an improved stability criterion for discrete time lur e systems with sector and slope restrictions this paper introduces an improved stability criterion for discrete time lur e systems with sector and slope restrictions for the stability criterion a lur e lyapunov functional candidate is constructed to include quadratic terms and integration terms with nonlinearities to handle the resulting integration terms in the lyapunov difference there is proposed a new bound lemma where the integration terms are relaxed into the second order terms by the geometric point of view in addition an appropriate weighting method uses slack variables from the sector and slope restrictions in the overall derivation the stabilization condition is formulated in terms of a parameterized linear matrix inequality plmi which is then converted into the linear matrix inequality an example illustrates that the proposed criterion presents a less conservative result than the previous criteria in the literature

system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

column reduction of polynomial matrices a few years ago beelen developed an algorithm to determine a minimal basis for the kernel of a polynomial matrix in this paper we use a modified version of this algorithm to find a column reduced polynomial matrix unimodularly equivalent to a given polynomial matrix

stability analysis of discrete time lur e systems a class of lyapunov functions is proposed for discrete time linear systems interconnected with a cone bounded nonlinearity using these functions we propose sufficient conditions for the global stability analysis in terms of linear matrix inequalities lmi only taking the bounded sector condition into account unlike frameworks based on the lur e type function the additional assumptions about the derivative or discrete variation of the nonlinearity are not necessary hence a wider range of cone bounded nonlinearities can be covered we also show that there is a link between global stability lmi conditions based on this new lyapunov function and a transfer function of an auxiliary system being strictly positive real in addition the novel function is considered in the local stability analysis problem of discrete time lur e systems subject to a saturating feedback a convex optimization problem based on sufficient lmi conditions is formulated to maximize an estimate of the basin of attraction another specificity of this new lyapunov function is the fact that the estimate is composed of disconnected sets numerical examples reveal the effectiveness of this new lyapunov function in providing a less conservative estimate with respect to the quadratic function

brief paper a revisited tsypkin criterion for discrete time nonlinear lur e systems with monotonic sector restrictions this paper revisits a well known tsypkin criterion for stability analysis of discrete time nonlinear luroe systems when nonlinearities are monotonic and sector restricted by 0 1 where 1 is positive de nite it is shown by kapila and haddad that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k ng z is strictly positive real wherek is nonnegative diagonal and g z represents a transfer function of the linear part of the luroe system with invertible or identically zero g 0 this paper extends this criterion when 1 is positive diagonal by choosing a new lyapunov function to obtain an lmi criterion from a frequency domain interpretation of this lmi criterion another su cient criterion is generated which establishes that the system is absolutely stable if a function g 0 z 1 1 mi 1 z 1 k 1 z k n g z is strictly positive real wherek andk are nonnegative diagonal and orthogonal to each other 1998 elsevier science ltd all rights reserved

deformed quantum calogero moser problems and lie superalgebras the deformed quantum calogero moser sutherland problems related to the root systems of the contragredient lie superalgebras are introduced the construction is based on the notion of the generalized root systems suggested by v serganova for the classical series a recurrent formula for the quantum integrals is found which implies the integrability of these problems the corresponding algebras of the quantum integrals are investigated the explicit formulas for their poincare series for generic values of the deformation parameter are presented

lmi searches for anticausal and noncausal rational zames falb multipliers abstract given a linear time invariant plant the search for a suitable multiplier over the class of zames falb multipliers is a challenging problem which has been studied for several decades recently a new linear matrix inequality search has been proposed over rational and causal zames falb multipliers this letter analyzes the conservatism of the restriction to causality on the multipliers and presents a complementary search for rational and anticausal multipliers the addition of a popov multiplier to the anticausal zames falb multiplier is implemented by analogy with the causal search as a result a search over a noncausal subset of zames falb multipliers is obtained a comparison between all the search methods proposed in the literature is given

relativistic lame functions the special case g 2 we study a class of eigenfunctions of an analytic difference operator generalizing the special lam operator paying particular attention to quantum mechanical aspects we show that in a suitable scaling limit the pertinent eigenfunctions lead to the eigenfunctions of the operator in a finite volume we establish various orthogonality and non orthogonality results by direct calculations generalize the one gap picture associated with the above lam operator and obtain duality properties for the hyperbolic trigonometric and rational specializations

lmi based stability criteria for discrete time lur e systems with monotonic sector and slope restricted nonlinearities this note presents new lmi based stability criteria for the discrete time multivariable lur e system with nonlinearities that are monotonic sector and slope restricted corresponding lur e lyapunov functions are constructed for such a system the new criteria are expressed in a reasonably general form that can be applied to both non diagonal and diagonal nonlinearities we explicitly compare the new approach to the existing lmi based popov like criteria in the literature and express the results in terms of an integral quadratic constraint iqc the applications of the new criteria to some control problems and strategies are briefly discussed numerical examples are included to show their performance and they are shown to be less conservative than the previous results

essential self adjointness of the elliptic ruijsenaars models we study the elliptic ruijsenaars models associated with arbitrary root systems which are difference analogs of the calogero moser model we give a dense subspace in the space of square integrable functions invariant under the action of the weyl group on a torus as a domain of its hamiltonian and prove its essential self adjointness by using perturbation theory it is also clarified that these models consist of pure point spectrum

elliptic algebro geometric solutions of the kdv and akns hierarchies an analytic approach we provide an overview of elliptic algebro geometric solutions of the kdv and akns hierarchies with special emphasis on floquet theoretic and spectral theoretic methods our treatment includes an effective characterization of all stationary elliptic kdv and akns solutions based on a theory developed by hermite and picard

input output and state space representations of finite dimensional linear time invariant systems in this paper we develop various representations for systems described by aset of high order differential equations of the form r0w r1w rsw s 0 with r0 r1 rs not necessarily square matrices the variables w are the external variables particular attention is paid to the problem of obtaining minimal state space realizations and input output or input state output representations of such systems

quantum integrability of the deformed elliptic calogero moser problem the integrability of the deformed quantum elliptic calogero moser problem introduced by chalykh feigin and veselov is proven explicit recursive formulas for the integrals are found for integer values of the parameter this implies the algebraic integrability of the systems

picard potentials and hill s equation on a torus hill s equation has drawn an enormous amount of consideration due to its ubiquity in applications as well as its structural richness of particular importance in the last 20 years is its connection with the kdv hierarchy and hence with integrable systems we show in this paper that regarding the independent variable as a complex variable yields a breakthrough for the problem of an efficient characterization of all elliptic finitegap potentials a major open problem in the field specifically we show that elliptic finitegap potentials of hill s equation are precisely those for which all solutions for all spectral parameters are meromorphic functions in the independent variable complementing a classical theorem of picard the intimate connection between picard s theorem and elliptic finite gap solutions of completely integrable systems is established in this paper for the first time in addition we construct the hyperelliptic riemann surface associated with a finitegap potential not necessarily elliptic i e determine its branch and singular points from a comparison of the geometric and algebraic multiplicities of eigenvalues of certain auxiliary operators associated with hill s equation these multiplicities are intimately correlated with the pole structure of the diagonal green s function of the operator h d2 dx2 q x in l2 r our construction is new in the present general complex valued periodic finite gap case before describing our approach in some detail we shall give a brief account of the history of the problem involved this theme dates back to a 1940 paper of ince 43 who studied what is presently called the lam6 ince potential

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

nonlinear small gain theorems for discrete time feedback systems and applications we derive in this work a local nonlinear small gain theorem in the framework of input to state stability for discrete time systems our primary objective is to show that as in the continuous time context these discrete time nonlinear small gain theorems are very effective in stability analysis and synthesis for various classes of discrete time control systems two converse lyapunov theorems for discrete exponential stability are developed to assist these applications new results in stability and stabilization presented in this paper are significant extensions of previous work by other authors ieee trans automat control 38 1993 1398 39 1994 2340 33 1988 1082

heat flow heat content and the isoparametric property let m be a riemannian manifold and omega a compact domain of m with smooth boundary we study the solution of the heat equation on omega having constant unit initial conditions and dirichlet boundary conditions the purpose of this paper is to study the geometry of domains for which at any fixed value of time the normal derivative of the solution heat flow is a constant function on the boundary we express this fact by saying that such domains have the constant flow property in constant curvature spaces known examples of such domains are given by geodesic balls and more generally by domains whose boundary is connected and isoparametric the question is are they all like that this problem is the analogous for the heat equation of the classical serrin s problem for harmonic domains in this paper we give an affirmative answer to the above question in fact we prove more generally that if a domain in an analytic riemannian manifold has the constant flow property then every component of its boundary is an isoparametric hypersurface for space forms we also relate the order of vanishing of the heat content with fixed boundary data with the constancy of the r mean curvatures of the boundary and with the isoparametric property finally we discuss the constant flow property in relation to other well known overdetermined problems involving the laplace operator like the serrin problem or the schiffer problem

stability of higher order discrete time lur e systems abstract we consider discrete time lur e systems obtained by applying nonlinear feedback to a system of higher order difference equations arma models the arma model relates the inputs and outputs of the linear system and does not involve any internal or state variables a stability theory subsuming results of circle criterion type is developed including criteria for input to output stability a concept which is very much reminiscent of input to state stability

a circle criterion for strong integral input to state stability abstract we present sufficient conditions for integral input to state stability iiss and strong iiss of the zero equilibrium pair of continuous time forced lur e systems where by strong iiss we mean the conjunction of iiss and small signal iss our main results are reminiscent of the complex aizerman conjecture and the well known circle criterion we derive a number of corollaries including a result on stabilisation by static feedback in the presence of input saturation in particular we identify classes of forced lur e systems with saturating nonlinearities which are strongly iiss but not iss

symmetry theorems for the overdetermined eigenvalue problems abstract the well known schiffer conjecture saying that for a smooth bounded domain r n if there exists a positive neumann eigenvalue such that the corresponding neumann eigenfunction u is constant on the boundary of then is a ball in this paper we shall prove that the schiffer conjecture holds if and only if the third order interior normal derivative of the corresponding neumann eigenfunction is constant on the boundary we also prove a similar result to the berenstein conjecture for the overdetermined dirichlet eigenvalue problem

systems with sector bound nonlinearities a behavioral approach the class of nonlinear dynamical systems obtained by interconnection of a linear time invariant dynamical system and a memoryless time invariant sector bound nonlinearity has been widely studied in literature in this paper it is shown that many important results for this class of systems can be elegantly unified using behavioral theory of dynamical systems systems with slope restricted nonlinearities are also considered

a less conservative lmi condition for stability of discrete time systems with slope restricted nonlinearities many conditions have been found for the absolute stability of discrete time lur e systems in the literature it is advantageous to find convex searches via lmis where possible in this technical note we construct two less conservative lmi conditions for discrete time systems with slope restricted nonlinearities the first condition is derived via lyapunov theory while the second is derived via the theory of integral quadratic constraints iqcs and noncausal zames falb multipliers both conditions are related to the jury lee criterion most appropriate for systems with such nonlinearities and the second generalizes it numerical examples demonstrate a significant reduction in conservatism over competing approaches

integrability of difference calogero moser systems a general class of n particle difference calogero moser systems with elliptic potentials is introduced besides the step size and two periods the hamiltonian depends on nine coupling constants we prove the quantum integrability of the model for n 2 and present partial results for n 3 in degenerate cases rational hyperbolic or trigonometric limit the integrability follows for arbitrary particle number from previous work connected with the multivariable q polynomials of koornwinder and macdonald liouville integrability of the corresponding classical systems follows as a corollary limit transitions lead to various well known models such as the nonrelativistic calogero moser systems associated with classical root systems and the relativistic calogero moser system

the perturbation of the quantum calogero moser sutherland system and related results the hamiltonian of the trigonometric calogero sutherland model coincides with a certain limit of the hamiltonian of the elliptic calogero moser model in other words the elliptic hamiltonian is a perturbed operator of the trigonometric one

semi global incremental input to state stability of discrete time lur e systems abstract we present sufficient conditions for semi global incremental input to state stability of a class of forced discrete time lur e systems the results derived are reminiscent of well known absolute stability criteria such as the small gain theorem and the circle criterion we derive a natural sufficient condition which guarantees that asymptotically almost periodic inputs generate asymptotically almost periodic state trajectories as a corollary we obtain sufficient conditions for the converging input converging state property to hold

a multivariable extension of the tsypkin criterion using a lyapunov function approach for analyzing the stability of discrete time systems containing a feedback nonlinearity the tsypkin criterion is the closest analog to the popov criterion which is used for analyzing such systems in continuous time traditionally the proof of this criterion is based upon input output properties and function analytic methods in this paper we extend the tsypkin criterion to multivariable systems containing an arbitrary number of monotonic sector bounded memoryless time invariant nonlinearities along with providing a lyapunov function proof for this classical result

complete integrability of relativistic calogero moser systems and elliptic function identities poincare invariant generalizations of the galilei invariant calogero moser a particle systems are studied a quantization of the classical integrals s1 sn is presented such that the operators si9 sn mutually commute as a corollary it follows that si9 9sn poisson commute these results hinge on functional equations satisfied by the weierstrass and 0 functions a generalized cauchy identity involving the function leads to an n x n matrix l whose symmetric functions are proportional to sl5 9sn

input to state stability of discrete time lur e systems an input to state stability theory which subsumes results of circle criterion type is developed in the context of discrete time lur e systems the approach developed is inspired by the complexified aizerman conjecture

second order counterexamples to the discrete time kalman conjecture the kalman conjecture is known to be true for third order continuous time systems we show that it is false in general for second order discrete time systems by construction of counterexamples with stable periodic solutions we discuss a class of second order discrete time systems for which it is true provided the nonlinearity is odd but false in general this has strong implications for the analysis of saturated systems

on elliptic calogero moser systems for complex crystallographic reflection groups abstract to every irreducible finite crystallographic reflection group i e an irreducible finite reflection group g acting faithfully on an abelian variety x we attach a family of classical and quantum integrable systems on x with meromorphic coefficients these families are parametrized by g invariant functions of pairs t s where t is a hypertorus in x of codimension 1 and s g is a reflection acting trivially on t if g is a real reflection group these families reduce to the known generalizations of elliptic calogero moser systems but in the non real case they appear to be new we give two constructions of the integrals of these systems an explicit construction as limits of classical calogero moser hamiltonians of elliptic dunkl operators as the dynamical parameter goes to 0 implementing an idea of v buchstaber g felder and a veselov 1994 bfv and a geometric construction as global sections of sheaves of elliptic cherednik algebras for the critical value of the twisting parameter we also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions

multidimensional baker akhiezer functions and huygens principle a notion of the rational baker akhiezer ba function related to a configuration of hyperplanes in c n is introduced it is proved that the ba function exists only for very special configurations locus configurations which satisfy a certain overdetermined algebraic system the ba functions satisfy some algebraically integrable schrodinger equations so any locus configuration determines such an equation some results towards the classification of all locus configurations are presented this theory is applied to the famous hadamard problem of description of all hyperbolic equations satisfying huygens principle we show that in a certain class all such equations are related to locus configurations and the corresponding fundamental solutions can be constructed explicitly from the ba functions

commuting partial differential operators and vector bundles over abelian varieties since then the case of rings of rank r were studied by several people 14 25 34 then the complete determination of geometrical object which classifies commutative rings of rank r for r 2 was accomplished by mulase 22 in 1989 it is thus natural to ask what happens if we replace ordinary differential operators by partial differential operators there are a few results on this problem 4 14 3 in these papers all operators are of scalar coefficients and spectral varieties are singular varieties curves or their products in 26 we have constructed commuting partial differential operators from line bundles on abelian varieties the particular feature of these operators is that they are of matrix coefficients the main purpose of this paper is to construct commuting partial differential operators with matrix coefficients from a class of vector bundles on abelian varieties in studying commuting operators their common eigenfunctions play a central role krichever 13 characterized such eigenfunctions as baker akhiezer ba functions in the case of ordinary differential operators the concept of ba functions are generalized to higher dimensions 26 7 ba functions are constructed from x d ri where x is a projective variety d an ample divisor on x and ri is a base of differentials of the second kind with poles on d modulo exact forms in this paper we introduce ba functions of coherent sheaves on abelian varieties which are our main objects of study in this paper

conserved operators of the generalized elliptic ruijsenaars models we construct the integrable families of the generalized elliptic ruijsenaars models by use of the yang baxter equation and the reflection equation the higher order conserved operators of these models are explicitly presented

stability and convergence properties of forced infinite dimensional discrete time lur e systems incremental stability and convergence properties for forced infinite dimensional discrete time lur e systems are addressed lur e systems have a linear and nonlinear component and arise as the fe

computer aided stability analysis renders popov criterion obsolete a class of multipliers that can be used for producing alternative stability tests for systems having a single incrementally conic nonlinearity f x has been nearly parameterized by zames and falb 1 the present note demonstrates how linear programming can be used to compute optimal multipliers leading to a frequency domain stability criterion that is less conservative than the standard graphical criteria

elliptic quantum many body problem and double affine knizhnik zamolodchikov equation the elliptic matrix quantum olshanetsky perelomov problem is introduced for arbitrary root systems by means of an elliptic version of the dunkl operators its equivalence with the double affine generalization of the knizhnik zamolodchikov equation in the induced representations is established

generalized lame operators we introduce a class of multidimensional schrodinger operators with elliptic potential which generalize the classical lame operator to higher dimensions one natural example is the calogero moser operator others are related to the root systems and their deformations we conjecture that these operators are algebraically integrable which is a proper generalization of the finite gap property of the lame operator using earlier results of braverman etingof and gaitsgory we prove this under additional assumption of the usual liouville integrability in particular this proves the chalykh veselov conjecture for the elliptic calogero moser problem for all root systems we also establish algebraic integrability in all known two dimensional cases a general procedure for calculating the bloch eigenfunctions is explained it is worked out in detail for two specific examples one is related to the b2 case another one is a certain deformation of the a2 case in these two cases we also obtain similar results for the discrete versions of these problems related to the difference operators of macdonald ruijsenaars type

zames falb multipliers for quadratic programming in constrained linear model predictive control a quadratic program must be solved on line at each control step and this constitutes a nonlinearity if zero is a feasible point for this quadratic program then the resultant nonlinearity is sector bounded we show that if the nonlinearity is static then it is also monotone and slope restricted hence we show the existence of zames falb multipliers for such a nonlinearity the multipliers may be used in a general and versatile analysis of the robust stability of input constrained model predictive control

bispectrality for the quantum ruijsenaars model and its integrable deformation an elementary construction of the eigenfunctions for the quantum rational ruijsenaars model with integer coupling parameter is presented as a by product we establish the bispectral duality between this model and the trigonometric calogero moser model in particular this gives a new way for calculating jack polynomials we propose also a certain one parameter deformation of the ruijsenaars model proving its integrability and bispectrality the generalizations related to other root systems and difference operators by macdonald are considered

huygens principle in minkowski spaces and soliton solutions of the korteweg de vries equation a new class of linear second order hyperbolic partial differential operators satisfying huygens principle in minkowski spaces is presented the construction reveals a direct connection between huygens principle and the theory of solitary wave solutions of the korteweg de vries equation