does it suffice to check a subset of multilinear parameters in robustness analysis for polynomials with general multilinear dependence on uncertain real parameters it does not suffice to check a subset for q for stability

an approach to robust stability of matrix polytopes through copositive homogeneous polynomials an approach to guaranteeing the stability of a polytope of matrices is proposed previous results in the literature have made the obvious connection between various robust stability problems and a test for positivity of a multivariable polynomial such results are extended in order to demonstrate that all matrices within a polytope are stable is and only if an associated homogeneous polynomial is strictly copositive the additional structure obtained by exploiting homogeneity of this multivariable polynomial leads to several computationally tractable sufficiency tests for establishing either robust stability or instability of a polytope of matrices

the jacobian determinant revisited this paper is devoted to the study of the jacobian determinant of a map g from a smooth bounded open subset of rn into rn n 2 more generally could be a smooth bounded open subset of an n dimensional manifold starting with the seminal work of c b morrey 27 y reshetnyak 32 and j ball 1 it has been known that one can define the distributional jacobian determinant det g under fairly weak assumption on g in particular it is defined for all maps g w 1 n2 n 1 and also for all

some counterexamples in robust stability theory this note deals with the problem of determining if a linear system whose characteristics polynomial depends multilinearly on n independent uncertain real parameters i i 1 n is robustly stable it is shown by example that a polynomial in n variables may have a unique real root and that this observation disposes of several natural conjectures in robust stability theory in particular we show that in a certain sense there are no edge or m dimensional face kharitonov like theorems for the general multilinear case the result holds even when restricted to that subset of multilinear functions which can be written in the form f 1 n det i diag 1 n m for some complex matrix m

lyapunov schmidt reduction algorithm for three dimensional discrete vortices abstract we address the persistence and stability of three dimensional vortex configurations in the discrete nonlinear schrodinger equation and develop a symbolic package based on wolfram s mathematica for computations of the lyapunov schmidt reduction method the lyapunov schmidt reduction method is a theoretical tool which enables us to study continuations and terminations of the discrete vortices for small coupling between lattice nodes as well as the spectral stability of the persistent configurations the method was developed earlier in the context of the two dimensional lattice and applied to the onsite and offsite configurations called the vortex cross and the vortex cell by using semianalytical computations d e pelinovsky p g kevrekidis d frantzeskakis physica d 212 2005 20 53 p g kevrekidis d e pelinovsky proc r soc a 462 2006 2671 2694 the present treatment develops a full symbolic computational package which takes a desired waveform at the anticontinuum limit of uncoupled sites performs a required number of lyapunov schmidt reductions and outputs the predictions on whether the configuration persists for finite coupling in the three dimensional lattice and whether it is stable or unstable it also provides approximations for the eigenvalues of the linearized stability problem we report a number of applications of the algorithm to important multisite three dimensional configurations such as the simple cube the double cross and the diamond for each configuration we identify exactly one solution which is stable for small coupling between lattice nodes

polynomially parameter dependent lyapunov functions for robust stability of polytopic systems an lmi approach in this note robust stability of state space models with respect to real parametric uncertainty is considered specifically a new class of parameter dependent quadratic lyapunov functions for establishing stability of a polytope of matrices is introduced i e the homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs the choice of this class which contains parameter dependent quadratic lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form is motivated by the property that a polytope of matrices is stable if and only there exists an hpd qlf the main result of the note is a sufficient condition for determining the sought hpd qlf which amounts to solving linear matrix inequalities lmis derived via the complete square matricial representation csmr of homogeneous matricial forms and the lyapunov matrix equation numerical examples are provided to demonstrate the effectiveness of the proposed approach

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

the minimal dimension of stable faces required to guarantee stability of a matrix polytope considers the problem of determining whether each point in a polytope n n matrices is stable the approach is to check stability of certain faces of the polytope for n or 3 the authors show that stability of each point in every 2n 4 dimensional face guarantees stability of the entire polytope furthermore they prove that for any k

lifting for manifold valued maps of bounded variation abstract let n be a smooth compact connected riemannian manifold without boundary let e n be the riemannian universal covering of n for any bounded smooth domain r d and any u bv n we show that u has a lifting v bv e our result proves a conjecture by bethuel and chiron

density of smooth functions between two manifolds in sobolev spaces abstract we give some results concerning density or non density of smooth functions between two compact manifolds m n and n k in sobolev spaces w 1 p m n n k in particular we study the case n s k

sharp commutator estimates via harmonic extensions abstract we give an alternative proof of several sharp commutator estimates involving riesz transforms riesz potentials and fractional laplacians our methods only involve harmonic extensions to the upper half space integration by parts and trace space characterizations the commutators we investigate are jacobians more generally coifman rochberg weiss commutators chanillo s commutator with the riesz potential coifman mcintosh meyer kato ponce vega type commutators the fractional leibniz rule and the da lio riviere three term commutators we also give a new limiting l 1 estimate for a double commutator of coifman rochberg weiss type and several intermediate estimates the beauty of our method is that all those commutator estimates which are originally proven by various specific methods or by general para product arguments can be obtained purely from integration by parts and trace theorems another interesting feature is that in all these cases the cancellation effect responsible for the commutator estimate simply follows from the product rule for classical derivatives and can be traced in a precise way

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

necessary and sufficient conditions for stability of a matrix polytope with normal vertex matrices abstract it is shown that the eigenvalue set of a matrix polytope with normal vertex matrices is a subset of the convex hull of eigenvalues of the vertex matrices

improved partial regularity for manifold constrained minimisers of subquadratic energies we consider minimising p harmonic maps from three dimensional domains to the real projective plane for 1 p 2 these maps arise as least energy configurations in variational models for nematic liquid crystals we show that the singular set of such a map decomposes into a 1 dimensional set which can be physically interpreted as a non orientable line defect and a locally finite set i e a collection of point defects

robust d stabilization of a polytope of matrices the problem of stabilization of a polytope of matrices in a subregion d r of the complex plane is revisited a new sufficient condition of robust d r stabilization is given it implies the solution of an lmi involving matrix variables constrained by a non linear algebraic relation some lmi relaxations are first proposed then it is shown that a cone complementarity formulation of this condition allows us to associate an efficient iterative numerical procedure which leads to a low computational burden this algorithm is tested on different numerical examples for which existing approaches in control literature fail

stability of discrete solitons in nonlinear schrodinger lattices we consider the discrete solitons bifurcating from the anti continuum limit of the discrete nonlinear schrodinger nls lattice the discrete soliton in the anti continuum limit represents an arbitrary finite superposition of in phase or anti phase excited nodes separated by an arbitrary sequence of empty nodes by using stability analysis we prove that the discrete solitons are all unstable near the anti continuum limit except for the solitons which consist of alternating anti phase excited nodes we classify analytically and confirm numerically the number of unstable eigenvalues associated with each family of the discrete solitons

some open problems in matrix theory arising in linear systems and control control theory has long provided a rich source of motivation for developments in matrix theory accordingly we discuss some open problems in matrix theory arising from theoretical and practical issues in linear systems theory and feedback control the problems discussed include robust stability matrix exponent induced norms stabilizability and pole assignability and nonstandard matrix equations a substantial number of references are included to acquaint matrix theorists with problems and trends in this application area feedback control theory has long provided a rich source of motivation for developments in matrix theory the purpose of this paper is to discuss several open problems in matrix theory that arise from theoretical and practical issues in feedback control theory and the associated area of linear systems theory many of these problems are remarkably simple to state are of intense interest in control theory and applications and yet remain unsolved besides leading to the resolution of these problems it is hoped that this paper will help to stimulate increased interaction between matrix and control theorists accord

on the spectrum of convex sets of matrices let k be an arbitrary compact convex set of square matrices this paper presents necessary and sufficient conditions for the spectrum of k to have no eigenvalues in a prespecified closed convex subset of the complex plane the obtained result implies different criteria for analysis of the spectral set of k in particular we have formulated criteria for nonsingularity inertia and hurwitz stability of k which can be used in the robustness analysis of linear control systems with uncertain parameters

a necessary and sufficient condition for robust asymptotic stability of time variant discrete systems a necessary and sufficient condition for the stability of time variant interval matrices is presented this condition allows stability to be tested by checking only products of vertex extreme matrices the implementation of the test in the form of an algorithm and two illustrative examples are provided

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

state space measures of robustness of pole locations for structured and unstructured perturbations abstract measures of pole location robustness for linear feedback systems are derived from a state space model of the system the robustness tests ensure that the eigenvalues of the perturbed systems matrix a e remain in a desired region d of the complex plane containing the eigenvalues of the nominal system matrix a the region d may be any open set of the complex plane whatsoever the results are expressed in terms of induced matrix norms and apply to structured perturbations of the form e b c where b and c define the structure of e and may be nonsquare matrices rank one perturbations e of minimal norm and with the given structure that will cause a e to have an eigenvalue outside of d are constructed for the cases when the matrix norm is induced by the vector 1 norm or the vector norm the advantages of having robustness measures for several matrix norms that can be computed are illustrated with a simple example that demonstrates how the conservatism of single tests can be reduced using several tests i e several matrix norms a method for computing numerically the robustness measures for particular norms is presented it can be used to compute with a guaranteed degree of accuracy the maximum of the norm of the frequency response of a system

on the distributional jacobian of maps from sn into sn in fractional sobolev and holder spaces h brezis and l nirenberg proved that if gk c 0 s n s n and g2 c 0 s n s n n 1 are such that gk g in bmo s n then deggk degg on the other hand if g 2 c 1 s n s n then kronecker s formula asserts that degg 1 jsnj r sn det rg d consequently r sn det rgk d converges to r sn det rg d provided gk g in bmo s n in the same spirit we consider the quantity j g r sn det rg d for all 2 c 1 s n r and study the convergence of j gk in particular we prove that j gk converges to j g for any 2 c 1 s n r if gk converges to g in c 0 s n for some n 1 n surprisingly this result is optimal when n 1 in the case n 1 we prove that if gk g almost everywhere and lim supk 1jgk gjbmo is suciently small then j gk j g for any 2 c 1 s 1 r we also establish bounds for j g which are motivated by the works of j bourgain h brezis and h m nguyen and h m nguyen we pay special attention to the case n 1

lifting in besov spaces abstract let be a smooth bounded simply connected domain in r n and let u be a complex valued measurable function on such that u x 1 a e assume that u belongs to a besov space b p q s we investigate whether there exists a real valued function b p q s r such that u e i this complements the corresponding study in sobolev spaces due to bourgain brezis and the first author the microscopic parameter q turns out to play an important role in some limiting situations the analysis of this lifting problem relies on some interesting new properties of besov spaces in particular a non restriction property when q p

the homological singularities of maps in trace spaces between manifolds we deal with mappings defined between riemannian manifolds that belong to a trace space of sobolev functions the homological singularities of any such map are represented by a current defined in terms of the boundary of its graph under suitable topological assumptions on the domain and target manifolds we show that the non triviality of the singular current is the only obstruction to the strong density of smooth maps moreover we obtain an upper bound for the minimal integral connection of the singular current that depends on the fractional norm of the mapping

lifting degree and distributional jacobian revisited let g i 0 1 s1 if g vmo we may write g ei for some vmo this is unique modulo 2 see 13 and the earlier work 14 there is no control of bmo in terms of g bmo since we always have g bmo 2 and bmo can be arbitrarily large recall however that when g bmo is sufficiently small there is a linear estimate bmo c g bmo see 13 theorem 4 14 and remark 0 2 below we are going to establish that a norm slightly stronger than g bmo does control bmo consider for 1 p 0 s 1 the fractional sobolev space ws p i equipped with its standard seminorm

robust stability under a class of nonlinear parametric perturbations this paper considers the robust stability verification of linear time invariant systems admitting a class of nonlinear parametric perturbations the general setting is one of determining the closed loop stability of systems whose open loop transfer functions consist of powers products and ratios of polytopes of polynomials apart from this general setting two special cases of independent interest are also considered the first special case concerns uncertainties in the open loop gain and real poles and zeros while the second special case treats uncertainties in the open loop gain and complex poles and zeros in light of the zero exclusion principle robust stability is equivalent to zero exclusion of the value sets of the system characteristic function a value set consists of the values of the characteristic functions at a fixed frequency the main results of this paper are as follows 1 the value set of the characteristic function at each fixed frequency is determined by the edges and some frequency dependent internal line segments 2 consequently hurwitz invariance verification simplifies to that of checking certain continuous scalar functions for avoidance of the negative real axis 3 for the case of real zero pole gain variations the critical lines are all frequency independent and therefore the determination of the robust stability is even simpler 4 for the case of complex zero pole gain variations the critical internal lines are shown to be either frequency independent or to be confined in certain two dimensional planes or three dimensional boxes

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