on vanishing stability regions in nonlinear systems with high gain feedback when local stabilization of nonlinear systems is achieved by linear feedback the resulting stability region may vanish as the feedback gains increase this is demonstrated by examples in which neglected nonlinearities create an unstable limit cycle around an asymptotically stable equilibrium as the feedback gains tend to infinity the unstable limit cycle shrinks to the equilibrium if a feedback linearization design is applied the same instability mechanism may occur when the nonlinearities are not precisely known

on posets and hopf algebras abstract we generalize the notion of the rank generating function of a graded poset namely by enumerating different chains in a poset we can assign a quasi symmetric function to the poset this map is a hopf algebra homomorphism between the reduced incidence hopf algebra of posets and the hopf algebra of quasi symmetric functions this work implies that the zeta polynomial of a poset may be viewed in terms hopf algebras in the last sections of the paper we generalize the reduced incidence hopf algebra of posets to the hopf algebra of hierarchical simplicial complexes

generalized bremmer series with rational approximation for the scattering of waves in inhomogeneous media the bremmer series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators in this paper sparse matrix representations of these pseudodifferential operators are derived the authors focus on designing sparse matrices keeping the accuracy high at the cost of ignoring any critical scattering angle phenomena such matrix representations follow from rational approximations of the vertical slowness and the transverse laplace operator symbols and of the vertical derivative as they appear in the parabolic equation method sparse matrix representations lead to a fast algorithm an optimization procedure is followed to minimize the errors in the high frequency limit for a given discretization rate the bremmer series solver consists of three steps directional decomposition into up and downgoing waves one way propagation and interaction of the counterpropagating constituents each of these steps is represented by a sparse matrix equation the resulting algorithm provides an improvement of the parabolic equation method in particular for transient wave phenomena and extends the latter method systematically for backscattered waves

wave front layer stripping approach to inverse scattering for the wave equation the inverse problem involving a point pulse source exterior to a scattering medium where the velocity c x is continuous is considered the layer stripping approach is applied to thin curvilinear layers whose surfaces are the primary wave fronts with c x continuous the reflected wave fronts will be secondary i e of lower order singularity it is shown that the layer stripping approach can be used in the time domain inverse problem without employing the added complexity of having to perform wave splitting furthermore the procedure to determine the normal derivative of c from the asymptotic short time behavior of the field quantities on an adjacent wave front surface has been simplified compared to earlier wave splitting methods uniqueness results are given

peak quasisymmetric functions and eulerian enumeration abstract via duality of hopf algebras there is a direct association between peak quasisymmetric functions and enumeration of chains in eulerian posets we study this association explicitly showing that the notion of cd index long studied in the context of convex polytopes and eulerian posets arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by stembridge thus eulerian posets having a nonnegative cd index for example face lattices of convex polytopes correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis we diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions and study the g polynomial for eulerian posets as an algebra homomorphism

exact constructions of square root helmholtz operator symbols the focusing quadratic profile operator symbols play a pivotal role in both the exact well posed one way reformulation of solving the elliptic helmholtz equation and the construction of the generalized bremmer coupling series the inverse square root and square root helmholtz operator symbols are the initial quantities of interest in both formulations in addition to providing the theoretical framework for the development and implementation of the parabolic equation pe method in wave propagation modeling exact standard left and weyl symbol constructions are presented for both the inverse square root and square root helmholtz operators in the case of the focusing quadratic profile in one transverse spatial dimension extending and ultimately unifying the previously published corresponding results for the defocusing quadratic case j math phys 33 1887 1914 1992 both i spectral modal summation representations and ii contour integral representations exploiting the underlying periodicity of the associated quantum mechanical harmonic oscillator problem are derived and ultimately related through the propagating and nonpropagating contributions to the operator symbol high and low frequency asymptotic operator symbol expansions are given along with the exact symbol representations for the corresponding operator rational approximations which provide the basis for the practical computational realization of the pe method moreover while the focusing quadratic profile is in some respects nonphysical the corresponding helmholtz operator symbols nevertheless establish canonical symbol features for more general profiles containing locally quadratic wells

a unifying framework for the semiglobal stabilization of nonlinear uncertain systems via measurement feedback we study the problem of semiglobally stabilizing an uncertain nonlinear system consisting of a linear nominal system perturbed by either nonlinearities or model uncertainties our approach relies on well known h sub spl infin linear control tools and allows one to recover and improve in the unifying framework of a semiglobal separation result existing results on the semiglobal stabilization via output feedback in particular we discuss the case of uncorrupted outputs input and output nonlinearities or model uncertainties which may include for example practical situations such as backlash hysteresis and saturations the key feature of our design procedure is given by the choice of two continuous functions the first one is instrumental in constructing a stabilizing controller the second one arises in the candidate lyapunov function for the closed loop system relying on our main theorem we give general tools for achieving large regions of attraction via bounded measurement feedback for a wide class of nonlinear uncertain interconnected systems

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

stochastic nonlinear stabilization i a backstepping design while the current robust nonlinear control toolbox includes a number of methods for systems affine in deterministic bounded disturbances the problem when the disturbance is unbounded stochastic noise has hardly been considered we present a control design which achieves global asymptotic lyapunov stability in probability for a class of strict feedback nonlinear continuous time systems driven by white noise in a companion paper we develop inverse optimal control laws for general stochastic systems affine in the noise input and for strict feedback systems a reader of this paper needs no prior familiarity with techniques of stochastic control

lyapunov like techniques for stochastic stability the purpose of this paper is to study the stabilizability problem for control stochastic nonlinear systems driven by a wiener process sufficient conditions for the existence of stabilizing feedback laws that are smooth except possibly at the equilibrium point of the system are provided by means of stochastic lyapunov like techniques the notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced and the stabilization by means of dynamic controllers is studied

shellable nonpure complexes and posets ii this is a direct continuation of shellable nonpure complexes and posets i which appeared in transactions of the american mathematical society 348 1996 1299 1327 8 interval generated lattices and dominance order in this section and the following one we will continue exemplifying the applicability of lexicographic shellability to nonpure posets let f i1 i2 in be a family of intervals of integers by which is meant sets of the form a b a a 1 b a b we assume that there are no containments among these intervals and that they are ordered so that their left and right endpoints are increasing let l f be the lattice of all sets that are unions of subfamilies of f ordered by inclusion such interval generated lattices l f were introduced and studied by greene g define an edge labeling of l f as follows if a b is a covering and a max b a then a b a if a 1 a and a is the left endpoint of some i f

uniform asymptotic expansion of the square root helmholtz operator and the one way wave propagator the bremmer coupling series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators such operators appear in the diagonalization process of the acoustic system s matrix of partial differential operators upon extracting a principal direction of one way propagation in this paper in three dimensions uniform asymptotic expansions of the schwartz kernels of these operators are derived also we derive a uniform asymptotic expansion of the one way propagator appearing in the series we focus on designing closed form representations valid in the high frequency limit taking into account critical scattering angle phenomena the latter phenomena are not dealt with in the standard calculus of pseudodifferential operators our expansion is not limited by propagation angle in principle the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away and near its diagonal

stochastic nonlinear stabilization ii inverse optimality after considering the stabilization of a specific class of stochastic nonlinear systems in a companion paper in this second part we address the classical question of when is a stabilizing in probability controller optimal and show that for every system with a stochastic control lyapunov function it is possible to construct a controller which is optimal with respect to a meaningful cost functional then we return to the problem from part i and design an optimal backstepping controller whose cost functional includes penalty on control effort and which has an infinite gain margin

generalization of the bremmer coupling series an operator formalism is developed to expand the acoustic wave field in a multi dimensionally smoothly varying medium generated by a source localized in space and time into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times this expansion is a generalization of the bremmer coupling series the condition of smoothness of the medium relates to the width of the signature of the source in the configuration both the existence and the convergence in the weak sense of the expansion are discussed the operator calculus involved leads to a natural generalization of the concept of slowness surface to multi dimensionally smoothly varying media the operator associated with the corresponding generalized vertical slowness induces the full one way wave operator in the type of media under consideration in addition a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents are derived

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

stabilization in probability of nonlinear stochastic systems with guaranteed cost we deal with nonlinear dynamical systems consisting of a linear nominal part plus model uncertainties nonlinearities and both additive and multiplicative random noise modeled as a wiener process in particular we study the problem of finding suitable measurement feedback control laws such that the resulting closed loop system is stable in some probabilistic sense and a given cost functional is minimized we give a lyapunov based separation result which splits the control design into a state feedback problem and a filtering problem finally we point out constructive algorithms for solving the state feedback and filtering problems with arbitrarily large region of attraction for a wide class of nonlinear systems which at least include feedback linearizable systems

stochastic h infty we consider stochastic linear plants which are controlled by dynamic output feedback and subjected to both deterministic and stochastic perturbations our objective is to develop an h infty type theory for such systems we prove a bounded real lemma for stochastic systems with deterministic and stochastic perturbations this enables us to obtain necessary and sufficient conditions for the existence of a stabilizing compensator which keeps the effect of the perturbations on the to be controlled output below a given threshhold gamma 0 in the deterministic case the analogous conditions involve two uncoupled linear matrix inequalities but in the stochastic setting we obtain coupled nonlinear matrix inequalities instead the connection between h infty theory and stability radii is discussed and leads to a lower bound for the radii which is shown to be tight in some special cases

3 d depth migration via mcclellan transformations three dimensional seismic wavefields may be extrapolated in depth one frequency at a time by two dimensional convolution with a circularly symmetric frequency and velocity dependent filter this depth extrapolation performed for each frequency independently lies at the heart of 3 d finite difference depth migration the computational efficiency of 3 d depth migration depends directly on the efficiency of this depth extrapolation mcclellan transformations provide an efficient method for both designing and implementing two dimensional digital filters that have a particular form of symmetry such as the circularly symmetric depth extrapolation filters used in 3 d depth migration given the coefficients of one dimensional frequency and velocity dependent filters used to accomplish 2 d depth migration mcclellan transformations lead to a simple and efficient algorithm for 3 d depth migration 3 d depth migration via mcclellan transformations is simple because the coefficients of two dimensional depth

on the shellability of the order complex of the subgroup lattice of a finite group we show that the order complex of the subgroup lattice of a finite group g is nonpure shellable if and only if g is solvable a by product of the proof that nonsolvable groups do not have shellable subgroup lattices is the determination of the homotopy types of the order complexes of the subgroup lattices of many minimal simple groups

the bremmer series for a multi dimensional acoustic scattering problem the bremmer series is used to reduce a complex scattering problem to a sequence of simpler single scattering problems in the bremmer series the wave equation is first decomposed into a coupled system of one way wave equations the system is then decoupled into a sequence of one way wave equations with a fixed point iteration in this paper a left symbol representation of the decomposition operator and the vertical propagation operator are used time domain convergence of the bremmer series is shown for a set of dispersive medium models the non dispersive case is treated with an approximation procedure

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

backstepping controller design for nonlinear stochastic systems under a risk sensitive cost criterion this paper develops a methodology for recursive construction of optimal and near optimal controllers for strict feedback stochastic nonlinear systems under a risk sensitive cost function criterion the design procedure follows the integrator backstepping methodology and the controllers obtained guarantee any desired achievable level of long term average cost for a given risk sensitivity parameter theta furthermore they lead to closed loop system trajectories that are bounded in probability and in some cases asymptotically stable in the large these results also generalize to nonlinear systems with strongly stabilizable zero dynamics a numerical example included in the paper illustrates the analytical results

robust output feedback control of nonlinear stochastic systems using neural networks we present an adaptive output feedback controller for a class of uncertain stochastic nonlinear systems the plant dynamics is represented as a nominal linear system plus nonlinearities in turn these nonlinearities are decomposed into a part obtained as the best approximation given by neural networks plus a remaining part which is treated as uncertainties modeling approximation errors and neglected dynamics the weights of the neural network are tuned adaptively by a lyapunov design the proposed controller is obtained through robust optimal design and combines together parameter projection control saturation and high gain observers high performances are obtained in terms of large errors tolerance as shown through simulations

global stabilization by output feedback examples and counterexamples abstract we show by means of examples that global complete observability and global stabilizability by state feedback are not sufficient to guarantee global stabilizability by dynamic output feedback we show that a main obstruction is related to unboundedness unobservability this is that some unmeasured state components may escape in finite time whereas the measurements remain bounded

uniform asymptotic expansion of the generalized bremmer series the bremmer coupling series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators in this paper in two dimensions uniform asymptotic expansions of the schwartz kernels of these operators are derived also we derive a uniform asymptotic expansion of the one way propagator appearing in the series we focus on designing closed form representations valid in the high frequency limit taking into account critical scattering angle phenomena our expansion is not limited by propagation angle in principle the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away from and near its diagonal the bremmer series solver consists of three steps directional decomposition into up and downgoing waves one way propagation and interaction of the counter propagating constituents each of these steps is represented here by a kernel for which a uniform asymptotic expansion is found the associated algorithm provid

further remarks on potentially global stabilizability given a control system with a nonlinear drift term it is proved that a linear like criterion for almost global stabilizability can be stated at least for control systems of the form x f x bu where f is a polynomial but not necessarily homogeneous vector field these stability criteria also allow for the control of the size of the region of attraction