survey paper transfer functions of distributed parameter systems a tutorial in this tutorial article the rich variety of transfer functions for systems described by partial differential equations is illustrated by means of several examples under various boundary conditions an important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations properness relative degree and minimum phase it is sometimes possible to design a controller using the irrational transfer function and several such techniques are outlined more often the irrational transfer function is approximated by a rational one for the purpose of controller design various approximation techniques and their underlying theory are briefly discussed

comparative lagrangian formulations for localized buckling an energy functional for a strut on a nonlinear softening foundation is worked into two different lagrangian forms in fast and slow space respectively the developments originate independently of the underlying differential equation and carry some quite general features in each case the kinetic energy is an indefinite quadratic form in fast space this leads to an escape phenomenon with fractal properties in slow space kinetic energy is added to a potential contribution that is familiar from modal formulations together and in conjunction with a recent set of numerical experiments they illustrate the extra complexities of localized as opposed to distributed periodic buckling

on the existence of solutions for a class of fourth order differential equations abstract we study the existence of the homoclinic solution of a class of nonlinear fourth order ordinary differential equations which involve particular cases of the extended fisher kolmogorov and swift hohenberg equations we address the case when the source term possesses a nonlinear dependence on the first order derivative of the solution the main result follows from an iterative method using the sequence of solutions obtained through the mountain pass technique in order to illustrate this method some numerical examples are shown

irregular dynamics and homoclinic orbits to hamiltonian saddle centers we consider 4 dimensional real analytic hamiltonian systems with a saddle center equilibrium related to a pair of real and a pair of imaginary eigenvalues and a homoclinic orbit to it we find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium we also consider one parameter families of reversible 4 dimensional hamiltonian systems we prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points we also present similar results for systems with heteroclinic orbits to saddle center equilibria 1997 john wiley sons inc

homoclinic orbits to a saddle center in a fourth order differential equation abstract we study homoclinic orbits to a saddle center of a fourth order ordinary differential equation which is invariant under the transformation x x involving an eigenvalue parameter q and an odd piece wise cubic type nonlinearity it is found that for a sequence of eigenvalues which tends to infinity homoclinic orbits exist whose complexity increases as the eigenvalue becomes larger these orbits are found to be embedded in branches of homoclinic orbits to periodic orbits as x

every storage function is a state function it is shown that for linear dynamical systems with quadratic supply rates a storage function can always be written as a quadratic function of the state of an associated linear dynamical system this dynamical system is obtained by combining the dynamics of the original system with the dynamics of the supply rate

semilinear elliptic variational inequalities with dependence on the gradient via mountain pass techniques abstract in this paper we consider a semilinear variational inequality with a gradient dependent nonlinear term obviously the nature of this problem is non variational nevertheless we study that problem associating a suitable semilinear variational inequality variational in nature with it and performing an iterative technique used in de figueiredo et al 2004 6 in order to treat semilinear elliptic equations when there is a gradient dependence on the nonlinearity we prove the existence of a non trivial non negative weak solution u for our problem using essentially variational methods a penalization technique and an iterative scheme via lewy stampacchia s estimates and regularity theory for elliptic equation we also show that u is differentiable and its gradient is h lder continuous on for any 0 1

on polynomial matrix spectral factorization by symmetric extraction we revise the 1963 davis algorithm 2 for the spectral factorization of a para hermitian nonnegative polynomial matrix phi by symmetric factor extraction this algorithm is careless about zeros at infinity by introducing the notion of diagonal reducedness of phi we obtain an easy sufficient test for the absence of zeros at infinity we show then how to get phi diagonally reduced by diagonal excess reduction steps similar to the oono and yasuura steps removing all zeros at infinity and then how to remove synunetrically finite zeros while keeping el diagonally reduced hence free of zeros at infinity this results in a revised symmetric extraction spectral factorization algorithm with monotone degree control an example shows the didactical conceptual simplicity of the method appropriate symmetric extraction is discovered by revising and discovering important particular one sided factor extraction properties of polynomial matrices

blow up oscillating solutions to some nonlinear fourth order differential equations abstract we give strong theoretical and numerical evidence that solutions to some nonlinear fourth order ordinary differential equations blow up in finite time with infinitely many wild oscillations we exhibit an explicit example where this phenomenon occurs we discuss possible applications to biharmonic partial differential equations and to the suspension bridges model in particular we give a possible new explanation of the collapse of bridges

some results on systems described by convolutional equations in this paper convolutional systems in the behavioral approach are considered convolutional equations generalize both differential and delay differential systems three fundamental problems are treated and solved the first consists of characterizing two kernel representations when they are equivalent i e to determine when two different convolutional equations provide the same set of solutions the second problem deals with the characterization of the spectral controllability for this class of systems the third problem provides a partial solution of the so called latent variable elimination problem which is one of the basic steps of the modeling procedure in the behavioral approach

on the periodic orbits of the fourth order differential equation u qu u ef u u u u we provide sufficient conditions for the existence of periodic solutions of the fourth order differential equation u qu u ef u u u u where q and e are real parameters e is small and f is a nonlinear function

dissipativity of pseudorational behaviors this paper studies dissipativity for a class of infinite dimensional systems called pseudorational in the behavioral context extending the finite dimensional counterpart we show that a pseudorational behavior is dissipative if and only if it admits a storage function or a dissipation function for its proof we derive a new necessary and sufficient condition for entire functions of exponential type in the so called paley wiener class to allow a symmetric factorization characterization of dissipative behaviors and linear quadratic optimal behaviors are also given

structural localization phenomena and the dynamical phase space analogy the localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints a from a modal perspective b from a formulation which allows the amplitude to modulate in an asymptotically defined slow space and c from a dynamical analogy in phase space suggested by the form of the underlying differential equation a simple strut on an asymmetric nonlinear foundation provides a typical illustrative example the three approaches emphasize different features of the localization phenomenon the modal view illustrates the distinctive effects of boundary conditions the modulated approach generates a convenient second order differential equation in the amplitude function and the dynamical phase space analogy suggests a useful interpretation of localization as a homoclinic connection comparisons are also made with nonlinear numerical solutions as the strut length approaches infinity it is shown that the fully localized solution represents the unstable post buckled state with the lowest energy allowing evaluation of the minimum energy barrier relevant to dynamical impact studies attention is drawn to the possibility of spatial chaos becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection

h sub spl infin control in a behavioral context the full information case the authors formulate the h sub spl infin control problem in a behavioral setting given a mathematical model say a set of higher order differential equations together with some static equations the vector of manifest variables is partitioned into yet to be controlled variables unknown exogenous variables and interconnection variables the interconnection variables are available for interconnection in the sense that they can be made to obey certain differential or static equations to be specified by the designer such a system of differential equations and static equations is called a controller the design problem that we consider is to find controllers such that the size of the to be controlled variables is less than a given tolerance for all disturbances in the unit ball and such that the interconnection is a stable system we find necessary and sufficient conditions for the existence of suitable controllers under the hypothesis that we have a full information problem these conditions involve indefinite factorizations of polynomial matrices and a test on a given pick matrix

dissipative systems synthesis a linear algebraic approach abstract in this paper we consider the problem of synthesis of dissipative systems for the case that first and higher order derivatives of the concerned variables also appear in the weighting function the problem is formulated and solved using the behavioral approach to systems and control we relate the problem of weighted h control as a special case of this synthesis problem the synthesis problem and its solution can be systematically understood when one notices that it is similar to finding a non negative subspace non negative with respect to a given constant matrix within a finite dimensional vector space satisfying certain inclusion and dimension constraints

on quadratic differential forms this paper develops a theory around the notion of quadratic differential forms in the context of linear differential systems in many applications we need to not only understand the behavior of the system variables but also the behavior of certain functionals of these variables the obvious cases where such functionals are important are in lyapunov theory and in lq and h infty optimal control with some exceptions these theories have almost invariably concentrated on first order models and state representations in this paper we develop a theory for linear time invariant differential systems and quadratic functionals we argue that in the context of systems described by one variable polynomial matrices the appropriate tool to express quadratic functionals of the system variables are two variable polynomial matrices the main achievement of this paper is a description of the interaction of one and two variable polynomial matrices for the analysis of functionals and for the application of higher order lyapunov functionals

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

global branches of multi bump periodic solutions of the swift hohenberg equation in this paper we study the existence of single and multi bump periodic solutions of a class of fourth order ordinary differential equations arising in problems of pattern formation measuring the tendency to form patterns by a parameter q we view the problem as a nonlinear eigenvalue problem with the use of analytical as well as numerical methods branches of periodic solutions are investigated both locally and globally

periodic solutions for extended fisher kolmogorov and swift hohenberg equations by truncature techniques abstract combining truncature techniques with a variational approach we establish an existence result for nontrivial periodic solutions for a class of fourth order ordinary differential equations involving extended fisher kolmogorov and swift hohenberg equations

a parametrisation for dissipative behaviours the matrix case we study linear time invariant dynamical systems that are dissipative with respect to a generalised power defined by a quadratic differential form we address several cases in an increasing order of complexity and show how dissipative systems can be parametrised in this process we also establish a number of results for polynomial matrices that are of independent interest the present article is a generalisation of our earlier work that dealt with single input single output dissipative systems pendharkar i and pillai h k 2004 a parametrization for dissipative behaviours systems and control letters 51 123 132

a fast recursive algorithm for the discrete sine transform a fast recursive algorithm for the discrete sine transform dst is developed an n point dst can be generated from two identical n 2 point dsts besides being recursive this algorithm requires fewer multipliers and adders than other dst algorithms it can be considered as a generalization of the cooley tukey fft fast fourier transform algorithm the structure of the algorithm is suitable for vlsi implementation

on the prime factor decomposition algorithm for the discrete sine transform presents a fast algorithm for computing the discrete sine transform dst when the transform size n is decomposable into mutually prime factors based on the lee algorithm for the discrete cosine transform dct and from the relations obtained between dst and dct a new set of input and output index mappings for dst is presented

homoclinic solutions for swift hohenberg and suspension bridge type equations we establish the existence of homoclinic solutions for a class of fourth order equations which includes the swift hohenberg model and the suspension bridge equation in the first case the nonlinearity has three zeros corresponding to a double well potential while in the second case the nonlinearity is asymptotically constant on one side the swift hohenberg model is a higher order extension of the classical fisher kolmogorov model its more complicated dynamics give rise to further possibilities of pattern formation the suspension bridge equation as studied by chen and mckenna j differential equations 136 1997 325 355 we give a positive answer to an open question raised by the authors c 2002 elsevier science usa

lossless and dissipative distributed systems this paper deals with linear shift invariant distributed systems by this we mean systems described by constant coefficient linear partial differential equations we define dissipativity with respect to a quadratic differential form i e a quadratic functional in the system variables and their partial derivatives the main result states the equivalence of dissipativity and the existence of a storage function or a dissipation rate the proof of this result involves the construction of the dissipation rate we show that this problem can be reduced to hilbert s 17th problem on the representation of a nonnegative rational function as a sum of squares of rational functions

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

input and output index mappings for a prime factor decomposed computation of discrete cosine transform a formal direct derivation of the prime factor decomposed computation algorithm is presented the derivation is direct in the sense that it is based on the real cosine function without resort to the discrete fourier transform expressions or the complex functions based on the equations obtained from the derivation input and output index mappings are introduced in the form of tables this tabulation enables any prime factor decomposable discrete cosine transform dct to be implemented in a straight forward manner the use of the index mapping tables is demonstrated for the 12 point dct

existence results for nonlinear elliptic equations with leray lions operator and dependence on the gradient the existence of positive solutions for nonlinear elliptic problems under dirichlet boundary condition is studied as well as the compactness and directness of the solution set the main novelties consist in the presence of a leray lions operator in the differential part and in the dependence of the reaction term on the gradient of the solution our approach relies on the method of sub supersolution nonlinear regularity theory and strong maximum principle

a sub supersolution approach for a quasilinear kirchhoff equation in this paper we establish an existence result for a quasilinear kirchhoff equation via a sub and supersolution approach by using the minty browder s theorem for pseudomonotone operators theory

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

positive and negative solutions of a quasi linear elliptic equation by a mountain pass method and truncature techniques abstract the existence of a positive and a negative solution of a problem of the type u f x u u 0 in u 0 on is proved when f has a growth at infinity depending both on u and on u the techniques are based on an iterative scheme of mountain pass approximated solutions and the use of a suitable truncature method

positive solutions for nonlinear elliptic problems with dependence on the gradient abstract we consider a quasilinear neumann problem with a differential operator and a reaction term both dependent on u and du using topological methods together with suitable truncation and comparison techniques we show that the problem has at least one positive smooth solution

a semilinear elliptic pde not in divergence form via variational methods in this paper we consider a semilinear equation driven by an operator not in divergence form precisely the principal part of the operator is in divergence form but it has also a lower order term depending on du while the right hand side of the equation satisfies superlinear and subcritical growth conditions at zero and at infinity the problem has not a variational structure but despite that we use variational techniques in order to prove an existence and regularity result for the equation

module theoretic approach to controllability of convolutional systems both the behavioral approach proposed by willems and the module theoretic approach proposed by fliess provide a representation free theory of dynamical systems according to these approaches controllability is an intrinsic system property independent of the system representation in the first part of this paper a comparison between the behavioral approach and the module theoretic approach to convolutional systems is proposed then controllability of this class of systems is analyzed according to these two approaches this concept is related to the existence of image representations finally the same analysis is proposed for systems described by delay differential equations

prime factor algorithm for multidimensional discrete cosine transform a prime factor fast algorithm is proposed for the computation of the multidimensional forward and inverse discrete cosine transform dct by using an example of a two dimensional 2 d dct it shows that an r dimensional dct can be obtained from a 2r dimensional dct with a post processing stage efficient method for input output mapping is reported to substantially reduce the computational overhead associated with the prime factor algorithm

existence and numerical approximations of periodic solutions of semilinear fourth order differential equations abstract a multiplicity result of existence of periodic solutions with prescribed wavelength for a class of fourth order nonautonomous differential equations related either to the extended fisher kolmogorov or to the swift hohenberg equation is proved variational approach is used some numerical solutions are calculated via the finite element method

homotopy classes for stable connections between hamiltonian saddle focus equilibria for a class of hamiltonian systems in 4 the set of homoclinic and heteroclinic orbits which connect saddle focus equilibria is studied using a variational approach the oscillatory properties of a saddle focus equilibrium and the variational nature of the problem give rise to connections in many homotopy classes of the configuration plane punctured at the saddle foci this variational approach does not require any assumptions on the intersections of stable and unstable manifolds such as transversality moreover these connections are shown to be local minimizers of an associated action functional this result has applications to spatial pattern formation in a class of fourth order bistable evolution equations

pattern selection of solutions of the swift hohenberg equation abstract we study the large time behaviour of solutions u x t of the swift hohenberg equation on a one dimensional domain 0 l focusing in particular on the role of the eigenvalue parameter and the length l of the domain on the selection of the limiting profiles v we show by means of numerical simulations how different values of these parameters may lead to qualitatively different final profiles and prove the existence of a collection n of disjoint intervals which depend on the value of such that if l n then solutions all converge to the trivial solution v 0 we show that branches of nontrivial solutions bifurcate from the trivial solution at the endpoints of these intervals and we study the local behaviour of these branches we establish global bounds for the solutions u x t and identify the global attractor finally we derive an estimate about the shape of the final state

stabilization lyapunov functions and dissipation for linear time invariant systems any stabilizing controller for a given plant can be associated with a supply rate with respect to which the plant in open loop is half line dissipative we also prove the equivalence between the stability of the interconnection of two systems and their half line dissipativity with respect to a supply rate and its negative

a dissipation inequality for the minimum phase property the minimum phase property is an important notion in systems and control theory in this paper a characterization of the minimum phase property of nonlinear control systems in terms of a dissipation inequality is derived it is shown that this dissipation inequality is equivalent to the classical definition of the minimum phase property in the sense of byrnes and isidori if the control system is affine in the input and the so called input output normal form exists

synthesis of dissipative systems using quadratic differential forms part ii the problem discussed is that of designing a controller for a linear system that renders a quadratic functional nonnegative our formulation and solution of this problem is completely representation free the system dynamics are specified by a differential behavior and the performance is specified through a quadratic differential form we view control as interconnection a controller constrains a distinguished set of system variables the control variables the resulting behavior of the to be controlled variables is called the controlled behavior the constraint that the controller acts through the control variables only can be succinctly expressed by requiring that the controlled behavior should be wedged in between the hidden behavior obtained by setting the control variables equal to zero and the plant behavior obtained by leaving the control variables unconstrained the main result is a set of necessary and sufficient conditions for the existence of a controlled behavior that meets the performance specifications the essential requirement is a coupling condition an inequality that combines the storage functions of the hidden behavior and the orthogonal complement of the plant behavior

minimum phase infinite dimensional second order systems in general better performance can be achieved with a controlled minimum phase system than a controlled nonminimum phase system we show that a wide class of second order infinite dimensional systems with either velocity or position measurements are minimum phase the results are illustrated by two examples

quasilinear equations with dependence on the gradient via mountain pass techniques in rn abstract we consider the quasilinear problem p u u p 2 u f u u p 2 u in r n where 1 p n and f is a nonlinearity depending also on the gradient of the solution the existence of a positive solution is stated through an iterative method based on mountain pass techniques