relevant geometrical and physical characteristics of the theory depend on the type of source in all the cases we obtain explicitly multivector fields which are solutions to the gravitational field equations finally a brief study of symmetries and conservation laws is done in this context we present a covariant multisymplectic formulation for the einstein hilbert model of general relativity as it is described by a second order singular lagrangian this is a gauge field theory with constraints the use of the unified lagrangian hamiltonian formalism is particularly interesting when it is applied to these kinds of theories since it simplifies the treatment of them in particular the implementation of the constraint algorithm the retrieval of the lagrangian description and the construction of the covariant hamiltonian formalism in order to apply this algorithm to the covariant field equations they must be written in a suitable geometrical way which consists of using integrable distributions represented by multivector fields of a certain type we apply all these tools to the einstein hilb

generalized hamiltonian dynamics i formulation on t q tq the dirac bergmann generalized hamiltonian dynamics for a degenerate lagrangian system is formulated on the whitney sum t q tq of the phase space t q and the velocity space tq over the configuration space q the formulation is related to those on t q and tq some ambiguities concerning generalized dynamics that have appeared in the literature are clarified

on homogeneous distributed parameter systems a geometric homogeneity is introduced for evolution equations in a banach space scalability property of solutions of homogeneous evolution equations is proven some qualitative characteristics of stability of trivial solution are also provided in particular finite time stability of homogeneous evolution equations is studied theoretical results are illustrated on important classes of partial differential equations

nonholonomic motion planning steering using sinusoids methods for steering systems with nonholonomic c onstraints between arbitrary configurations are investigated suboptimal trajectories are derived for systems that are not in canonical form systems in which it takes more than one level of bracketing to achieve controllability are considered the trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level a class of systems that can be steered using sinusoids claimed systems is defined conditions under which a class of two input systems can be converted into this form are given

dynamical realizations of finite volterra series in this paper realizations of finite volterra series are viewed as nonlinear analytic input output systems with state space described by an analytic manifold for a minimal realization guaranteed by h j sussmann the state space which is unique up to diffeomorphism is shown to have the homogeneous space structure of a nilmanifold the quotient of two nilpotent lie groups the structure of nilmanifolds as described by a malcev is used to show that for these systems the state space has a vector space structure as a consequence of this result it is shown that a minimal realization of a finite volterra series can be described as a cascade of linear subsystems with polynomial link maps in which the dimension o f each linear subsystem is independent of the realization considered

frequency domain synthesis of a filter using viete root functions this paper proposes a noniterative method to fit a rational transfer function to a specified frequency response a method using a smoothing of width modulated pulses of the phase asymptotic diagram is modified to provide an exact algebraic method this is then used to synthesize a robust controller and a nonrational transfer function with a fractional differentiation order

cyclic shifts and an exponential formula abstract with x and y noncommutative indeterminates a natural presentation of z log e x e y as a sum of group conjugates of x and y is shown to depend on the fact that the terms involving the cyclic shifts of any fixed word other than x or y in the power series presentation of z sum to zero

verification of iss iiss and ioss properties applying weighted homogeneity several conditions are proposed to check different robustness properties iss iiss ioss and oss for generic nonlinear systems applying the weighted homogeneity concept global or local the advantages of this result are that under some mild conditions the system robustness can be established as a function of the degree of homogeneity

convergence proof for goldberg s exponential series abstract the goldberg presentation of the formal series for log e x e y in noncommutative symbols x and y is shown to converge when x and y are replaced by normed elements in the unit sphere about zero provided the norm defining the unit sphere is compatible with multiplication

configuration controllability of mechanical systems underactuated by one control we investigate local configuration controllability for mechanical control systems within the affine connection formalism we rely on previous results on controllability and series expansions for the evolution of mechanical systems starting from rest extending the work by lewis for the single input case we are able to characterize local configuration controllability for systems with n degrees of freedom and n 1 input forces

nilpotent approximations of control systems and distributions a constructive method is given to approximate the vector fields in a nonlinear control system which is linear in the controls by a system of similar form and on the same state space with the describing vector fields of the approximating system generating a lie algebra which has a certain relevant sub algebra nilpotent given a k dimensional distribution on r n say locally defined near zero the method leads to an approximating k distribution which has nilpotent basis agrees with the original distribution at zero and each derived distribution of the approximating distribution and original distribution also agree at zero

an effective version of the lazard correspondence the lazard correspondence establishes an equivalence of categories between p groups of nilpotency class less than p and nilpotent lie rings of the same class and order the main tools used to achieve this are the baker campbell hausdorff formula and its inverse formulae here we describe methods to compute the inverse baker campbell hausdorff formulae using these we get an algorithm to compute the lie ring structure of a p group of class p furthermore the baker campbell hausdorff formula yields an algorithm to construct a p group from a nilpotent lie ring of order pn and class less than p at the end of the paper we discuss some applications of and practical experiences with the algorithms

invariants for the lagrangian equivalence problem abstract let m be a connected smooth manifold let aut p be the group automorphisms of the bundle p r m r and let q j 1 r m r j 1 r m be the canonical projection invariant functions on j r q under the natural action of aut p are discussed in relationship with the lagrangian equivalence problem the second order invariants are determined geometrically as well as some other higher order invariants for dim m 2

the number of functionally independent invariants of a pseudo riemannian metric the number of functionally independent scalar invariants of arbitrary order of a generic pseudo riemannian metric on an n dimensional manifold is determined

the algebra of norms and expanding maps on groups normed linear spaces and algebras equipped with norms are a familiar part of the mathematical landscape normed groups have achieved less prominence this may be due to the comparative youth of the subject it was only twenty years ago that milnor and others recognized that some of the geometry of manifolds could be captured by metric properties of the fundamental group much of the subsequent interest has been centered on the word length norm which we shall return to and growth properties this paper takes a retrospective view of some results about normed groups and endomorphisms initially proved in the context of global and harmonic analysis we ask whether the same results can be established using abstract algebraic techniques the model for this sensibility is found of course in the many elegant arguments devised by yitz to explain theorems about operators the exposition at hand alas fails to possess the clarity and polish which an appropriate tribute demands silhouettes and speculations will have to suffice when i began working on normed groups i despaired that there was not enough structure to get my hands on thanks to helpful clues provided by my colleagues marge murray and bill floyd i now lament that i cannot possibly master enough mathematics to understand the subject if my analysis and geometry are not up to snuff i request the indulgence of my readers by a norm on a group we shall mean a non negative real valued function 1 1 whose domain is the group g which satisfies

nice reachability for planar bilinear control systems with applications to planar linear switched systems corrected we consider planar bilinear control systems with measurable controls we show that any point in the reachable set can be reached by a ldquonicerdquo control specifically a control that is a concatenation of a bang arc with either 1 a bang bang control that is periodic after the third switch or 2 a piecewise constant control with no more than two discontinuities under the additional assumption that the bilinear system is positive or invariant for any proper cone we show that the reachable set is spanned by a concatenation of a bang arc with either 1 a bang bang control with no more than two discontinuities or 2 a piecewise constant control with no more than two discontinuities in particular any point in the reachable set can be reached using a piecewise constant control with no more than three discontinuities several known results on the stability of planar linear switched systems under arbitrary switching follow as corollaries of our result we demonstrate this with an example

the baker campbell hausdorff formula and nested commutator identities the coefficients of the baker campbell hausdorff expansion are calculated by using various methods comparison of the results yields several remarkable identities satisfied by multiple commutators which in turn allow us to greatly simplify the form of the expansion up to order eight

models for free nilpotent lie algebras if a lie algebra g can be generated by m of its elements e1 em and if any other lie algebra generated by m other elements f1 fm is a homomorphic image of g under the map ei fi we say that it is the free lie algebra on m generators the free nilpotent lie algebra gm r on m generators of rank r is the quotient of the free lie algebra by the ideal gr 1 generated as follows g1 g and gk gk 1 g let n denote dimension of gm r free lie algebras are those which have as few relations as possible only those which are a conseqence of the anti commutativity of the bracket and of the the jacobi identity free nilpotent lie algebras add the relations that any iterated lie bracket of more than r elements vanishes for further details see 18 or 13 in this paper we are interested in explicit computations of the lie algebras gm r it is well known 19 that that there is a representation of gm r on upper triangular n by n matrices the problem is that many computations are difficult using this representation in this paper we present an algorithm that yields vector fields e1 em defined in r with the property that they generate a lie algebra isomorphic to gm r see 6 for another approach to this problem in section 3 we restrict to two generators and give three consequences of this algorithm first the form of the vector fields is such that flows of the control system t e1 u t e2 x t may be computed explicitly research supported in part by postdoctoral research fellowships from the national science foundation

nonhomogeneous nilpotent approximations for nonholonomic systems with singularities nilpotent approximations are a useful tool for analyzing and controlling systems whose tangent linearization does not preserve controllability such as nonholonomic mechanisms however conventional homogeneous approximations exhibit a drawback in the neighborhood of singular points where the system growth vector is not constant the vector fields of the approximate dynamics do not vary continuously with the approximation point the geometric counterpart of this situation is that the sub riemannian distance estimate provided by the classical ball box theorem is not uniform at singular points with reference to a specific family of driftless systems we show how to build a nonhomogeneous nilpotent approximation whose vector fields vary continuously around singular points it is also proven that the privileged coordinates associated to such an approximation provide a uniform estimate of the distance

local asymptotic approximation of nonlinear control systems let g1 gm be smooth vector fields on rd using a canonical linear rescaling of time and space coordinates the control system is transformed into a system on the same state space which is arbitrarily close to a nilpotent one this approximation takes place in the c topology and is particularly useful for the study of local properties of which depend on the high order variational structure of the vector fields gi

frequency domain curve fitting with maximum amplitude criterion and guaranteed stability the problem is considered of fitting a stable rational transfer function to complex frequency response data minimizing a weighted maximum amplitude criterion a solution for this problem is provided in both discrete and continuous time and for both siso and mimo systems by parametrizing the transfer function and solving a smooth nonlinear constrained optimization problem the denominator is parametrized as a product of first and second order polynomials by adding linear constraints on the denominator parameters to the nonlinear programming problem the poles of the curve fit model are restricted to the stability region in the complex plane to provide an initial estimate for the nonlinear optimization problem an iterative procedure is proposed where in each step a linear programming problem has to be solved

goldberg s theorem and the baker campbell hausdorff formula abstract goldberg s theorem duke math j 23 1956 13 is applied to expressing the logarithm of exponential product in terms of free lie elements using hyperoperators namely inner derivations a symmetric property of the expansion coefficients in the formal power series is also discussed

dynkin s method of computing the terms of the baker campbell hausdorff series the infinite series for log exp x exp y for noncommuting x and y is expressible in terms of iterated commutators of x and y except for the linear term x y dynkin derived an explicit expression for the terms as a sum of iterated commutators over a certain set of sequences this paper presents a practical algorithm for applying dynkin s formula and gives several illustrative examples

a simple expression for the terms in the baker campbell hausdorff series a simple expression is derived for the terms in the baker campbell hausdorff series one formulation of the result involves a finite number of operations with matrices of rational numbers generalizations are discussed

control of nonlinear systems in singular problems abstract the minimum singular functional control problem is analyzed for a class of multi input affine nonlinear systems we develop an algorithm for solving the singular problem that is applicable whether or not singular sub arcs exist in the optimal control the optimal control of bilinear systems to obtain by considering the lie algebra generated by the system matrices

a global steering method for nonholonomic systems abstract in this paper we present an iterative steering algorithm for nonholonomic systems also called driftless control affine systems and we prove its global convergence under the sole assumption that the lie algebraic rank condition larc holds true everywhere that algorithm is an extension of the one introduced in jean et al 2005 21 for regular systems the first novelty here consists in the explicit algebraic construction starting from the original control system of a lifted control system which is regular the second contribution of the paper is an exact motion planning method for nilpotent systems which makes use of sinusoidal control laws and which is a generalization of the algorithm described in murray and sastry 1993 29 for chained form systems

higher order conditions for singular lagrangian dynamics for a kth order singular lagrangian the presymplectic equation plus the different mth order differential equation conditions that can be considered in the lagrangian space yield different dynamics which are studied this leads to a new classification of the constraints this study is also performed in the intermediate formalisms which can be defined between the lagrangian and the hamiltonian ones the corresponding classification schemes are then related

recursion formulas for the lie integral abstract an explicit definition is given of a non linear integral which is termed the lie integral this integral is defined for lie algebra valued functions in terms of the representations of the algebra various properties of the integral are investigated and applications are given to differential equations applications to differential geometry will be given elsewhere

exponential stabilization of driftless nonlinear control systems using homogeneous feedback this paper focuses on the problem of exponential stabilization of controllable driftless systems using time varying homogeneous feedback the analysis is performed with respect to a homogeneous norm in a nonstandard dilation that is compatible with the algebraic structure of the control lie algebra it can be shown that any continuous time varying controller that achieves exponential stability relative to the euclidean norm is necessarily non lipschitz despite these restrictions we provide a set of constructive sufficient conditions for extending smooth asymptotic stabilizers to homogeneous exponential stabilizers the modified feedbacks are everywhere continuous smooth away from the origin and can be extended to a large class of systems with torque inputs the feedback laws are applied to an experimental mobile robot and show significant improvement in convergence rate over smooth stabilizers

solving nonlinear equations from higher order derivations in linear stages for all polynomials a r of degree less than or equal to d and all higher order derivations e e s e 1 and f f s f 1 where 1 s and 1 i m of length s less or equal to r the reason for wanting such derivations e1 em is simple given several derivations f1 fm it is useful to have derivations e1 em which are good local approximations to the fi and which are easy to compute with notice that since the right hand sides of equations are known while the left hand sides involve the unspecified coefficients of the polynomial functions the equations are equivalent to a system of nonlinear algebraic equations involving the coefficients it is the purpose of this paper to give an algorithm which solves such a system when a solution exists in fact as the title suggests this algorithm will work by solving a sequence of r linear systems

control systems which generate decomposable lie algebras let x y be real analytic vector fields on an n manifold m ad x y x y denote the lie product inductively ad k 1 x y x ad k x y and 1 ad v x y v 0 the attainable set at time t denoted download full size image t for the control system 1 x x x u t y x x 0 p u t 1 is contained in the integral manifold through p of the lie algebra l 1 generated by 1 l 1 is said to be decomposable at p if essentially when dim l 1 p n the solution of 1 can be written as exp tx exp f 1 t u v 1 exp f n t u v n p with v 1 v n l 1 this is true for example at any p if l 1 is nilpotent the f i t u are shown to satisfy a system on n of the form 2 x i u t g i t x 1 x i 1 x i 0 0 i 1 n where if l 1 is nilpotent each g i is a polynomial in x 1 x i 1 in this case reachability properties of 2 for controls being countably additive measures impluses reduce to the study of algebraic equations homogeneity allows these to be considered on projective space n giving local results by bezout s theorem similar results hold for controls in the unit ball of krener s results on approximating systems of the form 1 by systems which generate nilpotent lie algebras can then be used to remove the condition that l 1 be nilpotent the method applies to show that if m denotes the set of all products of k elements of 1 with k m and dim span m 1 p dim span m for all odd m then exp tx p int download full size image t for all t 0

free nilpotent lie algebras admitting ad invariant metrics abstract in this work we find necessary and sufficient conditions for a free nilpotent or a free metabelian nilpotent lie algebra to be endowed with an ad invariant metric for such nilpotent lie algebras admitting an ad invariant metric the corresponding automorphisms groups are studied

a note on the zassenhaus product formula we provide a simple method for the calculation of the terms cn in the zassenhaus product ea b ea eb n 2 ecn for noncommuting a and b this method has been implemented in a computer program furthermore we formulate a conjecture on how to translate these results into nested commutators this conjecture was checked up to order n 17 using a computer

on minimum quadratic functional control of affine nonlinear systems abstract the minimization control problem of the quadratic functionals for the class of affine nonlinear systems in the hypothesis of nilpotent associated lie algebra is analyzed the optimal control correspondent to the first second and third order nilpotent operator is determined h bourdache siguerdinljane has applied the method of lie algebras to the study of optimal control regulation of satellites s banks and m yew have studied the optimal control of energy consumption minimization for a class of bilinear systems and j s liu et al have generalized this result for the class of affine nonlinear systems the optimal control determination of nonlinear system with a nilpotent structure minimizing the quadratic functionals generalizes the results of j s liu k yuan w s lin and s p banks and m k yew regarding the energy minimization of the affine nonlinear and bilinear systems