parameter dependent lmis in robust analysis characterization of homogeneous polynomially parameter dependent solutions via lmi relaxations this note investigates the robust stability of uncertain linear time invariant systems in polytopic domains by means of parameter dependent linear matrix inequality pd lmi conditions exploiting some algebraic properties provided by the uncertainty representation a systematic procedure to construct a family of finite dimensional lmi relaxations is provided the robust stability is assessed by means of the existence of a lyapunov function more specifically a homogeneous polynomially parameter dependent lyapunov hppdl function of arbitrary degree for a given degree if an hppdl solution exists a sequence of relaxations based on real algebraic properties provides sufficient lmi conditions of increasing precision and constant number of decision variables for the existence of an hppdl function which tend to the necessity alternatively if an hppdl solution of degree exists a sequence of relaxations which increases the number of variables and the number of lmis will provide an hppdl solution of larger degree the method proposed can be applied to determine homogeneous parameter dependent matrix solutions to a wide variety of pd lmis by transforming the infinite dimensional lmi problem described in terms of uncertain parameters belonging to the unit simplex in a sequence of finite dimensional lmi conditions which converges to the necessary conditions for the existence of a homogeneous polynomially parameter dependent solution of arbitrary degree illustrative examples show the efficacy of the proposed conditions when compared with other methods from the literature

cauchy problem for the multi dimensional boussinesq type equation abstract the paper studies the existence and non existence of global weak solutions to the cauchy problem for the multi dimensional boussinesq type equation u t t u 2 u u it proves that the cauchy problem admits a global weak solution under the assumptions that c r s is of polynomial growth order say p 1 either 0 s s 0 s d s r where 0 is a constant or the initial data belong to a potential well and the weak solution is regularized and the strong solution is unique when the space dimension n 1 in contrast any weak solution of the cauchy problem blows up in finite time under certain conditions and two examples are shown

global existence and nonexistence of solution for cauchy problem of two dimensional generalized boussinesq equations abstract in this paper we consider the cauchy problem of two dimensional generalized boussinesq type equation u t t u u t t 2 u f u 0 under the assumption that f u is a function with exponential growth at infinity and under some assumptions on the initial data we prove the existence and nonexistence of global weak solution there are very few works on boussinesq equation with nonlinear exponential growth term by potential well theory

two dimensional percolation logarithmic corrections to the critical behaviour from series expansions analyses several extant series for the mean cluster size the zeroth moment of the pair connectedness and the percolation probability for bond and site percolation on two dimensional lattices with a view to detecting possible logarithmic corrections the logarithmic correction exponent z is found to be in the range 0 or z or 0 15 and our analysis also provides some new information about the critical exponents gamma and beta

global existence and exponential growth of solution for the logarithmic boussinesq type equation abstract in this paper we consider the boussinesq type equation associated with logarithmic nonlinear term and initial boundary value conditions by using potential well method combined with the logarithmic sobolev inequality we obtain the existence of global solution at last we show that the l 2 norm of the solution will grow up as an exponential function as time goes to infinity under some suitable conditions for initial data the result shows that the polynomial nonlinearity is a critical condition of blow up in finite time for the solutions of boussinesq equations

affine parameter dependent lyapunov functions and real parametric uncertainty this paper presents new tests to analyze the robust stability and or performance of linear systems with uncertain real parameters these tests are extensions of the notions of quadratic stability and performance where the fixed quadratic lyapunov function is replaced by a lyapunov function with affine dependence on the uncertain parameters admittedly with some conservatism the construction of such parameter dependent lyapunov functions can be reduced to a linear matrix inequality lmi problem and hence is numerically tractable these lmi based tests are applicable to constant or time varying uncertain parameters and are less conservative than quadratic stability in the case of slow parametric variations they also avoid the frequency sweep needed in real spl mu analysis and numerical experiments indicate that they often compare favorably with spl mu analysis for time invariant parameter uncertainty

local stability conditions for t s fuzzy time delay systems using a homogeneous polynomial approach abstract local stability conditions for time delay t s fuzzy systems are proposed by use of a homogeneous polynomial approach lesser conservatism can be expected based on the derived stability criteria due to the following three factors a a new fuzzy lyapunov krasovskii functional is constructed which contains a homogeneous polynomially parameter dependent matrix of degree 2 b the off diagonal matrix in reciprocal convexity lemma is extended to a homogeneous polynomially parameter dependent matrix and c local stability conditions for time delay t s fuzzy systems are proposed to overcome the difficulty of dealing with the time derivative of membership functions finally two numerical examples illustrate the effectiveness of the proposed methods

series expansion analysis of corrections to scaling in the three state potts model several extant series for the d 2 q 3 potts model are analysed with a method that explicitly accounts for the effect of confluent corrections to scaling excellent agreement with the conjectured exponents for this model is found as well as confirmation of renormalisation group estimates of the sub leading eigenvalue

a less conservative lmi condition for the robust stability of discrete time uncertain systems abstract in this paper a less conservative condition for the robust stability of uncertain discrete time linear systems is proposed the uncertain parameters assumed to be time invariant are supposed to belong to convex bounded domains polytope type uncertainty the stability condition is formulated in terms of a set of linear matrix inequalities involving only the vertices of the polytope domain a simple and numerically efficient feasibility test provides a set of lyapunov matrices whose convex combination can be used to assess the stability of any dynamic matrix inside the uncertainty domain examples illustrate the results

robust performance analysis of uncertain lti systems dual lmi approach and verifications for exactness this paper addresses robust performance analysis problems of linear time invariant lti systems affected by real parametric uncertainties these problems known also as a special class of structured singular value computation problems are inherently intractable np hard problems as such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions where linear matrix inequality lmi plays an important nevertheless since lmi based conditions are expected to be conservative in general it is often the case that we cannot conclude anything if the lmi at hand turns out to be infeasible this motivates us to consider the dual of the lmi and examine the structure of the dual solution by pursuing this direction in this paper we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained in particular a set of uncertain parameters that violates the specified performance can be computed these results come from block moment matrix structure of the dual variable which is consistent with the recent results on polynomial optimization this particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained

robust absolute stability and nonlinear state feedback stabilization based on polynomial lur e functions this paper provides finite dimensional convex conditions to construct homogeneous polynomially parameter dependent lur e functions which ensure the stability of nonlinear systems with state dependent nonlinearities lying in general sectors and which are affected by uncertain parameters belonging to the unit simplex the proposed conditions are written as linear matrix inequalities parametrized in terms of the degree g of the parameter dependent solution and in terms of the relaxation level d of the inequality constraints based on the algebraic properties of positive matrix polynomials with parameters in the unit simplex as g and d increase progressively less conservative solutions are obtained the results in the paper include as special cases existing conditions for robust stability and for absolute stability analysis a convex solution suitable for the design of robust nonlinear state feedback stabilizing controllers is also provided numerical examples illustrate the efficiency of the proposed conditions c 2008 elsevier ltd all rights reserved

technical communique on the degree of polynomial parameter dependent lyapunov functions for robust stability of single parameter dependent lti systems a counter example to barmish s conjecture in this paper we consider the robust hurwitz stability analysis problems of a single parameter dependent matrix a q a email protected 1 over email protected 1 1 where a 0 a email protected r n x n with a 0 being hurwitz stable in particular we are interested in the degree n of the polynomial parameter dependent lyapunov matrix ppdlm of the form p q email protected i 0 email protected ip i that ensures the robust hurwitz stability of a q via p q 0 p q a q a t q p q 3 in general

asymptotically optimal orthonormal basis functions for lpv system identification a global model structure is developed for parametrization and identification of a general class of linear parameter varying lpv systems by using a fixed orthonormal basis function obf structure a linearly parametrized model structure follows for which the coefficients are dependent on a scheduling signal an optimal set of obfs for this model structure is selected on the basis of local linear dynamic properties of the lpv system system poles that occur for different constant scheduling signals the selected obf set guarantees in an asymptotic sense the least worst case modeling error for any local model of the lpv system through the fusion of the kolmogorov n width theory and fuzzy c means clustering an approach is developed to solve the obf selection problem for discrete time lpv systems based on the clustering of observed sample system poles

the cauchy problem for a class of the multidimensional boussinesq type equation abstract in this paper we study the cauchy problem for a class of the multidimensional boussinesq type equation u t t u 2 u 2 u t t f u where f u a u p or a u p 1 u a 0 is a constant first we establish a local existence theorem for the solution then for m 1 n 1 m 2 3 4 n 3 we prove the existence of a global h m solution finally we prove the global nonexistence and finite time blow up of the solution

on klauder s pseudo free oscillator studies the quantum oscillator on the half line defined by the hamiltonian h 1 2 delta 2 delta x2 1 2x2 together with vanishing boundary conditions at 0 the authors derives and discusses a new operator equation of motion of the form x x a for this system this equation is used to verify j r klauder s conjecture 1979 that x satisfy x x x 0 it is also used to obtain systematic results on the set of n point functions omega mod x t1 x t2 x tn mod omega it is suggested that these methods and results will act as a source of conjectures about the analogous pseudo free field theories postulated by klauder in his program for handling non renormalisable field theories

lmi characterization of structural and robust stability abstract this paper introduces several stability conditions for a given class of matrices expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable diagonal and simultaneous stability both characterized by polytopes of matrices are addressed using this approach a method particularly attractive to test a given matrix for d stability is proposed lyapunov parameter dependent functions are built in order to reduce conservativeness of the stability conditions the key idea is to relate hurwitz stability with a positive realness condition

new strategy for robust stability analysis of discrete time uncertain systems in this paper a new robust stability analysis approach is developed for uncertain discrete time linear time invariant systems with polytopic or affine parameter dependent uncertainty models the proposed approach is based on a combination of a branch and bound like strategy with linear matrix inequality lmi based analysis formulations two sufficient conditions are considered one for the robust stability of the uncertain system and other one for the contrary situation if both sufficient conditions fail to characterize the polytope then it is iteratively subdivided into subpolytopes until some one proves to be unstable or all ones are verified to be robustly stable the polytope subdivision is implemented by means of a specially developed simplex subdivision algorithm exhaustive numerical tests prove the efficiency of the proposed approach when compared with the most recent lmi based formulations

an lmi condition for the robust stability of uncertain continuous time linear systems a new sufficient condition for the robust stability of continuous time uncertain linear systems with convex bounded uncertainties is proposed in this note the results are based on linear matrix inequalities lmis formulated at the vertices of the uncertainty polytope which provide a parameter dependent lyapunov function that assures the stability of any matrix inside the uncertainty domain with the aid of numerical procedures based on unidimensional search and the lmis feasibility tests a simple and constructive way to compute robust stability domains can be established

lmi characterization of structural and robust stability the discrete time case abstract this paper extends to the discrete time case some robust stability conditions recently obtained for continuous time systems those conditions are expressed in terms of linear matrix inequalities lmi being thus simply and efficiently computable as in the continuous time case parameter dependent lyapunov functions can be constructed and consequently the new approach can yield much sharper and less conservative results than the simultaneous stability approach in particular well known stability problems namely d stability and robust stability in the presence of diagonally structured uncertainty can be more efficiently addressed numerical examples are included to illustrate the advantages of the new stability conditions

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

blowup of solutions for the bad boussinesq type equation abstract the paper studies the blowup of solutions to the initial boundary value problem for the bad boussinesq type equation u tt u xx bu xxxx u xx where b 0 is a real number and s is a given nonlinear function by virtue of the energy method and the fourier transform method respectively it proves that under certain assumptions on s and initial data the generalized solutions of the above mentioned problem blow up in finite time and a few examples are shown especially for the bad boussinesq equation two examples of blowup of solutions are obtained numerically

a note on the non existence of global solutions of initial boundary value problems for the boussinesq equation utt 3uxxxx uxx 12 u2 xx

longtime dynamics of boussinesq type equations with fractional damping abstract the paper investigates the well posedness and longtime dynamics of boussinesq type equations with fractional damping u t t 2 u u t f u g x with 1 2 the main results focus on the relations among the dissipative exponent the growth exponent p of nonlinearity f u and the well posedness and the longtime dynamics of the equations we find a new critical exponent p n 2 2 1 n 2 2 1 rather than p n 2 n 2 n 3 as known before and show that when 1 p p i the equations are like parabolic that is not only the ibvp of the equations are well posedness but also their weak solutions are of higher global regularity as t 0 ii the related solution semigroup has a global attractor a in natural energy space and also has an exponential attractor a e x p in the sense of partially strong topology in particular when 1 p p n 2 n 2 p the partially strong topology becomes the strong one iii for any 0 1 2 the family of global attractors a is upper semicontinuous at the point 0

coprime factors reduction methods for linear parameter varying and uncertain systems abstract we present a generalization of the coprime factors model reduction method of meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models a complete derivation of coprime factorizations for this class of systems is also given the reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured l 2 induced stability constraint required in the standard nonfactored case reduction error bounds in the l 2 induced norm of the factorized mapping are given

robust static output feedback control synthesis for linear continuous systems with polytopic uncertainties this paper studies the static output feedback sof control problem of continuous time linear systems with polytopic uncertainties novel lmi conditions with a line search over a scalar variable for designing robust sof controllers are proposed where the uncertain output matrix of the considered system is allowed to be not of full row rank in particular it is shown that the new method can give less or at least the same conservative results than those methods by inserting a matrix equality constraint between system output matrix and lyapunov matrix furthermore the result is extended to the case of h control numerical examples are given to illustrate the effectiveness of the proposed method

brief paper comparisons among robust stability criteria for linear systems with affine parameter uncertainties this paper is concerned with the problem of comparisons among robust stability criteria for a class of uncertain linear systems where the system state matrices considered are affinely dependent on the uncertain parameters at first a robust stability criterion for the class of systems to be affinely quadratically stable aqs is derived based on the vertex separator approach where affine parameter dependent lyapunov functions are exploited to prove stability then comparison results between the robust stability criterion and the existing tests for aqs are given in terms of degree of conservatism a numerical example is given to illustrate the results

non triviality and the tuning of the scaling parameter in scale covariant λ0 φ2 2 field theory the author generalises the results of rivers 1983 on an o n symmetric scale covariant pseudo free theory by considering the case of an explicit lambda 0 phi 2 2 interaction term in the lagrangian it is found that for general values of the scaling parameter beta the theory is trivial however in four and five dimensions there is a tuned form for beta at which the theory may be non trivial in this case the theory reduces to the pseudo free theory which has the same form as the canonical lambda phi 2 2 theory to leading order it thus appears that the hard core effects of scale covariant quantisation act as a lambda phi 4 type interaction

brief paper time varying discrete time linear systems with bounded rates of variation stability analysis and control design this paper investigates the problems of robust stability analysis and state feedback control design for discrete time linear systems with time varying parameters it is assumed that the time varying parameters lie inside a polytopic domain and have known bounds on their rate of variation a convex model is proposed to represent the parameters and their variations as a polytope and linear matrix inequality relaxations that take into account the bounds on the rates of parameter variations are proposed a feasible solution provides a parameter dependent lyapunov function with polynomial dependence on the parameters assuring the robust stability of this class of systems extensions to deal with robust control design as well as gain scheduling by state feedback are also provided in terms of linear matrix inequalities numerical examples illustrate the results

new similarity reductions of the boussinesq equation some new similarity reductions of the boussinesq equation which arises in several physical applications including shallow water waves and also is of considerable mathematical interest because it is a soliton equation solvable by inverse scattering are presented these new similarity reductions including some new reductions to the first second and fourth painleve equations cannot be obtained using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method that involves no group theoretical techniques

robust spl hscr sub spl infin filtering for discrete time linear systems with uncertain time varying parameters this paper deals with the problem of robust spl hscr sub spl infin filtering for linear discrete time state space models with uncertain time varying parameters the parameters enter affinely into the state space model matrices and their admissible values and variations are assumed to belong to given intervals a method is derived for designing a linear stationary asymptotically stable filter with a prescribed spl hscr sub spl infin performance in spite of large parameter uncertainty the proposed method incorporates information on available bounds on both the admissible values and variation of the uncertain parameters and is based on a lyapunov function with quadratic dependence on the parameters the filter design is given in terms of linear matrix inequalities

the continuous spin ising model g 0 φ 4 d field theory and the renormalization group we have used the method of high temperature series expansions to investigate the critical point properties of a continuous spin ising model and g0 4 d euclidean field theory we have computed through tenth order the hightemperature series expansions for the magnetization susceptibility second derivative of the susceptibility and the second moment of the spin spin correlation function on eight different lattices our analysis of these series is made using integral and pade approximants in three dimensions we find that hyperscaling fails for sufficiently ising like systems the strong coupling limit of g0 4 3 depends on how the ultraviolet cutoff is removed the level contours of the renormalized coupling constant for this model in theg0 correlation length plane exhibit a saddle point if the ultraviolet cutoff is removed beforeg0 the usual field theory results and the renormalization group fixed point with hyperscaling is obtained if the order of these limits is reversed the ising model limit where hyperscaling fails and the field theory is trivial is obtained in four dimensions we find that hyperscaling fails completely g0 4 4 is trivial for all g0 when the ultraviolet cutoff is removed

an improved lmi condition for robust d stability of uncertain polytopic systems a sufficient condition for robust d stability of linear systems with polytope type uncertainties is proposed the result is based on a linear parameter dependent lyapunov function obtained from the feasibility test of a set of linear matrix inequalities lmis defined at the vertices of the polytope this improved lmi condition encompasses previous results based on additional inequalities as well as results based on extra variables

brief paper gain scheduled output feedback controllers depending solely on scheduling parameters via parameter dependent lyapunov functions in this paper we propose a new design method for gain scheduled output feedback gsof controllers for continuous time linear parameter varying lpv systems via parameter dependent lyapunov functions pdlfs the gsof controllers depend solely on scheduling parameters although our method requires a line search to obtain suboptimal controllers it produces practical gsof controllers being independent of the derivatives of scheduling parameters our method is proved to be no more conservative than conventional design methods via constant lyapunov functions as well as particularly structured pdlfs

observer based control of discrete time lpv systems with uncertain parameters in this note linear matrix inequality based design conditions are presented for observer based controllers that stabilize discrete time linear parameter varying systems in the situation where the parameters are not exactly known but are only available with a finite accuracy the presented framework allows to make tradeoffs between the admissible level of parameter uncertainty on the one hand and the transient performance on the other in addition the level of parameter uncertainty can be maximized while still guaranteeing closed loop stability

finite element galerkin method for the good boussinesq equation recently it has been shown by manoranjan et al l that this class of equations possesses an interesting solitary wave interaction mechanism moreover using semi group theoretic approch they have established global existence of a generalized solution provided the initial data are in a potential well further it is shown that finite time blowup may occur for weak solution in r if the initial data are not in the potential well and the total energy at t 0 exceeds the depth of the well in the first part of this paper we shall discuss existence uniqueness and regularity results for the problem 1 i 1 3 using the faedo galerkin method in the literature very few results are available pertaining to the numerical study of 1 l 1 4 in 2 defrutos et al have analysed pseudo spectral fourier method for the equation i i i 2 with periodic boundary conditions they have also discussed the nonlinear stability and convergence results in the analytic framework introduced earlier by lopez marcos and sanz serna 3 for related results on finite difference schemes one may refer to ortega and sanz serna 4 recently manoranjan et al 5 have applied petrov galerkin finite element method for cauchy problem without rigorous error analysis and have presented some numerical results in the later part of the present paper we have derived optima1 error estimates for both semidiscrete

lmi tests for positive definite polynomials slack variable approach the considered problem is assessing non negativity of a function s values when indeterminates are in domains constrained by scalar polynomial inequalities the tested functions are multi indeterminates polynomial matrices which are required to be positive semi definite new tests based on linear matrix inequalities are provided in a slack variables type approach the results are compared to those obtained via the sum of squares approach they are proved to be equivalent in case of unbounded domains and less conservative if polytopic type bounds are known

stability of polytopes of matrices via affine parameter dependent lyapunov functions asymptotically exact lmi conditions abstract this paper investigates necessary and sufficient conditions for the existence of an affine parameter dependent lyapunov function assuring the hurwitz or schur stability of a polytope of matrices a systematic procedure for constructing a family of linear matrix inequalities conditions of increasing precision is given at each step a set of linear matrix inequalities provides sufficient conditions for the existence of the affine parameter dependent lyapunov function necessity is asymptotically attained through a relaxation based on a generalization of polya s theorem to the case of matrix valued functions numerical experiments illustrate the results

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

guaranteed properties of gain scheduled control for linear parameter varying plants abstract gain scheduling has proven to be a successful design methodology in many engineering applications however in the absence of a sound theoretical analysis these designs come with no guarantees on the robustness performance or even nominal stability of the overall gain scheduled design this paper presents such an analysis for one type of gain scheduled system namely a linear parameter varying plant scheduling on its exogenous parameters conditions are given which guarantee that the stability robustness and performance properties of the fixed operating point designs carry over to the global gain scheduled design these conditions confirm and formalize popular notions regarding gain scheduled design such as the scheduling variable should vary slowly

global attractor for a class of kirchhoff models the paper studies the longtime behavior of solutions to the initial boundary value problem ibvp for a class of kirchhoff models arising in elastoplastic flow utt div u m 1 u ut 2u h ut g u f x by combining the decomposition idea with the operate technique it proves that under rather mild conditions the dynamical system associated with above mentioned ibvp possesses in different phase spaces a global attractor which is connected respectively for application the fact shows that for the concerned elastoplastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space

pseudospectral method for the good boussinesq equation we prove the nonlinear stability and convergence of a fully discrete pseudospectral scheme for the good boussinesq equation utt uxxxx uxx u2 xx numerical comparisons with finite difference schemes are also reported

parameter dependent lyapunov functions and the discrete time popov criterion for robust analysis abstract parameter dependent lyapunov functions are developed for discrete time systems with constant real parameter uncertainty this construction of a family of lyapunov functions is then used to analyze robust stability with h 2 performance bounds for state space systems a special case of these results generalizes the classical discrete time popov criterion

on a conjecture about the norm of lyapunov mappings abstract the norm properties of lyapunov mappings and their restrictions on symmetric and skew symmetric subspaces are investigated for non negative non positive and tridiagonal matrices this paper gives an affirmative answer to an open conjecture which says that the norm of a lyapunov mapping over the space of real square matrices equals that of its restricted map over the subspace of real symmetric matrices in addition matrix transformation and norm relationships on certain matrices arising from the lyapunov mapping and its restrictions are provided

convergent lmi relaxations for robust analysis of uncertain linear systems using lifted polynomial parameter dependent lyapunov functions abstract this paper investigates the problems of checking robust stability and evaluating robust 2 performance of uncertain continuous time linear systems with time invariant parameters lying in polytopic domains the novelty is the ability to check robust stability by constructing a particular parameter dependent lyapunov function which is a polynomial function of the uncertain system matrices as opposed to a general polynomial function of the uncertain parameter the degree of the polynomial is tied to a certain integer the existence of such lyapunov function can be proved by solving parameter dependent linear matrix inequalities lmis which are guaranteed to be solvable for a sufficiently large yet finite value of whenever the system is robustly stable extensions to guaranteed 2 cost computation are also provided numerical aspects concerning the programming and the evaluations of the proposed tests are discussed and illustrated by examples

a convex approach to robust stability for linear systems with uncertain scalar parameters in this paper robust stability for linear systems with several uncertain complex and or real scalar parameters is studied a countable family of conditions sufficient for robust stability is given in terms of solvability of some simple linear matrix inequalities lmis these conditions are of increasing precision and it is shown conversely that robust stability implies solvability of these lmis from a certain rank and beyond this result constitutes an extension of the characterization by solvability of lyapunov inequality of the asymptotic stability for usual linear systems it is based on the search of parameter dependent quadratic lyapunov functions polynomial of increasing degree in the parameters

towards non trivial scalar theories in d 4 space time dimensions shows in the large n limit how non canonical quantisation of an o n invariant pseudo free scalar theory in d or 4 dimensions can give non trivial behaviour implications for recent results on the triviality of canonical scalar theories in d or 4 dimensions are discussed

fixed structure lpv discrete time controller design with induced l2 norm and h2 performance abstracta new method for the design of fixed structure dynamic output feedback linear parameter varying lpv controllers for discrete time lpv systems with bounded scheduling parameter variations is presented sufficient conditions for the stability h2 and induced l2 norm performance of a given lpv system are represented through a set of linear matrix inequalities lmis these lmis are used in an iterative algorithm with monotonic convergence for lpv controller design extension to the case of uncertain scheduling parameter value is considered as well controller parameters appear directly as decision variables in the optimisation program which enables preserving a desired controller structure in addition to the low order efficiency of the proposed method is illustrated on a simulation example with an iterative convex optimisation scheme used for the improvement of the control system performance

a new class of universal lyapunov functions for the control of uncertain linear systems the authors analyze the problem of synthesizing a state feedback control for the class of uncertain continuous time linear systems affected by time varying memoryless parametric uncertainties they consider as candidate lyapunov functions the elements of the class spl sigma sub p sup z which is formed by special homogeneous positive definite functions they show that this class is universal in the sense that a lyapunov function exists if and only if there exists a lyapunov function in spl sigma sub p sup z they prove this result in a constructive way showing that such a lyapunov function can always be obtained by smoothing a polyhedral function for which construction algorithms are available the authors show that unlike the polyhedral lyapunov functions these functions allow us to derive explicit formulas for the stabilizing controller

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

longtime dynamics of the damped boussinesq equation abstract the paper studies the longtime dynamics of the damped boussinesq equation u t t 2 u u t g u f x first the existence of global solutions to the initial boundary value problem of the equation is obtained provided that the growth exponent of g u say p is either non supercritical subcritical and critical or supercritical especially the stability of solutions is established when p is non supercritical second the existence of a global attractor and an exponential attractor for the related solution semigroup s t are respectively established in the non supercritical case

a new robust d stability condition for real convex polytopic uncertainty abstract the problem of robust d stability analysis with respect to real convex polytopic uncertainties is tackled a new lmi based sufficient condition for the existence of parameter dependent lyapunov functions is proposed this condition generalises previously published conditions numerical comparisons with quadratic stability results as well as previous results based on parameter dependent lyapunov functions illustrate the relevance of this new condition finally this result appears to be promising for robust multi objective performance analysis and control synthesis purposes

scale covariant field theories iii the augmented scalar theory for pt ii see ibid vol 15 p 2929 45 1982 the authors examine the augmented formalism for a scalar pseudo free field theory motivated by a diagrammatic expansion for mass renormalisation they justify a constraint equation for the theory that gives additional information over the subtracted scale invariant formalism

scale covariant field theories ii the pseudo perturbation expansion for pt i see ibid vol 15 p 2911 27 1982 the authors examine the possibility of developing a perturbation expansion about the pseudo free scalar theory the degeneracy of the subtracted scale covariant equations requires that additional information be provided they suggest two ways in which this can be done

a new iterative approach for multi objective fault detection observer design and its application to a hypersonic vehicle abstractthis paper addresses the multi objective fault detection observer design problems for a hypersonic vehicle owing to the fact that parameters variations modelling errors and disturbances are inevitable in practical situations system uncertainty is considered in this study by fully utilising the orthogonal space information of output matrix some new understandings are proposed for the construction of lyapunov matrix sufficient conditions for the existence of observers to guarantee the fault sensitivity and disturbance robustness in infinite frequency domain are presented in order to further relax the conservativeness slack matrices are introduced to fully decouple the observer gain with the lyapunov matrices in finite frequency range iterative linear matrix inequality algorithms are proposed to obtain the solutions the simulation examples which contain a monte carlo campaign illustrate that the new methods can effectively reduce the design conservativeness compared with the existing methods

global existence to generalized boussinesq equation with combined power type nonlinearities abstract the cauchy problem to the generalized boussinesq equation with combined power type nonlinearities is studied global solvability or finite time blow up of the solutions with subcritical initial energy is proved by means of the sign preserving property of the nehari functional for generalized lienard or generalized bernoulli nonlinear terms the critical energy constant is explicitly evaluated a new method that can be considered as a modification of the potential well method is developed the performed numerical experiments support the theoretical results

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

on strict positive real systems design guaranteed cost and robustness issues this paper deals with unbounded nonlinear real uncertainty and proposes a solution to the strict positive real control problem by dynamic output feedback besides a guaranteed cost problem solved by lmi optimization combined with line search is introduced in order to provide robust performance

non canonical quantization as a source of symmetry breaking with chiral theories and gravity in mind the authors use block renormalization group rg methods and rg equations to examine the proposal by de alfaro fubini and furlan 1984 that symmetry breaking can be induced by the choice of conformal invariant measures

stability region analysis using polynomial and composite polynomial lyapunov functions and sum of squares programming we propose using bilinear sum of squares programming for obtaining inner bounds of roas for dynamical systems with polynomial vector fields we search for polynomial as well as composite lyapunov functions comprised of pointwise maximums of polynomial functions results for several examples from the literature are presented using the proposed methods and the penbmi solver

generalized robust gain scheduled pid controller design for affine lpv systems with polytopic uncertainty in the paper a generalized guaranteed cost output feedback robust gain scheduled pid controller synthesis is presented for affine linear parameter varying systems under polytopic model uncertainty the controller synthesis is generalized in a sense that it covers robust robust gain scheduled and robust switched with arbitrary switching algorithm pid controller design the proposed centralized decentralized controller method is based on bellman lyapunov equation guaranteed cost and parameter dependent quadratic stability the proposed sufficient robust stability and performance conditions are derived in the form of bilinear matrix inequalities bmi which can efficiently be solved or further linearized as the main result the suggested performance and stability conditions without any restriction on the controller structure are convex functions of the scheduling and uncertainty parameters hence there is no need for applying multi convexity or other relaxation techniques and consequently the proposed solution delivers a less conservative design method the viability of the novel design technique is demonstrated and evaluated through numerical examples

relaxed stability conditions for continuous time t s fuzzy control systems via augmented multi indexed matrix approach this paper is concerned with the problem of developing an advanced strategy to reduce the conservatism in stability analysis and control synthesis of continuous time takagi sugeno t s fuzzy systems a novel augmented multi indexed matrix approach is proposed to implement new right hand side slack variables technique for the homogenous polynomial setting combining with the finsler lemma with homogenous matrix lagrange multipliers convergent linear matrix inequality lmi relaxations for stability analysis are proposed by using the generalization of the polya theorem for the case of positive polynomials with matrix valued coefficients a new type of state feedback controller namely the homogeneous polynomially nonquadratic control law hpnqcl is developed to conceive less conservative stabilization conditions the obtained stability and stabilization conditions are further relaxed by using the proposed right hand side slack variables technique moreover the advantages over the existing control schemes are certificated in theory three numerical examples are also provided to illustrate the effectiveness of the proposed methods

distributed control design for spatially interconnected systems this paper deals with analysis synthesis and implementation of distributed controllers designed for spatially interconnected systems we develop a state space framework for posing problems of this type and focus on systems whose model is spatially discrete in this paper analysis and synthesis results are developed for this class of systems using the l sub 2 induced norm as the performance criterion the results are stated in terms of linear matrix inequalities and are thus readily amenable to computation a special implementation of the resulting controllers is presented which is particularly attractive for distributed operation of the controller several examples are provided to further illustrate the application of the results

stability analysis and gain scheduled state feedback control for continuous time systems with bounded parameter variations the problems of robust stability analysis and state feedback control based on gain scheduling for continuous time systems with time varying parameters that have bounded rates of variation and lie inside a polytope are addressed in this article with respect to previous results in the literature two main contributions of the article are i the robust stability analysis conditions are less conservative and demand less computational effort than the existing ones ii the conditions can be extended to cope with the problem of control design for this class of system parameter dependent linear matrix inequality lmi conditions are given for the existence of a parameter dependent lyapunov function quadratic in the state and homogeneous polynomially of arbitrary degree in the parameter assuring robust stability two convex procedures based on lmis exhibiting distinct complexities are proposed to solve the problem of robust stability an extension to deal with the computation of a stabilising parameter dependent state feedback gain for this class of time varying systems is also provided as a sequence of lmis of increasing precision examples illustrate the results including comparisons with other techniques from the literature

decay estimates for a dissipative dispersive linear semigroup and application to the viscous boussinesq equation abstract at the core of this article is the new estimate for a class of dissipative dispersive linear semigroup e t i p t arising in the study of the viscous boussinesq equation we combine the decay estimate with introducing a set of time weighted sobolev spaces where the time weights and the regularity of the sobolev spaces are determined by our decay estimate to show the global existence and asymptotic behavior of solutions to the viscous boussinesq equation in r n

analysis and synthesis of robust control systems via parameter dependent lyapunov functions in this paper the problem of robust stability of systems subject to parametric uncertainties is considered sufficient conditions for the existence of parameter dependent lyapunov functions are given in terms of a criterion which is reminiscent of but less conservative than popov s stability criterion an equivalent frequency domain criterion is demonstrated the relative sharpness of the proposed test and existing stability criteria is then discussed the use of parameter dependent lyapunov functions for robust controller synthesis is then considered it is shown that the search for robustly stabilizing controllers may be limited to controllers with the same order as the original plant a possible synthesis procedure and a numerical example are then discussed

finite dimensional attractors for the kirchhoff models the paper studies the existence of the finite dimensional global attractor and exponential attractor for the dynamical system associated with the kirchhoff models arising in elasto plastic flow utt div u m 1 u ut 2u h ut g u f x by using the method of l trajectories and the operator technique it proves that under subcritical case 1 m n 2 n 2 the above mentioned dynamical system possesses in different phase spaces a finite dimensional weak global attractor and a weak exponential attractor respectively for application the fact shows that for the concerned elasto plastic flow the permanent regime global attractor can be observed when the excitation starts from any bounded set in phase space and the fractal dimension of the attractor that is the number of degree of freedom of the turbulent phenomenon and thus the level of complexity concerning the flow is finite

nonsmooth h_infty synthesis we develop nonsmooth optimization techniques to solve h infty synthesis problems under additional structural constraints on the controller our approach avoids the use of lyapunov variables and therefore leads to moderate size optimization programs even for very large systems the proposed framework is versatile and can accommodate a number of challenging design problems including static fixed order fixed structure decentralized control design of pid controllers and simultaneous design and stabilization problems our algorithmic strategy uses generalized gradients and bundling techniques suited for the h infty norm and other nonsmooth performance criteria we compute descent directions by solving quadratic programs and generate steps via line search convergence to a critical point from an arbitrary starting point is proved and numerical tests are included to validate our methods the proposed approach proves to be efficient even for systems with several hundreds of states

on the corrections to scaling in three dimensional ising models the leading correction to scaling amplitudes for the spin 1 2 nearest neighbor sc bcc and fee ising models are considered with the particular aim of determining their signs on the basis of previous two variable series analyses by chen fisher and nickel and renormalization group 4 d expansions it is concluded that the correction amplitudes for the susceptibility correlation length specific heat and spontaneous magnetization arenegative for all three lattices thus for example the effective exponent eff t asymptotically approaches the true susceptibility exponent fromabove other earlier and more recent high temperature series and field theoretic analyses are seen to be consistent with this result however the usual nonasymptotic perturbative field theoretic approaches are essentially committed to positive correction amplitudes the question of the signs therefore relates directly to the applicability of these non asymptotic field theoretic calculations to three dimensional ising models as well as to different experimental systems

an existence result for polynomial solutions of parameter dependent lmis we show in this paper that any system of linear matrix inequalities depending continuously upon scalar parameters and solvable for any value of the latter in a fixed compact set admits a branch of solutions polynomial with respect to the parameters this result is useful for studying e g parametric robustness or gain scheduling issues

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

parameter dependent lyapunov functions for discrete time systems with time varying parametric uncertainties abstract in this paper we consider discrete time systems with polytopic time varying uncertainty we look for a class of parameter dependent lyapunov functions which are quadratic on the system state and depend in a polytopic way on the uncertain parameter we show that extending the new discrete time stability condition proposed by de oliveira et al systems control lett 36 1999 135 to the case of time varying uncertainty leads to a necessary and sufficient condition for the computation of such a lyapunov function this allows to check asymptotic stability of the system under study the obtained linear matrix inequalities condition can also be used to cope with the control synthesis problem

convex liftings based robust control design this paper presents a new approach for control design of constrained linear systems affected by bounded additive disturbances and polytopic uncertainties this method hinges on so called convex liftings which emulate control lyapunov function by providing a constructive framework for optimization based control implementation it will be shown that this method can guarantee the recursive feasibility and robust stability finally a numerical example will be presented to illustrate this method

a new method for robust schur stability analysis this article is concerned with robust stability of uncertain discrete time linear systems the matrix defining the linear system system matrix is assumed to depend affinely on a set of time invariant unknown parameters lying on a known polytope robust stability is investigated by checking whether a certain integer power of the uncertain system matrix has spectral norm less than one this peculiar stability test is shown to be equivalent to the positivity analysis of a homogeneous symmetric matrix polynomial with precisely known coefficients and degree indexed by a unique feature is that no extra variables need to be added to the problems being solved numerical experiments reveal that the value of needed to test robust stability is mostly independent of the system dimension but grows sharply as the eigenvalues of the uncertain system approach the unit circle by identifying the proposed stability test with a particular choice of a parameter dependent lyapunov function extra variables can be int