master symmetries for differential difference equations of the volterra type abstract it is demonstrated that in the case of integrable differential difference equations similar to the well known volterra equation unlike the korteweg de vries and nonlinear schrodinger equations there are many instances in which local master symmetries can be found those master symmetries are new interesting examples of local evolution chains explicitly depending on the time and discrete variable and integrable in a special sense the examples are constructed by a direct and elementary approach which enables one to get new integrable equations using miura type transformations

on the general structure of nonlinear evolution equations integrable by the two dimensional matrix spectral problem a generalization of the akns technique to the two dimensional arbitrary order matrix spectral problem is given the general form of the integrable equations and their backlund transformations in 1 2 dimensions are found the reduction problem is discussed

a recursive generation of local higher order sine gordon equations and their backlund transformation a new hierarchy of local nonlinear evolution equations is generated by a recursion operator and its explicit inverse it is shown that this hierarchy satisfies a canonical geometrical scheme and that it contains as special cases the sine gordon and liouville equations in laboratory coordinates a generalization of the well known backlund transformation and nonlinear superposition formula for the sine gordon equation is also obtained

oscillation criteria of yan type for linear hamiltonian systems abstract new oscillation criteria of yan type are established for the linear hamiltonian system x x a t x b t y t t0 y c t x a t y where a t b t and c t are continuous n n matrix valued functions with b t b t 0 c t c t the criteria obtained generalize and improve some previous results for the second order linear matrix differential systems in literature

oscillation results for linear hamiltonian systems for linear hamiltonian systems this paper established some new oscillation criteria which improve previous results even for self adjoint second order differential systems the main tool used is a generalized riccati transformation

lp parameter perturbation and design of robust controllers for linear systems in this paper linear systems with characteristic equations that are linearly dependent on some uncertain parameters are considered a new procedure is proposed for the calculation of d stability margins in the parameter space in general lp norms the procedure is simple and computationally efficient for several important regions of interest in the complex plane e g left hand plane sectors and circles centred at the origin it requires the solution of only one equation based on this result a new method is presented for the design of robust controllers robust placement of the closed loop poles inside the regions of interest is addressed the method provides a simple and computationally tractable solution to the robust controller design problem it can be used to synthesize robust controllers with general rational transfer functions for plants modelled as the ratio of two uncertain polynomials subjected to lp parameter perturbation

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

necessary and sufficient conditions for robust stability of linear systems with multiaffine uncertainty structure this correspondence is concerned with an image f b sup m of a box b sup m spl sub r sup m under a multiaffine transformation f r sup m spl rarr c the authors introduce a notion of a principal point of b sup m and prove that the boundary of f b sup m is covered by images of principal points the authors exploit this result to provide necessary and sufficient robust stability conditions for polynomials whose coefficients are multiaffine functions of parameters an application of the general criterion obtained in the correspondence to the particular case of systems with a cascade of first order uncertain blocks leads to a computationally tractable procedure that verifies stability of the systems

oscillation of linear second order matrix differential equations in this paper sufficient conditions are obtained for oscillation of all nontrivial prepared symmetric solutions of a class of nonlinear second order matrix differential equations of the form p t y q t f y 0 t geqslant 0 and y q t f y 0 t geqslant 0

baklund transformations and singular integral equations abstract a systematic method for deriving backlund transformations for singular linear integral equations is presented the method leads in a natural way to the backlund transformations for the corresponding integrable nonlinear partial differential equations by repeated use of the method a hierarchy of singular integral equations and backlund transformations can be derived corresponding to so called multi modified partial differential equations specific examples include the korteweg de vries equation and the first second and third modified korteweg de vries equation the nonlinear schrodinger equation and the anisotropic heisenberg spin chain the sine gordon equation and the modified sine gordon equation

brief recursive algorithms for inner ellipsoidal approximation of convex polytopes in this paper fast recursive algorithms for the approximation of an n dimensional convex polytope by means of an inscribed ellipsoid are presented these algorithms consider at each step a single inequality describing the polytope and under mild assumptions they are guaranteed to converge in a finite number of steps for their recursive nature the proposed algorithms are better suited to treat a quite large number of constraints than standard off line solutions and have their natural application to problems where the set of constraints is iteratively updated as on line estimation problems nonlinear convex optimization procedures and set membership identification

the prolongation structure of the inhomogeneous equation of the reaction diffusion type the inhomogeneous extension of equations of the reaction diffusion type is investigated by means of the covariant prolongation structures theory we construct the sl 2 r r t prolongation structure for an inhomogeneous equation of the reaction diffusion type and give the corresponding akns type

on miura transformations among nonlinear partial differential equationsa in this paper we study miura transformations u v from partial differential equations uxxx f u ux ut to nonlinear partial differential equations g v vx vt xlv t1v 0 defined using integrable systems on v we classify all such miura transformations under some restrictions and hence generalize the classical miura transformation to a large class of nonlinear partial differential equations for some examples by applying miura transformations found in this paper we derive exact solutions v from known solutions u in particular kink and soliton kink solutions of vt 32vxsin2v 12vx3 vxxx are obtained from constant solutions and soliton solutions of the mkdv equation as another application of miura transformations of this paper we deduce a new backlund transformation for each of vt 32vxsin2v 12vx3 vxxx and vt 32vxsinh2v 12vx3 vxxx from the known backlund transformations for the mkdv equations

oscillation criteria for linear matrix hamiltonian systems abstract in this paper we present oscillation criteria in terms of the coefficient functions for the matrix linear hamiltonian system u a x u b x v v c x u a x v under the hypothesis h a x b x b x 0 and c x c x are n n matrices of real valued continuous functions on the interval i a a these criteria generalize earlier results due to etgen and pawlowski 1976 erbe qingkai kong and shigui ruan 1993 and fanwei meng jizhong wang and zhaowen zheng 1998 for the system p x u q x u 0 or its special cases

gauge transformations and symmetries of integrable systems we analyse several integrable systems in zero curvature form within the framework of invariant gauge theory in the drinfeld sokolov gauge we derive a two parameter family of nonlinear evolution equations which as special cases include the korteweg de vries kdv and harry dym hd equations we find residual gauge transformations which lead to infinitesimal symmetries of this family of equations for kdv and hd equations we find an infinite hierarchy of such symmetry transformations and we investigate their relation with local conservation laws constants of the motion and the bi hamiltonian structure of the equations applying successive gauge transformations of miura type we obtain a sequence of gauge equivalent integrable systems among them the modified kdv and calogero kdv equations

interval oscillation criteria for matrix differential systems with damping abstract this paper concerns the oscillation of solutions to leighton type for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t and r t r t are n n matrices of real valued continuous functions on the interval t 0

global optimization of nonconvex factorable programming problems in this paper we consider a special class of nonconvex programming problems for which the objective function and constraints are defined in terms of general nonconvex factorable functions we propose a branch and bound approach based on linear programming relaxations generated through various approximation schemes that utilize for example the mean value theorem and chebyshev interpolation polynomials coordinated with a reformulation linearization technique rlt a suitable partitioning process is proposed that induces convergence to a global optimum the algorithm has been implemented in c and some preliminary computational results are reported on a set of fifteen engineering process control and design test problems from various sources in the literature the results indicate that the proposed procedure generates tight relaxations even via the initial node linear program itself furthermore for nine of these fifteen problems the application of a local search method that is initialized at the lp relaxation solution produced the actual global optimum at the initial node of the enumeration tree moreover for two test cases the global optimum found improves upon the solutions previously reported in the source literature

robust gamma stability analysis in a plant parameter space abstract given a characteristic polynomial whose coefficients depend polynomially on l uncertain parameters the following robustness problem arises determine whether all the roots of the polynomial are located in a prescribed region in the complex plane for all admissible parameter values to this end the boundary of is mapped into the parameter space a necessary and sufficient condition for stability of an operating domain in parameter space is that it contains at least one stable point and is not intersected by the image of this condition may be tested graphically by gridding l 2 parameters and projecting all boundaries into a two dimensional subspace of the parameter space finally the method is applied to a track guided bus with uncertain mass and velocity

explicit solutions for the harry dym equation the well known link between korteweg de vries and hd harry dym equations is improved in this paper by the introduction of a suitable spectral parameter in the corresponding backlund relationship this additional spectral parameter allows the complete transfer of the group theoretical structure between the two equations as an application the authors give new solutions of the hd on finite intervals several plots are included the method of solution is also applied to the kawamoto equation

hamiltonian structure of the general integrable equations under reductions the general form of the nonlinear evolution equations integrable by the linear matrix spectral problem and their hamiltonian structures are presented for potentials with nonzero constant asymptotic values the reduction problem is discussed within the framework of the akns method a method is proposed for the calculation of the hamiltonian structures for the integrable equations under reductions matrix zn and n reductions and some other reductions are considered as examples general forms of the corresponding reduced equations and their hamiltonian structures are found

robustness analysis for linear dynamical systems with linearly correlated parametric uncertainties the authors propose an approach for robust pole location analysis of linear dynamical systems with parametric uncertainties linear control systems with characteristic polynomials whose coefficients are affine in a vector of uncertain physical parameters are considered a design region in complex plane for system pole placement and a nominal parameter vector generating a characteristic polynomial with roots in that region are given the proposed method allows the computation of maximal domains bounded by linear inequalities and centered at the nominal point in system parameter space preserving system poles in the given region the solution of this problem is shown to also solve the problem of testing robot location of a given polytope of polynomials in parameter space it is proved that for stability problems for continuous time systems with independent perturbations on polynomial coefficients this method generates the four extreme kharitonov polynomials

oscillation theorems for nonlinear differential equations of second order abstract in this paper we present some new criteria for the oscillation of the differential equation a t x t x t p t x t q t f x t 0 the obtained results extend improve and correlate a number of existing criteria

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

a comparison of complete global optimization solvers results are reported of testing a number of existing state of the art solvers for global constrained optimization and constraint satisfaction on a set of over 1000 test problems in up to 1000 variables collected from the literature the test problems are available online in ampl and were translated into the input formats of the various solvers using routines from the coconut environment these translators are available online too

an algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations an algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations compared with most existing tanh methods the jacobi function expansion method or other sophisticated methods the proposed method not only gives new and more general solutions but also provides a guideline to classify the various types of the solutions according to the values of some parameters the solutions obtained in this paper include a polynomial solutions b exponential solutions c rational solutions d triangular periodic wave solutions e hyperbolic and solitary wave solutions and f jacobi and weierstrass doubly periodic wave solutions the efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper new 2 1 dimensional calogero kdv equation 3 1 dimensional jimbo miwa equation symmetric regular long wave equation drinfel d sokolov wilson equation 2 1 dimensional generalized dispersive long wave equation double sine gordon equation calogero degasperis fokas equation and coupled schrodinger boussinesq equation in addition the links among our proposed method the tanh method the extended method and the jacobi function expansion method are also clarified generally

phase space analysis for the direct algebraic method for nonlinear evolution and wave equations abstract the basis of the direct algebraic method employing a series in real exponential functions for solving nonlinear evolution and wave equations is examined in terms of the general theory of autonomous ordinary differential equations odes and phase space analysis the closed form solutions obtained by the direct algebraic method are identified with special trajectories in phase space that connect a critical point to itself or other critical points the phase portrait is investigated for odes resulting from a combined kdv and mkdv equation a kdv like equation with fifth degree nonlinearity a special case of the calogero degasperis fokas modified mkdv equation and a generalized fisher equation the phase space analysis serves to clarify and partially justify the direct algebraic method

hodograph transformations of linearizable partial differential equations in this paper an algorithmic method is developed for transforming quasilinear partial differential equations of the form u t g u u nx f u u x cdots u n 1 x u mx equiv partial m u partial x m where dg du not equiv 0 into semilinear equations i e equations of the above form with g u 1 this crucially involves the use of hodograph transformations i e transformations involving the interchange of dependent and independent variables furthermore the most general quasilinear equation of the above form is found that can be mapped via a hodograph transformation to a semilinear form this algorithm provides a method for establishing whether a given quasilinear equation is linearizable i e is solvable in terms of either a linear partial differential equation or of a linear integral equation in particular this method is used to show how the painleve tests may be applied to quasilinear equations this appears to resolve the problem that solutions of lineariz

exact explicit traveling wave solutions for the cdf equation abstract the traveling wave solutions of the calogero degasperis fokas cdf equation are studied by using the approach of dynamical systems and the theory of bifurcations with the aid of maple all bifurcations and phase portraits in the parametric space are obtained all possible explicit parametric representations of the bounded traveling wave solutions solitary wave solutions kink and anti kink wave solutions periodic wave solutions and breaking wave solutions are given moreover the reason for appearance of breaking waves and for persistence of smoothness of smooth traveling waves is explained

nonlinear evolution equations and nonabelian prolongations a systematic analysis of the class of nonlinear evolution equations ut uxxx u ux 0 is carried out within the estabrook wahlquist prolongation scheme

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

reduction technique for matrix nonlinear evolution equations solvable by the spectral transform the main purpose of this paper is to describe a technique of reduction whereby from the class of evolution equations for matrices of order n solvable via the spectral transform associated to the matrix linear schrodinger eigenvalue problem one derives subclasses of nonlinear evolution equations involving less than n2 fields to illustrate the method from the equations for matrices of order 2 two subclasses of equations for 2 fields rather than 4 are obtained the first class coincides or rather includes that solvable via the spectral transform associated to the generalized zakharov shabat spectral problem further reduction to nonlinear evolution equations for a single field reproduces a number of well known equations but also yields a novel one highly nonlinear the second class also yields highly nonlinear equations some examples are given including another novel evolution equation for a single field

riccati techniques and variational principles in oscillation theory for linear systems on etablit des conditions suffisantes pour que toutes les solutions soient oscillatoires on utilise les techniques de riccati et les principes variationnels

specializations of integrable systems and affine lie algebras we analyze a number of restrictions on the evolution systems associated with the zero curvature equations corresponding to the extended dynkin diagram a 1 n 1 the resulting specialized evolution systems contain exponential terms like a third order differential equation previously derived in ref 5

from singular manifold equations to integrable evolution equations the singular manifold equations are used to construct nonlinear integrable partial differential equations a number of well known integrable equations hierarchies of korteveg de vries caudrey dodd gibbon kaup kuperschmidt and harry dym are obtained some new integrable equations are presented a possible approach to classification of integrable evolution equations is discussed

positive polynomials and robust stabilization with fixed order controllers recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial an application to the design of fixed order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed based on linear matrix inequality optimization the key ingredient in the design procedure resides in the choice of the central polynomial several numerical examples illustrate the relevance of the approach

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

robust stability manifolds for multilinear interval systems the stability of a class of multilinearly perturbed families of systems is considered it is shown how the problem of checking the stability of the entire family can be reduced to that of checking certain subsets that are independent of the degrees of the polynomials involved the extremal property of these subsets is established the results point to the need for a complete study of the stability of manifolds of polynomials composed of products of simple surfaces

linear transformation and oscillation criteria for hamiltonian systems abstract using a linear transformation similar to the kummer transformation some new oscillation criteria for linear hamiltonian systems are established these results generalize and improve the oscillation criteria due to i s kumari and s umanaheswaram i sowjaya kumari s umanaheswaram oscillation criteria for linear matrix hamiltonian systems j differential equations 165 2000 174 198 q yang et al q yang r mathsen s zhu oscillation theorems for self adjoint matrix hamiltonian systems j differential equations 190 2003 306 329 and s chen and z zheng shaozhu chen zhaowen zheng oscillation criteria of yan type for linear hamiltonian systems comput math appl 46 2003 855 862 these criteria also unify many of known criteria in literature and simplify the proofs

oscillation criteria for second order linear matrix differential systems with damping using a generalization of sturm s comparison theorem some new oscillation criteria are established for the matrix differential system with damping p t y r t y q t y 0 under the hypothesis p t p t 0 q t q t y t are nxn matrices of real valued continuous functions on the interval t 0 and r t r t c 1 t 0 r n 2 our results are sharper than some previous results

computation of the frequency response of multilinear affine systems deals with the problem of computing the frequency response of an uncertain transfer function whose numerator and denominator polynomials are multiples of independent uncertain polynomials of the form p s q l sub o q l sub 1 q s spl middot spl middot spl middot l sub n q s sup n whose coefficients depend linearly on q q sub 1 q sub 2 q sub q sup t and the uncertainty box is q q q sub i spl epsiv q sub i q sub i i 1 2 q using the geometric structure of the value set of p s q a powerful edge elimination procedure is proposed for computing the bode nyquist and nichols envelopes of these uncertain systems a numerical example is included to illustrate the benefit of the method presented

computation of parameter stability margins using polynomial programming techniques in this paper we consider linear time invariant lti systems with parameter uncertainty for such systems we present global optimization techniques to determine permissible perturbations of the parameters of the system that maintain stability the so called parameter stability margins for cases in which the coefficients of the characteristic equation of the system are polynomial functions of the uncertain parameters the parameter uncertainty domains for maintaining stability are characterized as hypersolids defined with respect to lp norms for various values of p 1 algorithms are devised based on the reformulation linearization convexification technique rlt in concert with branch and bound methods to solve the underlying parametric non convex subproblems for computing the stability margins several illustrative examples are solved to demonstrate the efficacy of the proposed approach towards producing global optimal solutions we also present comparative computational experience with the