improved convergence towards generalized euler mascheroni constant we propose new simple sequences approximating the euler mascheroni constant and its generalization which converge faster towards their limits than those considered by detemple d w detemple a quicker convergence to euler s constant am math monthly 100 5 1998 468 470 sintamarian a sintamarian a generalization of euler s constant numer algorithms 46 2 2007 141 151 and vernescu a vernescu a new accelerate convergence to the constant of euler gazeta matem ser a bucharest xvii xcvi 4 1999 273 278

on some euler mascheroni type sequences the aim of this paper is to introduce new sequences which extend the classical sequence convergent to the euler mascheroni constant some new techniques for obtaining faster converging sequences of euler mascheroni type are given we show how these sequences can be combined with the harmonic sequence to improve the rate of convergence

ramanujan formula for the generalized stirling approximation abstract the aim of this paper is to improve ramanujan s formula for approximation of the factorial function starting from burnside s formula in contradistinction with the classical formula that starts from stirling s formula

a class of integral approximations for the factorial function we introduce stirling s formula in a more general class of approximation formulas to extend the integral representation of liu z liu a new version of the stirling formula tamsui oxf j math sci 23 4 2007 389 392 finally an accurate approximation for the factorial function is established

interconnection and damping assignment passivity based control of port controlled hamiltonian systems passivity based control pbc is a well established technique that has shown to be very powerful to design robust controllers for physical systems described by euler lagrange el equations of motion for regulation problems of mechanical systems which can be stabilized shaping only the potential energy pbc preserves the el structure and furthermore assigns a closed loop energy function equal to the difference between the energy of the system and the energy supplied by the controller thus we say that stabilization is achieved via energy balancing unfortunately these nice properties of el pbc are lost when used in other applications which require shaping of the total energy for instance in electrical or electromechanical systems or even some underactuated mechanical devices our main objective in this paper is to develop a new pbc theory which extends to a broader class of systems the aforementioned energy balancing stabilization mechanism and the structure invariance towards this end we depart from the el description of the systems and consider instead port controlled hamiltonian models which result from the network modelling of energy conserving lumped parameter physical systems with independent storage elements and strictly contain the class of el models

on the synthesis of boundary control laws for distributed port hamiltonian systems this paper is concerned with the energy shaping of 1 d linear boundary controlled port hamiltonian systems the energy casimir method is first proposed to deal with power preserving systems it is shown how to use finite dimensional dynamic boundary controllers and closed loop structural invariants to partially shape the closed loop energy function and how such controller finally reduces to a state feedback when dissipative port hamiltonian systems are considered the casimir functions do not exist anymore dissipation obstacle and the immersion via a dynamic controller reduction through invariants method cannot be applied the main contribution of this paper is to show how to use the same ideas and state functions to shape the closed loop energy function of dissipative systems through direct state feedback i e without relying on a dynamic controller and a reduction step in both cases the existence of solution and the asymptotic stability by additional damping injection of the closed loop system are proven the general theory and achievable closed loop performances are illustrated with the help of a concluding example the boundary stabilization of a longitudinal beam vibrations

well posedness of boundary controlled and observed stochastic port hamiltonian systems in this article stochastic port hamiltonian systems sphs on infinite dimensional spaces governed by ito stochastic differential equations sdes are introduced and some properties of this new class of systems are studied they are an extension of sphss defined on a finite dimensional state space the concept of well posedness in the sense of weiss and salamon is generalized to the stochastic context under this extended definition sphss are shown to be well posed the theory is illustrated on an example of a vibrating string subject to a hilbert space valued gaussian white noise process

computation of the gamma digamma and trigamma functions this paper gives an approximation for the gamma function that while different has the same form as one by lanczos siam j numer anal b1 1964 pp 86 96 both approximations correct stirlin

dirac structures and boundary control systems associated with skew symmetric differential operators associated with a skew symmetric linear operator on the spatial domain a b we define a dirac structure which includes the port variables on the boundary of this spatial domain this dirac structure is a subspace of a hilbert space naturally associated with this dirac structure is an infinite dimensional system we parameterize the boundary port variables for which the c 0 semigroup associated with this system is contractive or unitary furthermore this parameterization is used to split the boundary port variables into inputs and outputs similarly we define a linear port controlled hamiltonian system associated with the previously defined dirac structure and a symmetric positive operator defining the energy of the system we illustrate this theory on the example of the timoshenko beam

weak form of stokes dirac structures and geometric discretization of port hamiltonian systems we present the mixed galerkin discretization of distributed parameter port hamiltonian systems due to the inherent definition of boundary interconnection ports this system representation is particularly useful for the modeling of interconnected multi physics systems and control at the prototypical example of a system of two conservation laws in arbitrary spatial dimension we derive the main contributions i a weak formulation of the underlying geometric stokes dirac structure ii its geometric approximation by a finite dimensional dirac structure using a mixed galerkin approach and power preserving maps on the space of discrete power variables and iii the approximation of the hamiltonian to obtain finite dimensional port hamiltonian state space models the power preserving maps on the discrete bond space offer design degrees of freedom for the discretization which is illustrated at the example whitney finite elements on a 2d simplicial triangulation the resulting schemes can be considered as trade offs between centered approximations and upwinding

energy based lyapunov functions for forced hamiltonian systems with dissipation in this paper we propose a constructive procedure to modify the hamiltonian function of forced hamiltonian systems with dissipation in order to generate lyapunov functions for nonzero equilibria a key step in the procedure which is motivated from energy balance considerations standard in network modeling of physical systems is to embed the system into a larger hamiltonian system for which a series of casimir functions can be easily constructed interestingly enough for linear systems the resulting lyapunov function is the incremental energy thus our derivations provide a physical explanation to it an easily verifiable necessary and sufficient condition for the applicability of the technique in the general nonlinear case is given some examples that illustrate the method are given

exponential stability of a class of boundary control systems we study a class of partial differential equations with variable coefficients on a one dimensional spatial domain with control and observation at the boundary for this class of systems we provide simple tools to check exponential stability this class is general enough to include models of flexible structures traveling waves heat exchangers and bioreactors among others the result is based on the use of a generating function the energy for physical systems and an inequality condition at the boundary furthermore based on the port hamiltonian approach we give a constructive method to reduce this inequality to a simple matrix inequality

exponential stabilization of port hamiltonian boundary control systems via energy shaping this article is concerned with the exponential stabilization of a class of linear boundary control systems bcss in port hamiltonian form through energy shaping starting from a first feedback loop that is in charge of modifying the hamiltonian function of the plant a second control loop that guarantees exponential convergence to the equilibrium is designed in this way a major limitation of standard energy shaping plus damping injection control laws applied to linear port hamiltonian bcss namely the fact that only asymptotic convergence is assured has been removed

pseudo spectral methods for the spatial symplectic reduction of open systems of conservation laws a reduction method is presented for systems of conservation laws with boundary energy flow it is stated as a generalized pseudo spectral method which performs exact differentiation by using simultaneously several approximation spaces generated by polynomials bases and suitable choices of port variables the symplecticity of this spatial reduction method is proved when used for the reduction of both closed and open systems of conservation laws for any choice of collocation points i e for any polynomial bases the symplecticity of some more usual collocation schemes is discussed and finally their accuracy on approximation of the spectrum on the example of the ideal transmission line is discussed in comparison with the suggested reduction scheme

precise calculation of the dynamical exponent of two dimensional percolation we report numerical data obtained on the special purpose computer percola for the exponent tilde t of the electrical conductivity of 2d percolation the extrapolation yields tilde t 0 9745 pm 0 0015 and a correction to the scaling exponent 1 2 0 2

dirac structures on hilbert spaces and boundary control of distributed port hamiltonian systems abstract aim of this paper is to show how the dirac structure properties can be exploited in the development of energy based boundary control laws for distributed port hamiltonian systems usually stabilization of non zero equilibria has been achieved by looking at or generating a set of structural invariants namely casimir functions in closed loop since this approach fails when an infinite amount of energy is required at the equilibrium dissipation obstacle this paper illustrates a novel approach that enlarges the class of stabilizing controllers the starting point is the parametrization of the dynamics provided by the image representation of the dirac structure that is able to show the effects of the boundary inputs on the state evolution in this way energy balancing and control by state modulated source methodologies are extended to the distributed parameter scenario and a geometric interpretation of these control techniques is provided the theoretical results are discussed with the help of a simple but illustrative example i e a transmission line with an rlc load in both serial and parallel configurations in the latter case energy balancing controllers are not able to stabilize non zero equilibria because of the dissipation obstacle the problem is solved thanks to a boundary state modulated source

on representations and integrability of mathematical structures in energy conserving physical systems in the present paper we elaborate on the underlying hamiltonian structure of interconnected energy conserving physical systems it is shown that a power conserving interconnection of port controlled generalized hamiltonian systems leads to an implicit generalized hamiltonian system and a power conserving partial interconnection to an implicit port controlled hamiltonian system the crucial concept is the notion of a generalized dirac structure defined on the space of energy variables or on the product of the space of energy variables and the space of flow variables in the port controlled case three natural representations of generalized dirac structures are treated necessary and sufficient conditions for closedness or integrability of dirac structures in all three representations are obtained the theory is applied to implicit port controlled generalized hamiltonian systems and it is shown that the closedness condition for the dirac structure leads to strong conditions on the input vector fields

dissipativity based boundary control of linear distributed port hamiltonian systems the main contribution of this paper is a general synthesis methodology of exponentially stabilising control laws for a class of boundary control systems in port hamiltonian form that are dissipative with respect to a quadratic supply rate being the total energy the storage function more precisely general conditions that a linear regulator has to satisfy to have at first a well posed and secondly an exponentially stable closed loop system are presented the methodology is illustrated with reference to two specific stabilisation scenarios namely when the distributed parameter plant is in impedance or in scattering form moreover it is also shown how these techniques can be employed in the analysis of more general systems that are described by coupled partial and ordinary differential equations in particular the repetitive control scheme is studied and conditions on the finite dimensional linear plant to have asymptotic tracking of generic periodic reference signals are determined

stability and passivity preserving petrov galerkin approximation of linear infinite dimensional systems this contribution presents two approximation methods for linear infinite dimensional systems that ensure the preservation of stability and passivity the first approach allows one to approximate internal source free infinite dimensional systems such that the resulting approximation is a port controlled hamiltonian system with dissipation the second method deals with the class of systems that are not required to have conjugated outputs but only a dissipative system operator it yields approximations with a dissipative system matrix for which bounds of their stability margin are provided both approaches are based on a state space formulation of the infinite dimensional system this makes it possible to use the petrov galerkin approximation whose free parameters are partly used for achieving the structure preservation since still free parameters remain further application specific objectives such as e g moment matching can be achieved both approaches are applied to the approximation of an euler bernoulli beam

on the asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas in this paper we present new asymptotic expansions of the gamma function related to the nemes gosper and burnside formulas

the asymptotic series of the generalized stirling formula the aim of this paper is to construct some asymptotic expansions which produce increasingly accurate approximations of the gamma function

a characterization of boundary conditions yielding maximal monotone operators abstract we provide a characterization for maximal monotone realizations for a certain class of nonlinear operators in terms of their corresponding boundary data spaces the operators under consideration naturally arise in the study of evolutionary problems in mathematical physics we apply our abstract characterization result to port hamiltonian systems and a class of frictional boundary conditions in the theory of contact problems in visco elasticity

best estimates of the generalized stirling formula the aim of this paper is to introduce an approximations family of the factorial function that contains stirling s formula burnside s formula and gosper s formula the parameters which provide the best approximations are indicated finally numerical computations are made to show the superiority of our formulas over other known formulas

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

energy shaping of distributed parameter port hamiltonian systems based on finite element approximation the main contribution of this paper is a procedure for the control by energy shaping via casimir generation of infinite dimensional port hamiltonian systems based on a particular finite element approximation the proposed approach is justified by the fact that the adopted spatial discretization technique is able to preserve casimir functions in the closed loop system when going from the distributed to the approximated lumped parameter system besides the intrinsic difficulties related to the large number of state variables the finite element model is generally given in terms of a dirac structure and is completely a causal which implies that the plant dynamics is not given in standard input state output form but as a set of daes consequently the classical energy casimir method has to be extended in order to deal with dynamical systems with constraints usually appearing in the form of lagrangian multipliers the general methodology is illustrated with the help of an example in which the distributed parameter system is a lossless transmission line

dirac structures and their composition on hilbert spaces dirac structures appear naturally in the study of certain classes of physical models described by partial differential equations and they can be regarded as the underlying power conserving structures we study these structures and their properties from an operator theoretic point of view in particular we find necessary and sufficient conditions for the composition of two dirac structures to be a dirac structure and we show that they can be seen as lagrangian hyper maximal neutral subspaces of krein spaces moreover special emphasis is laid on dirac structures associated with operator colligations it turns out that this class of dirac structures is linked to boundary triplets and that this class is closed under composition

transport on percolation clusters with power law distributed bond strengths the simplest transport problem namely finding the maximum flow of current or maxflow is investigated on critical percolation clusters in two and three dimensions using a combination of extremal statistics arguments and exact numerical computations for power law distributed bond strengths of the type p ensuremath sigma ensuremath sim ensuremath sigma ensuremath ensuremath alpha assuming that only cutting bonds determine the flow the maxflow critical exponent v is found to be v ensuremath alpha d ensuremath 1 ensuremath nu 1 1 ensuremath ensuremath alpha this prediction is confirmed with excellent accuracy using large scale numerical simulation in two and three dimensions however in the region of anomalous bond capacity distributions 0l ensuremath alpha l 1 we demonstrate that due to cluster structure fluctuations it is not the cutting bonds but the blobs that set the transport properties of the backbone this blob dominance avoids a crossover to a regime where structural details the distribution of the number of red or cutting bonds would set the scaling the restored scaling exponents however still follow the simplistic red bond estimate this is argued to be due to the existence of a hierarchy of so called minimum cut configurations for which cutting bonds form the lowest level and whose transport properties scale all in the same way we point out the relevance of our findings to other scalar transport problems i e conductivity

exponential stability for infinite dimensional non autonomous port hamiltonian systems abstract we study the non autonomous version of an infinite dimensional linear port hamiltonian system on an interval a b employing abstract results on evolution families we show c 1 well posedness of the corresponding cauchy problem and thereby existence and uniqueness of classical solutions for sufficiently regular initial data further we demonstrate that a dissipation condition in the style of the dissipation condition sufficient for uniform exponential stability in the autonomous case also leads to a uniform exponential decay rate of the energy in this non autonomous setting

stabilization of infinite dimensional port hamiltonian systems by nonlinear dynamic boundary control the conditions for existence of solutions and stability asymptotic and exponential of a large class of boundary controlled systems on a 1d spatial domain subject to nonlinear dynamic boundary actuation are given the consideration of such class of control systems is motivated by the use of actuators and sensors with nonlinear behavior in many engineering applications these nonlinearities are usually associated to large deformations or the use of smart materials such as piezo actuators and memory shape alloys including them in the controller model results in passive dynamic controllers with nonlinear potential energy function and or nonlinear damping forces first it is shown that under very natural assumptions the solutions of the partial differential equation with the nonlinear dynamic boundary conditions exist globally secondly when energy dissipation is present in the controller then it globally asymptotically stabilizes the partial differential equation finally it is shown that assuming some additional conditions on the interconnection and on the passivity properties of the controller consistent with physical applications global exponential stability of the closed loop system is achieved

exponential stabilization of boundary controlled port hamiltonian systems with dynamic feedback it is shown that a strictly input passive linear finite dimensional controller exponentially stabilizes a large class of partial differential equations actuated at the boundary of a one dimensional spatial domain this follows since the controller imposes exponential dissipation of the total energy the result can by use for control synthesis and for the stability analysis of complex systems modeled by sets of coupled pde s and ode s the result is specialized to port hamiltonian control systems and a simplified dna manipulation process is used to illustrate the result

dissipative extensions and port hamiltonian operators on networks abstract in this article we study port hamiltonian partial differential equations on certain one dimensional manifolds we classify those boundary conditions that give rise to contraction semigroups as an application we study port hamiltonian operators on networks whose edges can have finite or infinite length in particular we discuss possibly infinite networks in which the edge lengths can accumulate zero and port hamiltonian operators with hamiltonians that neither are bounded nor bounded away from zero we achieve this by first providing a new description for maximal dissipative extensions of skew symmetric operators the main technical tool used for this is the notion of boundary systems the latter generalizes the classical notion of boundary triple t s and allows to treat skew symmetric operators with unequal deficiency indices in order to deal with fairly general variable coefficients we develop a theory of possibly unbounded non negative injective weights on an abstract hilbert space

a generalization of the scattering transformation for conic systems a generalized version of the scattering transformation is introduced which allows for rendering of the input output characteristics of a nonlinear conic system into an arbitrary prescribed conic sector the transformation is consequently applied to the problem of stabilization of conic systems interconnections the stability is achieved by rendering the input output characteristics of the subsystems into complementary cones such that the graph separation stability condition is satisfied in the presence of communication delays the scattering based technique for stabilization of feedback interconnections is generalized to the case of arbitrary conic subsystems

new universality class for the permeability problem in a percolation cluster abstract the maximum flow that a percolation cluster can sustain with a gaussian distribution of individual bond capacities is calculated we show the existence of a new universality class if the probability distribution of bond capacities has a finite value in the limit of zero capacity