a critical p biharmonic system with negative exponents abstract in this paper we are devoted to the study of critical p biharmonic system with a parameter which involves strongly coupled critical nonlinearities and negative exponents we prove that there exists a positive constant such that the above problem admits at least two solutions if 0

three solutions for a dirichlet boundary value problem involving the p laplacian abstract in this paper we prove the existence of at least three weak solutions for the problem p u f x u a x u p 2 u in u 0 on our main tool is a recent three critical points theorem of b ricceri on a three critical points theorem arch math basel 75 2000 220 226

grassmannian central projection and output feedback pole assignment of linear systems a sufficient rank condition for arbitrary pole assignability by static output feedback of m input p output linear systems of mcmillan degree n mp is given this condition is easy to check and is satisfied by the generic system an effective computational method to compute the feedback laws assigning any self conjugate set of closed loop poles is also given

assigning frequencies via determinantal equations new counterexamples and invariants the problem of arbitrary pole zero assignment by fixed structure compensators has been addressed so far in terms of either necessary or sufficient conditions the strongest existing condition is based on the rank of the differential of a related map on a degenerate controller and holds true generically when the degrees of freedom of the compensator exceed the number of frequencies to be assigned mp n for the output feedback problem the complete nongeneric solvability of the problem is still open even when complex controllers are considered a simple necessary solvability condition is that the linear map appearing as a factor of the main determinantal map defining the assignment problem is onto here we examine the special problem of assignment of matrix pencil zeros via diagonal perturbations and we present a new necessary and sufficient condition for complex solvability in terms of a new invariant involving the minors of this linear map based on this result we demonstrate that for the important case where the degrees of freedom of the controller are equal to the number of frequencies to be assigned the surjectivity of this linear map although it is necessary it is not sufficient for the solvability of the problem

pole placement by static output feedback for generic linear systems we consider linear systems with m inputs p outputs and mcmillan degree n such that n mp if both m and p are even we show that there is a nonempty open in the usual topology subset u of such systems where the real pole placement map is not surjective it follows that for each system in u there exists an open set of pole configurations symmetric with respect to the real line which cannot be assigned by any real static output feedback

some remarks on real and complex output feedback we provide some new necessary and sufficient conditions which guarantee arbitrary pole placement of a particular linear system over the complex numbers we exhibit a nontrivial real linear system which is not controllable by real static output feedback and discuss a conjecture from algebraic geometry concerning the existence of real linear systems for which all static feedback laws are real

a minimax inequality and its applications to ordinary differential equations the aim of this paper is to investigate the minimax inequality which plays a fundamental role in the critical points theorem of b ricceri below equivalent formulations are shown and characterization is proved in particular for a special class of functionals as an application a multiplicity result for an ordinary dirichlet problem is emphasized

some results on the genericity of the pole placement problem abstract in this note the pole placement problem for a linear mimo systems with p outputs and m inputs is studied from the algebraic point of view a formulation is proposed that allows to analyze both theoretical and numerical aspects of the case min m p 2 with more sharpness moreover here it is shown that arbitrary pole placement by static output feedback of unitary rank is generically not possible even if m p n holds true

a new sufficient condition for arbitrary pole placement by real constant output feedback abstract a new sufficient condition for generic pole assignment is given based on the height of the first whitney class of a certain grassmannian this result contains the condition m p 1 n and is stronger than lscat p m n this test is equivalent to the factorial test in 7 but it has the advantage that it is testable for most of the cases

a three critical points theorem revisited abstract in this paper the following result is proved let x be a reflexive real banach space i r an interval x r a sequentially weakly lower semicontinuous c 1 functional bounded on each bounded subset of x whose derivative admits a continuous inverse on x j x r a c 1 functional with compact derivative assume that lim x x j x for all i and that there exists r such that sup i inf x x x j x inf x x sup i x j x then there exist a non empty open set a i and a positive real number r with the following property for every a and every c 1 functional x r with compact derivative there exists 0 such that for each 0 the equation x j x x 0 has at least three solutions in x whose norms are less than r

multiplicity theorems for the dirichlet problem involving the p laplacian abstract in this paper we establish some results on the existence of at least three weak solutions for dirichlet problems involving the p laplacian by a variational approach

least squares pole assignment by memoryless output feedback in this paper a pole assignment problem for a linear time invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed the cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum this approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters the relation between the modified cost functions and the original pole assignment problem is revealed the proposed approach is compared with other existing approaches and illustrated by numerical results

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

on a p biharmonic problem with no flux boundary condition the study of fourth order partial differential equations has flourished in the last years however a p biharmonic problem with no flux boundary condition has never been considered before not even for constant p this is an important step further since surfaces that are impermeable to some contaminants are appearing quite often in nature hence the significance of such boundary condition by relying on several variational arguments we obtain the existence and the multiplicity of weak solutions to our problem we point out that although we use a mountain pass type theorem in order to establish the multiplicity result we do not impose an ambrosettirabinowitz type condition nor a symmetry condition on our nonlinearity f

three solutions for a class of quasilinear elliptic systems involving the p q laplacian abstract the existence of at least three weak solutions is established for a class of quasilinear elliptic systems involving the p q laplacian with dirichlet boundary condition our technical approach is based on the three critical points theorem obtained by b ricceri

on linear solutions of the output feedback pole assignment problem a new linear method for solving the output feedback pole assignment problem of linear systems is introduced and new sufficient conditions are obtained

new results in pole assignment by real output feedback this paper considers the problem of tuning natural frequencies of a linear system by a memoryless controller using algebro geometric methods it is shown how it is possible to improve current sufficiency conditions the main result is an exact combinatorial characterization of the nilpotency index of the bmod 2 cohomology ring of the real grassmannian using this characterization new sufficiency results for generic pole assignment for the linear system with m inputs p outputs and mcmillan degree n are given among other results it is shown that 2 25 cdot max m p min m p 3 geq n is a sufficient condition for generic real pole placement provided min m p geq 4

generic eigenvalue assignment by memoryless real output feedback by extensive use of methods from algebraic geometry x wang proved that arbitrary pole placement by static output feedback is generically possible for strictly proper plants with n states m inputs and p outputs if n mp here we show the same result using no more of algebraic geometry than the definition of genericity

some remarks on a three critical points theorem abstract some remarks on a strict minimax inequality which plays a fundamental role in ricceri s three critical points theorem are presented as a consequence some recent applications of ricceri s theorem to nonlinear boundary value problems are revisited by obtaining more precise conclusions

pole placement via static output feedback is np hard this note proves that the problem of pole placement via static output feedback for linear time invariant systems is np hard

multiplicity results for solutions of p biharmonic problems abstract in this paper we deal with the multiplicity existence of solutions for the following p biharmonic equation p 2 u u p 2 u u q 2 u in u u 0 on where is a bounded domain in r n p 2 u u p 2 u p q p n p n 2 p r is a parameter when p q p we prove that the above problem possesses infinitely many solutions while when q p a multiplicity existence result is obtained

existence and multiplicity of solutions for fourth order boundary value problems with parameters abstract in this paper we study the existence and multiplicity of the solutions for the fourth order boundary value problem bvp u 4 t u t u t f t u t 0 t 1 u 0 u 1 u 0 u 1 0 where f 0 1 r r is continuous r and r are parameters by means of the idea of the decomposition of operators shown by chen w y chen a decomposition problem for operators xuebao of dongbei renmin university 1 1957 95 98 see also m krasnosel skii topological methods in the theory of nonlinear integral equations gostehizdat moscow 1956 and the critical point theory we obtain that if the pair is on the curve 2 4 satisfying 2 2 then the above bvp has at least one two three and infinitely many solutions for being in different interval respectively

complex static skew symmetric output feedback control we study the problem of feedback control for skew symmetric and skew hamiltonian transfer functions using skew symmetric controllers this extends work of helmke et al who studied static symmetric feedback control of symmetric and hamiltonian linear systems we identify spaces of linear systems with symmetry as natural subvarieties of the moduli space of rational curves in a grassmannian give necessary and sufficient conditions for pole placement by static skew symmetric complex feedback and use schubert calculus for the orthogonal grassmannian to count the number of complex feedback laws when there are finitely many of them finally we also construct a real skew symmetric linear system with only real feedback for any set of real poles

output feedback pole assignment for transfer functions with symmetries this paper studies the problem of pole assignment for symmetric and hamiltonian transfer functions a necessary and sufficient condition for pole assignment by complex symmetric output feedback transformations is given moreover in the case where the mcmillan degree coincides with the number of parameters appearing in the symmetric feedback transformations we derive an explicit combinatorial formula for the number of pole assigning symmetric feedback gains the proof uses intersection theory in projective space as well as a formula for the degree of the complex lagrangian grassmann manifold

nonuniformly nonlinear elliptic equations of p biharmonic type this paper deals with the existence and multiplicity of weak solutions to nonlinear differential equations involving a general p biharmonic operator in particular p biharmonic operator under dirichlet boundary conditions or navier boundary conditions our method is mainly based on variational arguments

existence of three solutions for a quasilinear two point boundary value problem abstract in this paper we investigate a quasilinear second order differential equation with dirichlet boundary conditions the existence of an open interval of parameters which ensures this problem admits at least three solutions is determined by using the critical point theory

output feedback eigenstructure assignment using a new reduced orthogonality condition this paper presents a new method of output feedback eigenstructure assignment a new reduced orthogonality condition is derived which is less restrictive on the design degrees of freedom than others in the literature from this a novel general formulation for the gain matrix is admitted which utilizes a two stage design process in the first stage a subset of desired eigenvectors is assigned either left or right in the second stage a dual set is assigned which adheres to the necessary and sufficient conditions for output feedback eigenstructure assignment proved earlier in the paper a numerical example is presented to illustrate the whole procedure

a necessary condition for pole assignment by constant output feedback in this paper we shall investigate the pole assignment problem by constant output feedback using transfer function techniques in particular a necessary condition is given for the existence of constant output feedback which assigns the eigenvalues of closed loop system to any desired positions

multiple nontrivial solutions for some fourth order semilinear elliptic problems abstract in this paper we consider the existence of multiple nontrivial solutions for some fourth order semilinear elliptic boundary value problems the weak solutions are sought by means of morse theory and local linking

global asymptotic linearisation of the pole placement map a closed form solution for the constant output feedback problem abstract the problem of pole assignment by constant output feedback controllers is studied for minimal systems described by a proper transfer function matrix g s r m p s with mcmillan degree n a new method is presented based on asymptotic linearisation around a degenerate point of the pole placement map related to the problem the essence of the present approach is to reduce the multilinear nature of the problem to one of solving a linear set of equations and this is achieved without losing any of the degrees of freedom in the controller the solution is given in closed form in terms of a one parameter family of static feedback compensators for which the closed loop poles approach the required ones as 0 conditions for the generic as well as exact solvability of the arbitrary pole placement problem are given in terms of the numbers m p n and the rank of a new system invariant the d restricted plucker matrix it is shown that the method works generically when mp n which along with the boundary case mp n is the best possible condition as far as the number of states of the open loop system is concerned for achieving arbitrary pole placement via constant output feedback

existence of three solutions for p q biharmonic systems abstract in this paper we establish the existence of at least three solutions to a navier boundary problem involving the p q biharmonic systems the technical approach is mainly based on a three critical points theorem of b ricceri

stabilizability of multivariable systems and the ljusternik snirel mann category of real grassmannians abstract in this paper a numeric criterion k m p n is given for stabilizability by constant gain output feedback of the generic linear multivariable system with m inputs n states and p outputs this criterion is defined in terms of a topological invariant of the space of gains both finite and infinite arising from an interpretation of the problem of stabilizability as a problem concerning open covers of this space this invariant originally considered by ljusternik and snirel mann in the calculus of variations can be estimated from below by using a theorem of eilenberg together with the methods of the schubert calculus thus leading to some very explicit corollaries concerning generic stabilizability as far as i am aware these corollaries cannot be derived from existing results concerning pole assignability after passing to a problem on coverings of the compactified gain space the main technical problem which remains in a high gain stability analysis stated here as a high gain lemma which appears to be of independent interest that is the topological argument implies that the closed loop system is stable for some gain possibly infinite if the root locus map were defined and continuous at all infinite gains then the conclusion that the system is stabilizable by finite gain could be deduced from a simple continuity argument however it is known that if mp n then the root locus map has points of discontinuity at infinity and since mp n is now known to be necessary for generic stabilizability in most cases of interest one requires a far more subtle argument than one would give for example in the scalar case

static output feedback pole placement via a trust region approach this technical note presents two closely related algorithms for the problem of pole placement via static output feedback the algorithms are based on two different trust region methods and utilize the derivatives of the closed loop poles extensive numerical experiments show the effectiveness of the algorithms in practice though convergence to a solution is not guaranteed for either algorithm while desired poles must be distinct strategies for dealing with repeated poles are also presented

on some fourth order semilinear elliptic problems in r n the existence of two solutions for a fourth order semilinear elliptic problem involving critical growth from the viewpoint of sobolev embedding was established the basic tools used in the analysis were the mountain pass theorem constrained minimization and concentration compactness principle the existence of positive principle eigenvalues for the corresponding linear elliptic problem was also investigated

infinitely many solutions for a perturbed nonlinear navier boundary value problem involving the p biharmonic abstract by using critical point theory we establish the existence of infinitely many weak solutions for a class of elliptic navier boundary value problems depending on two parameters and involving the p biharmonic operator

multiplicity results for perturbed fourth order kirchhoff type elliptic problems using variational methods and critical point theory we establish multiplicity results of nontrivial and nonnegative solutions for a perturbed fourth order kirchhoff type elliptic problem

multivariable nyquist criteria root loci and pole placement a geometric viewpoint classical control theory is concerned with the topics in our title in the context of single input single output systems there is now a large and growing literature on the extension of these ideas to the multiinput multioutput case this development has posed certain difficulties some due to the intrinsic nature of the problem and some we would argue due to an inadequate reflection on what the multivariable problem calls for in this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory a rather satisfactory generalization of the nyquist criterion will be described and a clear analog of the asymptotic properties of the root locus will be obtained in the multiparameter case however an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains this calls for a new ingredient a compactification of the space of gains and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement this problem is difficult in the multivariable case but by establishing a correspondence with a classical set of problems in geometry we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field

a boundary value problem for fourth order elastic beam equations multiplicity results for a fourth order nonlinear boundary value problem are presented the proof of the main result is based on the critical point theory

on a fourth order elliptic equation with critical nonlinearity in dimension six abstract this paper is devoted to the study of the following nonlinear elliptic problem 2 u ku 5 u 0 in u u 0 on where k is a positive function and is a bounded and smooth domain in r 6 we prove a version of morse lemma at infinity for this problem using this morse lemma at infinity we characterize the critical points at infinity of the associated functional and we give an existence result

linear output feedback compensation of rank one systems a new method is presented for arbitrarily assigning the closed loop poles of a linear multi input multi output system using linear output feedback the method employs a new notion of rank one system representations and involves the use of state transformation matrices which preserve such representations while reducing the degree of the feedforward matrix

on pole placement via eigenstructure assignment approach this note comes back to the hard problem of pole placement by static output feedback let a triplet of matrices a b c be given with n state variables m inputs and p ouputs find a matrix k such that the spectrum of a bkc equals a specified set when mp m p a simple noniterative technique based upon the notion of eigenstructure that in most cases assigns m p roots is proposed it therefore enables the designer to assign the whole of the desired spectrum when m p n mp

three solutions for a navier boundary value problem involving the p biharmonic the existence of at least three weak solutions is established for a class of navier boundary value problem involving the p biharmonic our technical approach is based on the three critical points theorem obtained by b ricceri

a sufficient condition for arbitrary eigenvalue assignment in linear descriptor systems by output feedback the problem of eigenvalue assignment in the linear descriptor system e mathdotx ax bu y cx via output feedback is considered it is shown that mp rank e m and p are respectively the numbers of inputs and outputs of the system is a sufficient condition for generic real output feedback eigenvalue assignability the result is the best possible for general m p rank e in general the assignment is achieved not in an exact sense but in arbitrarily small neighborhoods around a given set of eigenvalues

counterexamples to pole placement by static output feedback abstract we consider linear systems with inner state of dimension n with m inputs and p outputs such that n mp min m p 2 and max m p is even we show that for each m n p satisfying these conditions there is a non empty open subset u of such systems where the real pole placement map is not surjective it follows that for systems in u there exist open sets of pole configurations which cannot be assigned by any real output feedback

eigenvalue assignment by static output feedback on a new solvability condition and the computation of low gain feedback matrices in this article the static output feedback problem for linear time invariant systems is considered for arbitrary assignability of the roots of the characteristic polynomial by static output feedback a new necessary and sufficient condition is derived although the proof is based on simple analysis the known sufficient conditions derived by techniques of algebraic geometry are directly covered furthermore an algorithm for the calculation of feedback matrices assigning a desired set of eigenvalues is proposed this algorithm does not require the desired eigenvalues to be distinct and it explicitly exploits the available degrees of freedom for reducing the feedback gain the presented approach is illustrated on computational examples

output feedback pole placement with dynamic compensators derives a new rank condition which guarantees the arbitrary pole assignability of a given system by dynamic compensators of degree at most q by using this rank condition the authors establish several new sufficiency conditions which ensure the arbitrary pole assignability of a generic system the authors proofs also come with a concrete numerical procedure to construct a particular compensator which assigns a given set of closed loop poles

multiple solutions for p laplacian type equations abstract in this paper we establish the existence of three weak solutions of an equation which involves a general elliptic operator in divergence form in particular a p laplacian operator while the nonlinearity has a p 1 sublinear growth at infinity this result completes some recent papers where mountain pass type solutions were obtained providing the nonlinear term via a p 1 superlinear growth at infinity fulfilling an ambrosetti rabinowitz type condition in our case an abstract critical point result is applied proved by g bonanno g bonanno some remarks on a three critical points theorem nonlinear analysis 54 2003 651 665

numerical solution of the schrodinger equation in a wavelet basis for hydrogen like atoms an iterative method is proposed to solve the schrodinger eigenvalue problem in a wavelet framework orthonormal wavelets are used to represent the corresponding operator as a sparse band matrix this representation called the nonstandard ns form is obtained by means of the beylkin coifman rokhlin bcr algorithm and simplifies the numerical calculations problems due to the one dimensional mathematical model and to the discretization process receive special attention