the p cover problem abstract the modified p center called here p cover problem was discussed recently in several papers a practical and effective procedure to solving it is presented in this note

algebraic theory for robust stability of interconnected systems necessary and sufficient conditions we consider an interconnected system s o made of linear mulrivariable subsystems which are specified by matrix fractions with elements in a ring of stable scalar transfer functions h given that the k th subsystem is perturbed from g k n rk d k 1 to tilde g k n rk delta n rk d k delta d k 1 and that the system s o is h stable we derive a computationally efficient necessary and sufficient condition for the h stability of the perturbed system these fractional perturbations are more general than the conventional additive and multiplicative perturbations the result is generalized to handle simultaneous perturbations of two or more subsystems

chains of darboux transformations for the matrix schrodinger equation chains of darboux transformations for the matrix schr odinger equation are considered a matrix generalization of the well known for the scalar equation crum krein formulae for the resulting action of such chains is given

inverse problem for harmonic oscillator perturbed by potential characterization consider the perturbed harmonic oscillator ty y x2y q x y in l2 where the real potential q belongs to the hilbert space h q xq l2 the spectrum of t is an increasing sequence of simple eigenvalues n q 1 2n n n 0 such that n 0 as n let n x q be the corresponding eigenfunctions define the norming constants n q lim x log n x q n x q we show that open image in new window for some real hilbert space open image in new window and some subspace open image in new window furthermore the mapping q q n q 0 n q 0 is a real analytic isomorphism between h and open image in new window is the set of all strictly increasing sequences s s n 0 such that open image in new window the proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials we use ideas from the analysis of the inverse problem for the operator y py p l2 0 1 with dirichlet boundary conditions on the unit interval there is no literature about the spaces open image in new window we obtain their basic properties using their representation as spaces of analytic functions in the disk

design of multivariable feedback systems with stable plant this paper considers in a general algebraic framework the design of a unity feedback multivariable system with a stable plant the method is based on a simple parameterization of the four closed loop transfer functions in terms of p the plant transfer function and q h y 1 u 1 in particular the i o transfer function q h y 2 u 1 pq using the framework of rational transfer functions we show that the closed loop system will be exponentially stable if and only if q is exponentially stable furthermore if both p and q are strictly proper then the controller is also strictly proper the basic result is design theorem 2 an algorithm is given for obtaining strictly proper controllers such that the resulting i o map is decoupled all its poles can be chosen by the designer and the same holds for zeros except of course for the c zeros prescribed by the c zeros of the plant a discussion is included to temper these results by the constraints imposed by noise and plant saturation

inverse spectral problems and closed exponential systems consider the inverse eigenvalue problem of the schr odinger operator de fined on a finite interval we give optimal and almost optimal conditions for a set of eigenvalues to determine the schr odinger operator these conditions are simple closedness properties of the exponential system corresponding to the known eigenvalues the statements contain nearly all former results of this topic we give also conditions for recovering the weyl titchmarsh m function from its values m en

linear stable unity feedback system necessary and sufficient conditions for stability under nonlinear plant perturbations the authors consider a linear not necessarily time invariant stable unity feedback system where the plant and the compensator have normalized right coprime factorizations they study two cases of nonlinear plant perturbations additive and feedback with four subcases resulting from 1 allowing exogenous input to delta p or not 2 allowing the observation of the output of delta p or not the plant perturbation delta p is not required to be stable using the factorization approach the authors obtain necessary and sufficient conditions for all cases in terms of two pairs of nonlinear pseudostate maps simple physical considerations explain the form of these necessary and sufficient conditions finally the authors obtain the characterization of all perturbations delta p for which the perturbed system remain stable

trace formulae for the matrix schrodinger equation with energy dependent potential abstract in this paper we consider the eigenvalue problems for the matrix schrodinger equation with energy dependent potential and with separated boundary conditions on the finite interval and find new trace formulae for the matrix schrodinger operator

inverse spectral theory for a singular sturm liouville operator on 0 1 xer3 acting on functions in p 1x1 1 vanishing on 1x1 1 we set x 1x1 and assume q x is a real valued function in l 0 11 the closure h of i is a self adjoint operator with compact resolvent hence h has pure point spectrum without finite accumulation points moreover using the mapping u x xu x and expansion in spherical harmonics one shows that h is unitarily equivalent to a direct sum of sturm liouville operators acting on l o 11 cf 11 the first operator in this sum is

weyl titchmarsh functions of vector valued sturm liouville operators on the unit interval the matrix valued weyl titchmarsh functions m of vector valued sturm liouville operators on the unit interval with the dirichlet boundary conditions are considered the collection of the eigenvalues i e poles of m and the residues of m is called the spectral data of the operator the complete characterization of spectral data or equivalently n n weyl titchmarsh functions corresponding to n n self adjoint square integrable matrix valued potentials is given if all n eigenvalues of the averaged potential are distinct

borg type theorems for matrix valued schrodinger operators abstract a borg type uniqueness theorem for matrix valued schrodinger operators is proved more precisely assuming a reflectionless potential matrix and its spectrum a half line 0 we derive the triviality of the potential matrix our approach is based on trace formulas and matrix valued herglotz representation theorems as a by product of our techniques we obtain an extension of borg s classical result from the class of periodic scalar potentials to the class of reflectionless matrix valued potentials

inverse spectral analysis with partial information on the potential ii the case of discrete spectrum we discuss results where the discrete spectrum or partial information on the discrete spectrum and partial information on the potential q of a one dimensional schrodinger operator h d 2 dx 2 q determine the potential completely included are theorems for finite intervals and for the whole line in particular we pose and solve a new type of inverse spectral problem involving fractions of the eigenvalues of h on a finite interval and knowledge of q over a corresponding fraction of the interval the methods employed rest on weyl m function techniques and densities of zeros of a class of entire functions

the inverse sturm liouville problem ii soit q q 0 2 l r 2 0 1 le probleme de sturm liouville y q x y y 0 x 1 y 0 cos y 0 sin 0 y 1 cos y 1 sin 0 a un spectre discret de valeurs propres simples 0 q 1 q avec des fonctions propres h 0 x q h 1 x q on fixe p dans 0 2 l r 2 0 1 et soit m p q 0 2 l r 2 0 1 n q n p n 0 l espace de tous les points q de meme spectre que p on etudie m p comme une sous variete de 0 2 l r 2 0 1

an inverse problem for the matrix schrodinger equation abstract by modifying and generalizing some old techniques of n levinson a uniqueness theorem is established for an inverse problem related to periodic and sturm liouville boundary value problems for the matrix schrodinger equation

principal gains and principal phases in the analysis of linear multivariable feedback systems the concepts of principal gain and principal phase are introduced for linear multivariable systems and their use in the analysis of feedback behavior is demonstrated a sufficient nyquist type stability criterion is presented in terms of these quantities and is used to characterize the robustness of the closed loop stability property when the system model is subjected to a linear perturbation either multiplicative or additive at any point in the feedback configuration the results presented are less conservative than those obtained via the small gain theorem

note on a modified one center model the objective of the one center model single facility minimax location in the plane is changed to be the maximum weighted number of demand points within a given weighted distance from the facility the paper presents an on2 log n algorithm for the model along with some computational experience

the p maximal cover p partial center problem on networks abstract the p maximal cover problem is to find a set of locations for p facilities so as to maximize the total demand covered that is at most t units away from a closest facility the p partial center problem is to find a set of locations for the p facilities that minimizes the maximum distance between a closest facility and the demand covered in this paper we show the relationship between the two problems on networks for a tree network with one facility we present an algorithm to obtain all pareto locations with respect to the two objectives maximum cover and minimax distance for general networks we discuss the p maximal cover problem requiring that the optimal solution is a pareto location

heuristic procedures for the m partial cover problem on a plane abstract the m partial cover problem on a plane is defined as follows given npoints located at cartesian coordinates xj yj j 1 n each with an associated weight wj a critical distance r and an integer number m determine the location of m centres so that the sum of the weights of those points lying within distance r of at least one centre is maximised five heuristic procedures to solve the m partial cover problem are presented and computational experience reported the use of the procedures for some related problems is discussed

h robust stabilizability abstract in this paper we improve on the results of robust stabilizability obtained by kimura we do this in a constructive way by using an h approach and exhibit an upper bound for the order of robust controllers

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

parametrization of the isospectral set for the vector valued sturm liouville problem abstract we obtain a parametrization of the isospectral set of matrix valued potentials for the vector valued sturm liouville problem on a finite interval

inverse spectral problems for dirac operators on a finite interval abstract we consider the direct and inverse spectral problems for dirac operators that are generated by the differential expressions t q 1 i i 0 0 i d d x 0 q q 0 and some separated boundary conditions here q is an r r matrix valued function with entries belonging to l 2 0 1 c and i is the identity r r matrix we give a complete description of the spectral data eigenvalues and suitably introduced norming matrices for the operators under consideration and suggest an algorithm of reconstructing the potential q from the corresponding spectral data

the spectral class of the quantum mechanical harmonic oscillator the purpose of this paper is to study the so calledspectral classq of anharmonic oscillatorsq d2 q having the same spectrum n 2n n 0 as the harmonic oscillatorq0 d2 x2 1 thenorming constants t n mathop lim limits x uparrow infty ell g 1 n e n x mathord left vphantom e n x e n right kern nulldelimiterspace e n x of the eigenfunctions ofq form a complete set of coordinates inq in terms of which the potential may be expressed asq x2 1 2d2 lg with theta det left delta ij e ti 1 int limits x infty e i 0 e j 0 0 leqq i j infty right e n 0 being thenth eigenfunctionq0 the spectrum and norming constants are canonically conjugate relative to the bracket f g fd gdx to wit i j 0 ti 2 j 1 or 0 according to whetheri j or not and ti tj 0 this prompts an investigation of the symplectic geometry ofq the function is related to the theta function of a singular algebraic curve numerical results are also presented

robust stability under additive perturbations we consider a mimo linear time invariant feedback system1 s p c which is assumed to be u stable the plant p is subjected to an additive perturbation delta p which is proper but not necessarily stable we prove that the perturbed system is u stable if and only if delta p i q delta p 1 is u stable here q c i pc 1

a relationship between sensitivity and stability of multivariable feedback systems both comparison sensitivity and robust stability analysis attempt to extend concepts which are well understood in scalar feedback systems to multivariable control system design although the two bodies of work have developed about a decade apart the basic philosophies are similar both attempt to maintain a system property sensitivity reduction versus stability in the presence of plant modeling errors this paper demonstrates how the similarity in philosophy translates into a similarity in results and interpretations

algebraic theory of linear multivariable feedback systems this paper presents an algebraic theory for analysis and design of linear multivariable feedback systems the theory is developed in an algebraic setting sufficiently general to include as special cases continuous and discrete time systems both lumped and distributed designs are implemented by construction of a controller with two vector inputs and one vector output use of controllers of this type is shown to generate convenient stability results and convenient global parametrizations of all i o maps and all disturbance to output maps achievable for a given plant by a stabilizing compensator these parametrizations are then used to show that any such i o map and any such disturbance to output map may be simultaneously realized by choice of an appropriate controller in the special case of lumped systems it is shown that the design theory can be reduced to manipulations involving polynomial matrices only the resulting design procedure is thus shown to be more efficient computationally finally the problem of asymptotically tracking a class of input signals is considered in the general algebraic setting it is shown that the classical results on asymptotic tracking can be generalized to this setting additionally sufficient conditions for robustness of asymptotic tracking and robustness of stability are developed

the gradual covering decay location problem on a network abstract in covering problems it is assumed that there is a critical distance within which the demand point is fully covered while beyond this distance it is not covered at all in this paper we define two distances within the lower distance a demand point is fully covered and beyond the larger distance it is not covered at all for a distance between these two values we assume a gradual coverage decreasing from full coverage at the lower distance to no coverage at the larger distance

the inverse sturn liouville problem iii on considere la theorie spectrale inverse du probleme de sturm liouville y q x y y 0 x 1 avec les conditions aux limites y 0 0 by 1 y 1 0 q l r 2 0 1 et b r

the variable radius covering problem in this paper we propose a covering problem where the covering radius of a facility is controlled by the decision maker the cost of achieving a certain covering distance is assumed to be a monotonically increasing function of the distance i e it costs more to establish a facility with a greater covering radius the problem is to cover all demand points at a minimum cost by finding optimal number locations and coverage radii for the facilities both the planar and discrete versions of the model are considered heuristic approaches are suggested for solving large problems in the plane these methods were tested on a set of planar problems mathematical programming formulations are proposed for the discrete problem and a solution approach is suggested and tested