a time dependent theory of quantum resonances in this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues we apply this theory to many body schrouml dinger operators and to the problem of quasiclassical tunneling

well posedness of feedback systems insights into exact robustness analysis and approximate computations this paper establishes a framework for robust stability analysis of linear time invariant uncertain systems the uncertainty is assumed to belong to an arbitrary subset of complex matrices the concept used here is well posedness of feedback systems leading to necessary and sufficient conditions for robust stability based on this concept some insights into exact robust stability conditions are given in particular frequency domain and state space conditions for well posedness are provided in terms of hermitian form inequalities it is shown that these inequalities can be interpreted as small gain conditions with a generalized class of scalings given by linear fractional transformations lft using the lft scaled small gain condition in the state space setting the duality is established between the h sub spl infin norm condition with frequency dependent scalings and the parameter dependent lyapunov condition connections to the existing results including the structured singular value and the integral quadratic constraints are also discussed finally we show that our well posedness conditions can be used to give a less conservative yet computable bound on the real structured singular value this result is illustrated by numerical examples

a new methodology for observer design and implementation observers are usually formulated as explicit systems of differential equations and implemented using standard ode solvers in this paper we show that there can be advantages in formulating the observer as a differential algebraic equation dae we propose a canonical dae observer and show that it can be implemented using standard dae solvers the dae implementation of the observer can be considered regardless of the design philosophy used we present a simple design strategy for this canonical dae observer based on the extended linearization method

dark soliton generation for the intermediate nonlinear schrodinger equation the generation of dark solitons due to a small change of the initial data is studied within the framework of the intermediate nonlinear schrodinger inls equation in particular we analyze the spectral problem associated with the inls equation when the potential consists of a small perturbation imposed on a constant background we derive a criterion for the perturbation to generate a pair of new discrete eigenvalues as well as their explicit expressions in terms of the perturbation in addition we demonstrate that the eigenvalues appear without a threshold on the magnitude of the perturbation we also consider both the shallow and deep water limits of various results obtained for the inls spectral problem in the former case the limiting procedure can be performed smoothly whereas in the latter case the spectral equation exhibits a new feature of bound states showing that the eigenvalue is exponentially small compared with the magnitude of the perturbation

robustness in the presence of mixed parametric uncertainty and unmodeled dynamics continuing the development of the structured singular value approach to robust control design the authors investigate the problem of computing mu in the case of mixed real parametric and complex uncertainty the problem is shown to be equivalent to a smooth constrained finite dimensional optimization problem in view of the fact that the functional to be maximized may have several local extrema an upper bound on mu whose computation is numerically tractable is established this leads to a sufficient condition of robust stability and performance a historical perspective on the development of the mu theory is included

generalized mixed μ bounds for real and complex multiple block uncertainty with internal matrix structure new absolute stability results that unify and extend existing structured singular value bounds for mixed uncertainty are developed using frequency domain arguments these bounds generalize prior upper bounds for mixed by permitting the treatment of non diagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty several specializations to well known absolute stability criteria from the classical literature are given which allow for a systematic comparison of these results in addressing the problem of robust stability for real parameter uncertainty

duality in multiplier based robustness analysis frequency domain conditions involving multipliers are useful for robustness analysis the resulting analysis problem is generally convex but infinite dimensional and numerical solutions restricted to finite dimensional subspaces need to be considered the resulting finite dimensional problem can be transformed into a linear matrix inequality which can be solved using efficient algorithms the paper presents a format for the dual of the infinite dimensional problem the dual problem can be used to estimate the conservatism of particular finite dimensional subspaces

convergence of petviashvili s iteration method for numerical approximation of stationary solutions of nonlinear wave equations we analyze a heuristic numerical method suggested by v i petviashvili in 1976 for approximation of stationary solutions of nonlinear wave equations the method is used to construct numerically the solitary wave solutions such as solitons lumps and vortices in a space of one and higher dimensions assuming that the stationary solution exists we find conditions when the iteration method converges to the stationary solution and when the rate of convergence is the fastest the theory is illustrated with examples of physical interest such as generalized korteweg de vries benjamin ono zakharov kuznetsov kadomtsev petviashvili and klein gordon equations

discrete time extensions of mixed μ bounds to monotonic and odd monotonic nonlinearities using the parameter dependent lyapunov function framework of haddad and bernstein this paper extends and connects some of the classical absolute stability results on discrete time systems with monotonic and odd monotonic feedback nonlinearities to modern robust stability and performance problems for constant real parameter uncertainty as in the continuous time case it is shown that an immediate application of this framework provides a generalization and reformulation of discrete time mixed theory in terms of lyapunov functions and riccati equations while blending h 2 robust performance within the framework

eigenfunctions and eigenvalues for a scalar riemann hilbert problem associated to inverse scattering a complete set of eigenfunctions is introduced within the riemann hilbert formalism for spectral problems associated to some solvable nonlinear evolution equations in particular we consider the time independent and time dependent schrodinger problems which are related to the kdv and kpi equations possessing solitons and lumps respectively non standard scalar products orthogonality and completeness relations are derived for these problems the complete set of eigenfunctions is used for perturbation theory and bifurcation analysis of eigenvalues supported by the potentials under perturbations we classify two different types of bifurcations of new eigenvalues and analyze their characteristic features one type corresponds to thresholdless generation of solitons in the kdv equation while the other predicts a threshold for generation of lumps in the kpi equation

m form numerical range and the computation of the structured singular value an equivalent expressions for the structures singular value is proposed leading to an alternative algorithm for its computation the approach is based to the geometric properties of the m form numerical range of certain tuples of matrices similar to previously considered schemes the algorithm proposed is proved to give the correct value for block structures of size up to three for larger sizes insight is gained into the question of the possible gap between the structured singular value and its known upper bound

control of lpv systems with partly measured parameters the control of linear parameter varying lpv systems with time varying real uncertain parameters where only some of the parameters are measured and available for feedback is considered the control objectives are internal stability and disturbance attenuation in the sense of a bounded induced l sub 2 norm from the disturbance to the controlled output using parametric lyapunov functions the solvability conditions for dynamic output feedback controllers that depend on the measured parameters are expressed in terms of a set of linear matrix inequalities lmis and an additional coupling constraint which destroys the convexity of the overall problem by transforming the coupling constraint into a rank condition the problem is recast into a rank minimization problem with lmi constraints an example is included that demonstrates the application of the results

direct and inverse scattering problems of the nonlinear intermediate long wave equation the inverse scattering transformation method associated with a nonlinear singular integrodifferential equation is discussed the equation describes long internal gravity waves in a stratified fluid of finite depth and reduces to the korteweg de vries equation as shallow water limit and the benjamin ono equation as deep water limit both limits of the method and novel aspects of the theory are also discussed

perturbations of embedded eigenvalues for the planar bilaplacian operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum in this paper we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite dimensional manifold with infinite codimension in an appropriate space of potentials

brief paper robust finite word length controller design a novel finite word length fwl controller design is proposed in the framework of a mixed m theory a robust fwl controller performance measure is first developed which takes into account the standard robust control requirements as well as the fwl implementation considerations and the corresponding controller design problem is naturally reformulated as a mixed m problem which can be treated effectively with the results of the mixed m theory

fast computation of the multivariable stability margin for real interrelated uncertain parameters an alternative implementation of an algorithm for the computation of the multivariable stability margin to check the robust stability of feedback systems with real parametric uncertainty is proposed this method eliminates the need for the frequency search by reducing it to the testing of a finite number of conditions these conditions have a special structure that allows a significant improvement in the speed of computation

characterization and efficient computation of the structured singular value the concept of structured singular value was recently introduced by doyle 1 as a tool for the analysis and synthesis of feedback systems with structured uncertainties it is a key to the design of control systems under joint robustness and performance specifications and it very nicely complements the h infty approach to control system design in this paper it is shown that the structured singular value can be obtained as the solution of several smooth optimization problems properties of these optimization problems are exhibited leading to a fast algorithm that always yields the structured singular value for block structures of size no larger than 3 and often does for block structures of larger size

morse index and bifurcation for figure eight choreographies of the equal mass three body problem we report on the morse index and periodic solutions bifurcating from the figure eight choreography for the equal mass three body problem under homogeneous potential 1 r a for and under lennard jones lj type potential where r is a distance between bodies it is shown that the morse index changes at a bifurcation point and all solutions bifurcating are approximated by variational functions responsible for the change of the morse index inversely we observed bifurcation occurs at every point where the morse index changes for the figure eight choreography under 1 r a and for solution under the lj type potential where solution is a figure eight choreography tending to that under 1 r 6 for an infinitely large period thus to our numerical studies change of the morse index is not only necessary but also a sufficient condition for bifurcation for these choreographies further we observed that the change of the morse index is equal to the number of bifurcated solutions regarding solutions with congruent orbits as the same solution

rank preservation of matrices with structured uncertainties and its applications in robust control theory matrix rank is determined by the nonsingularity of its submatrices for matrices in which entries are quadratic functions of some uncertain parameters this paper derives sufficient conditions on parameters to that ensure the matrices preserve to some degrees the ranks they have when the parameters are all zero the rank preservation problem is converted to the nonsingularity analysis problem of the minors of the matrix in discussion and suitable tools such as the spl mu analysis method are used to solve the problem applications in robust control theory including tests for robust controllability observability minimum phaseness coprimeness and schur stability are given together with illustrative examples

continuation of periodic orbits in conservative and hamiltonian systems we introduce and justify a computational scheme for the continuation of periodic orbits in systems with one or more first integrals and in particular in hamiltonian systems having several independent symmetries our method is based on a generalization of the concept of a normal periodic orbit as introduced by sepulchre and mackay nonlinearity 10 1997 679 we illustrate the continuation method on some integrable hamiltonian systems with two degrees of freedom and briefly discuss some further applications

nondifferentiable optimization algorithm for designing control systems having singular value inequalities it has been known for some time that many control system design requirements can be expressed as differentiable inequalities more recently it has been shown that important structural properties such as robustness and low noise sensitivity can be expressed as nondifferentiable inequalities involving the singular values of a system or return difference transfer function matrix this paper presents an optimization algorithm which permits all these constraints to be considered

interaction of lumps with a line soliton for the dsii equation abstract exact solutions of the focussing davey stewartson ii equation are presented which describe the interaction of n lumps with a line soliton these solutions are constructed by analysing the inverse spectral problem of the associated lax pair the dynamical properties of these solutions are also discussed

on the discrete spectrum of the nonstationary schrodinger equation and multipole lumps of the kadomtsev petviashvili i equation the discrete spectrum of the nonstationary schrodinger equation and localized solutions of the kadomtsev petviashvili i kpi equation are studied via the inverse scattering transform it is shown that there exist infinitely many real and rationally decaying potentials which correspond to a discrete spectrum whose related eigenfunctions have multiple poles in the spectral parameter an index or winding number is asssociated with each of these solutions the resulting localized solutions of kpi behave as collection of individual humps with nonuniform dynamics

an implicit small gain condition and an upper bound for the real structured singular value in this paper we develop an upper bound for the real structured singular value that has the form of an implicit small gain theorem the implicit small gain condition involves a shifted plant whose dynamics depend upon the uncertainty set bound and unlike prior bounds does not require frequency dependent scales or multipliers numerical results show that the implicit small gain bound compares favorably with real bounds that involve frequency dependent scales and multipliers

a normal form for nonlinear resonance of embedded solitons a normal form for nonlinear resonance of embedded solitons is derived for a coupled two wave system that generalizes the second harmonic generating model this wave system is non hamiltonian in general an embedded soliton is a localized mode of the nonlinear system that coexists with the linear wave spectrum it occurs as a result of a codimension one bifurcation of non local wave solutions nonlinearity couples the embedded soliton and the linear wave spectrum and induces a one sided radiation driven decay of embedded solitons the normal form shows that the embedded soliton is semi stable i e it survives under perturbations of one sign but is destroyed by perturbations of the opposite sign when a perturbed embedded soliton sheds continuous wave radiation the radiation amplitude is generally not minimal even if the wave system is hamiltonian the results of the analytical theory are confirmed by numerical computations

controller design with real parametric uncertainty a number of techniques have been developed in recent years for the analysis and design of controllers which are robust with respect to structured complex uncertainty in particular the complex synthesis procedure has been successfully applied to a number of engineering problems however the presence of real parametric uncertainty in the problem description substantially complicates matters so that standard complex synthesis techniques are no longer adequate in this paper we develop a procedure to tackle the mixed real and complex synthesis problem this procedure involves a d g k iteration between computing the mixed upper bound and solving an h optimal control problem and has guaranteed convergence to a local minimum of the nonconvex problem the procedure has been implemented in software and several controller designs are compared with the corresponding complex synthesis designs

efficient computation of a guaranteed lower bound on the robust stability margin for a class of uncertain systems sufficient conditions for the robust stability of a class of uncertain systems with several different assumptions on the structure and nature of the uncertainties can be derived in a unified manner in the framework of integral quadratic constraints these sufficient conditions in turn can be used to derive lower bounds on the robust stability margin for such systems the lower bound is typically computed with a bisection scheme with each iteration requiring the solution of a linear matrix inequality feasibility problem we show how this bisection can be avoided altogether by reformulating the lower bound computation problem as a single generalized eigenvalue minimization problem which can be solved very efficiently using standard algorithms we illustrate this with several important commonly encountered special cases diagonal nonlinear uncertainties diagonal memoryless time invariant sector bounded popov uncertainties structured dynamic uncertainties and structured parametric uncertainties we also present a numerical example that demonstrates the computational savings that can be obtained with our approach

robustness analysis with correlated highly structured complex uncertainties it is shown that the results reported by the author ibid vol 35 no 3 pp 350 352 mar 1990 on robustness analysis with highly structured independent perturbations is directly extendable to the case of correlated perturbations a direct consequence of this extension is that the phase sensitive structured singular value and the multiloop phase margin can be computed exactly

time evolution of gaussian type initial conditions associated with the davey stewartson equations the cauchy problem of the davey stewartson equations with non trivial boundaries is studied initial conditions for the equations are chosen to have gaussian type envelope shapes and the time evolution is investigated both theoretically and numerically it is found that an initial packet grows and oscillates radiating ripples then a localized structure called a dromion appears asymptotically it is also observed that the ripples run away mainly along the mean flows the results of numerical simulations and the analysis by the inverse scattering transform show good agreement with each other

stability and bifurcations of the figure 8 solution of the three body problem the stability properties of a recently discovered solution of the general three body problem with equal masses and the shape of a figure 8 are analyzed as the masses are varied it is shown by numerical continuation and the evaluation of the characteristic multipliers that the solution is stable only in a narrow mass interval other less symmetrical and unstable solutions with equal masses in the same homotopy class as the figure 8 orbit have been found the branching behavior is also analyzed

bifurcations of new eigenvalues for the benjamin ono equation a criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the benjamin ono bo equation this bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the bo equation the asymptotic approximation of an exponentially small new eigenvalue is derived the method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions explicit expressions are obtained for the soliton potentials

representation of uncertain time delays in the h framework abstract how to represent an uncertain time delay in a form suitable for robust control in terms of the h norm and the structured singular value is discussed to use synthesis the uncertain delay has to be approximated to yield a proper rational interconnection matrix m s and a purely complex perturbation matrix s the parametric average of the delay is usually included in the nominal model while the uncertain perturbation covers variation around the average delay it is proposed to model the nominal time delay as uncertainty i e using a larger uncertainty set this yields a delay free nominal model which simplifies the controller synthesis for the cases studied the delay free nominal model does not yield a more conservative design than the average delay model despite its larger uncertainty set

integral quadratic constraint approach vs multiplier approach integral quadratic constraints iqc arise in many optimal and or robust control problems the iqc approach can be viewed as a generalization of the classical multiplier approach in the absolute stability theory in this paper we study the relationship between the two approaches for robust stability analysis using a slightly modified multiplier approach we show that the existence of an iqc is equivalent to the existence of a multiplier in most known cases it is hoped that this result provides some new insight into both approaches and makes them more useful in robust control applications

an input output approach to structured stochastic uncertainty we consider linear time invariant systems with exogenous stochastic disturbances and in feedback with structured stochastic uncertainties this setting encompasses linear systems with both additive and multiplicative noise our concern is to characterize second order properties such as mean square stability mss and performance a purely input output treatment of these systems is given without recourse to state space models and thus the results are applicable to certain classes of distributed systems we derive necessary and sufficient conditions for mss in terms of the spectral radius of a linear matrix operator whose dimension is that of the number of uncertainties rather than the dimension of any underlying state space models our condition is applicable to the case of correlated uncertainties and reproduces earlier results for uncorrelated uncertainties for cases where state space realizations are given linear matrix inequality equivalents of the input output conditions are given

on the norms used in computing the structured singular value different norms are considered to replace the euclidean norm in an algorithm given by m h k fan and a l tits ibid vol 33 p 284 9 1988 which is used for the computation of the structured singular value of any matrix the algorithm is explained and it is shown that the l sub 1 norm is the best possible norm in a certain sense

structure of model uncertainty for a weakly corrupted plant in this paper we investigate the structure of the transfer function set which includes all the transfer functions deduced from the plant available information it is shown that when an upper bound of the plant transfer function s h sup spl infin norm has been supplied and the noise contaminating the time domain identification experiment data is not too significant such a transfer function set can be parameterized by a linear fractional transformation of two transfer function matrices one of them is a fixed transfer function matrix which is completely determined by the plant available information and the noise magnitude the other is a norm bounded structure fixed free transfer function matrix moreover it is shown that the problem of analytically obtaining the fixed complexity nominal model that best approximates this transfer function set is as difficult as the spl mu synthesis problem

exact calculation of the multiloop stability margin a mapping theorem by l a zadeh and c a desoer 1963 serves as the basis for an algorithm that computes the stability margin k sub m of diagonally perturbed multivariable feedback systems without conservatism the stability margin determination not only verifies system stability but also quantifies how much further the plan uncertainties can be extended before instability occurs the essence of the computational approach is the realization that the true image of a given domain can be approximated with arbitrary accuracy by first subdividing the domain into the requisite number of subdomains and then forming the union of the convex hulls of their images the technique is illustrated in an example with a plant model having three uncertainties in general the method is applicable to either siso or mimo lti systems whose uncertainties can be modeled with noninteracting coefficients in the open loop transfer function representations

a convex parameterization of robustly stabilizing controllers this paper treats synthesis of robust controllers for linear time invariant systems uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial the main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations with the new parameterization certain robust performance problems can be stated in terms of quasi convex optimization

exchange orbits in the planar 1 4 body problem abstract we study some doubly symmetric orbits in the planar 1 2 n body problem that is the mass of the central body is significantly bigger than the other 2 n equal masses the necessary and sufficient conditions for periodicity of the orbits are discussed we also study numerically these kinds of orbits for the case n 2 the system under study corresponds to one conformed by a planet and four satellites of equal mass we determine a 1 parameter family of time reversible invariant tori related with the reversing symmetries of the equations of motion the initial conditions of the orbits were determined by means of solving a boundary value problem with one free parameter the numerical solution of the boundary value problem was obtained using the software auto for the numerical analysis we have used the value of 3 5 1 0 4 as mass ratio of some satellite and the planet in the computed solutions the satellites are in mean motion resonance 1 1 and they librate around a relative equilibria that is a solution where the distances between the bodies remain constant for all time

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

gauge transformation and spectral decomposition for the ishimori ii equations we study the gauge transformation between spectral problems and their adjoints for the ishimori ii is ii and davey stewartson ii ds ii equations the commutativity between the gauge and adjoint transformations is proved the commutativity is used for spectral decomposition and surface representation theorems for the ishimori ii equations

a µ based optimal finite word length controller design a novel optimal finite word length fwl controller design is proposed in the framework of m theory a computationally tractable close loop stability measure with fwl implementation considerations of the controller is derived based on the m theory and the optimal fwl controller realizations are obtained by solving the resulting optimal fwl realization problem using linear matrix inequality techniques

computational complexity of real structured singular value in l sub p setting this paper studies a generalized real structured singular value spl mu problem where uncertain parameters are bounded by an l sub p norm two results are presented the first one shows that this generalized spl mu problem is np hard for any given rational number p spl isin 1 spl infin the np hardness holds as long as le the size of the largest repeated block exceeds one this result generalizes the known np hardness result for the conventional spl mu problem with p spl infin the second result which strengthens the first one considers the approximability problem of the generalized spl mu we show that the problem of obtaining an estimate for the generalized spl mu with some guaranteed bounds on the relative error remains to be np hard regardless how large this bound is

real m bounds based on fixed shapes in the nyquist plane parabolas hyperbolas cissoids nephroids and octomorphs in this paper we introduce new bounds for the real structured singular value the approach is based on absolute stability criteria with plant dependent multipliers that exclude the nyquist plot from fixed plane curve shapes containing the critical point jo unlike half plane and circle based bounds the critical feature of the fixed curve bounds is their ability to differentiate between the real and imaginary components of the uncertainty since the plant dependent multipliers have the same functional form at all frequencies the resulting graphical interpretation of the absolute stability criteria are frequency independent in contrast to the frequency dependent off axis circles that arise in standard real bounds