parametrization of all strictly causal stabilizing controllers within the framework of the factorization approach we present parametrizations of all strictly causal stabilizing controllers

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

feedback stabilization over commutative rings the matrix case this paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring for stabilizable plants certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other in the case of rings with irreducible spectrum this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs if the maxspectrum of the ring is noetherian and of zero krull dimension then this result shows that the stabilizable plants have doubly coprime fractions over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant certain invariants of these matrices known as elementary factors are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring this condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty a formula for the class of all stabilizing controllers is then developed that generalizes the previous well known formula in factorization theory for multidimensional transfer functions these results provide concrete conditions for stabilizabilty finally it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions

an algebraic approach to strong stabilizability of linear nd mimo systems although some necessary conditions for the strong stabilizability of linear multidimensional nd multiple input multiple output mimo systems have been available recently very little is known about sufficient conditions for the same problem this note presents two sufficient conditions for strong stabilizability of some classes of linear nd mimo systems obtained using an algebraic approach a simple necessary and sufficient condition is also given for the strong stabilizability of a special class of linear nd mimo systems an advantage of the proposed algebraic approach is that a stable stabilizing compensator can be constructed for an nd plant satisfying the sufficient conditions for the strong stabilizability presented in this note

feedback stabilization of multivariable two dimensional linear systems a constructive algorithm is presented for solving a two dimensional 2 d polynomial matrix equation that plays an important role in the output feedback stabilization of multivariable 2 d linear systems it is shown that this algorithm can have some computational advantages over the previous known ones based on the grobner basis approach as an application of the algorithm the class of all stabilizing strictly causal 2 d compensators is parametrized for a given stabilizable causal not necessarily strictly causal 2 d plant

on the universal module of p adic spherical hecke algebras let widetilde g be a split connected reductive group with connected center z over a local non archimedean field f of residue characteristic p let widetilde k be a hyperspecial maximal compact open subgroup in widetilde g let r be a commutative ring let v be a finitely generated r free r widetilde k module for an r algebra b and a character chi frak h v widetilde g widetilde k rightarrow b of the spherical hecke algebra frak h v widetilde g widetilde k rm end r widetilde g rm ind widetilde k widetilde g v we consider the specialization m chi v rm ind widetilde k widetilde g v otimes frak h v widetilde g widetilde k chi b of the universal frak h v widetilde g widetilde k module rm ind widetilde k widetilde g v for large classes of r including cal o f and overline bbb f p v b and chi arguing geometrically on the bruhat tits building we give a sufficient criterion for m chi v to be b free and to admit a widetilde g equivariant resolution by a koszul complex built from finitely many copies of rm ind widetilde k z widetilde g v this criterion is the exactness of certain fairly small and explicit frak n equivariant r module complexes where frak n is the group of cal o f valued points of the unipotent radical of a borel subgroup in widetilde g we verify it if f bbb q p and if v is an irreducible overline bbb f p widetilde k representation with highest weight in the closed bottom p alcove or a lift of it to cal o f we use this to construct p adic integral structures in certain locally algebraic representations of widetilde g

support varieties and representation types for basic classical lie superalgebras abstract let be an algebraically closed field of characteristic p 3 and g a restricted lie superalgebra over we introduce the definition of restricted cohomology for g and show its cohomology ring is finitely generated provided g is a basic classical lie superalgebra as a consequence we show that the restricted enveloping algebra of a basic classical lie superalgebra g is always wild except g sl 2 or g osp 1 2 or g c 2

algebraic aspects of linear control system stability after formulating the problem of linear control system stability in very general terms the problem is resolved by 1 establishing a necessary and sufficient condition for stability 2 exploring in some detail the nature of this condition and 3 developing a parametric characterization of all controllers which stabilize a given plant the problem formulation is based on a ring theoretic setting for linear systems and is distinguished by the use of a novel control system configuration any interconnection of two distinct systems can be represented in terms of this configuration while preserving the distinction between the two systems furthermore no other configuration has this property several benefits and extensions are accrued as a result of using this configuration first a completely general yet physically meaningful problem formulation results consequently state space and operator theoretic approaches to linear control system stability are thoroughly reconciled this reconciliation embraces continuous time as well as discrete time systems lumped as well as distributed systems and 1 d as well as n d systems second the controller parametrization affords four degrees of freedom two more than any existing parametrization the additional degrees of freedom make explicit the design opportunity associated with controller implementation and thus determine several important control system characteristics this result is expected to provide a theoretical basis for the development of comprehensive control system design methodologies wherein design considerations related to controller implementation are addressed

representation type of the blocks of category os abstract in this paper the authors investigate the representation type of the blocks of the relative parabolic category o s for complex semisimple lie algebras a complete classification of the blocks corresponding to regular weights is given the main results of the paper provide a classification of the blocks in the mixed case when the simple roots corresponding to the singular set and s do not meet

constrained traffic equilibrium in routing we study noncooperative routing in which each user is faced with a multicriterion optimization problem formulated as the minimization of one criterion subject to constraints on others we address the questions of existence and uniqueness of equilibrium we show that equilibria indeed exist but uniqueness may be destroyed due to the multicriteria nature of the problem we obtain uniqueness in some weaker sense under appropriate conditions we show that the link utilizations are uniquely determined at equilibrium we further study the normalized constrained equilibrium and apply it to pricing

controller parameterization of anantharam s example we consider the controller parameterization of anantharam s example 1985 it is shown that the parameterization is achieved by using only one parameter even though the plant does not have a coprime factorization

i inverness for polynomial matrices of non constant rank abstract let p1 pt be polynomials in c n with a variety v of common zeros contained in a suitable open set u explicit formulas are provided to construct rational functions 1 s such that i 1spi i 1 and such that the singularities of the i are contained in u this result is applied to compute rational functions valued 1 inverses of matrices with polynomial coefficients which do not have constant rank while retaining control over the location of the singularities of the rational functions themselves

on explicit solutions to the bezout equation explicit constructions are given for the solution of the bezout equation for matrices whose coefficients are multivariate polynomials or entire functions of paley wiener type this procedure also provides the 1 inverse when the bezout equation has no solutions

complexity for modules over finite chevalley groups and classical lie algebras 1 1 let g be a connected reductive algebraic group defined over a finite prime field fp of p elements for a given rational g module m one can consider the restrictions of m to the restricted lie algebra lie g and to the finite group g fp of fp rational points even though there is no direct functorial connection between the categories of restricted lie g modules and g fp modules the ambient algebraic group g can serve as a bridge linking these two important module categories early results of curtis in the 1960s provided a one to one correspondence between the simple restricted modules over k fp of lie g and those of g fp this correspondence is given by simply restricting certain classes of simple g modules more importantly these results along with work of steinberg allow one to easily transfer the questions pertaining to the computation of irreducible characters among these three categories for over thirty years this approach has been one of the main ideas used to relate the representation and cohomology theory of algebraic groups lie algebras and finite groups of lie type in the defining characteristic see e g cpsvan jan2 hum3 and jan1 given the previous results in this area a natural course of investigation should be to find relationships between canonical modules like projective modules for lie g and g fp more precisely

robustness analysis of uncertain linear singular systems with output feedback control the robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered the problem is transformed into a robust nonsingularity problem based on the linear fractional transformation lft approach the robustness bounds to preserve regularity impulse immunity and stability are found in terms of the structured singular value spl mu with respect to parametric uncertainties the lft approach provides a unified framework for robustness analysis of both uncertain linear continuous discrete time singular systems

geometric invariants of representations of finite groups abstract for a finite group and a field k of characteristic p dividing the order of we construct a map p u p y of varieties over k with unions of affine spaces as fibers and with p y p isogenous to proj h k on u we construct a universal p nilpotent operator p which leads to the construction of equivariant coherent sheaves associated to finite dimensional k modules m these coherent sheaves are algebraic vector bundles on u if m has constant jordan type for any finite dimensional k module m these coherent sheaves are vector bundles when restricted to the complements of the generalized support varieties of m for a linear algebraic group g we compare these constructions for the finite group g f p to previous constructions for the infinitesimal group scheme g r our comparison of bundle invariants for a rational g module m of exponential type p r upon restriction to g f p and to g r is a strengthened form of earlier comparisons of support varieties for m our constructions extend to semi direct products g of an infinitesimal group scheme g and a finite group thus to all finite group schemes if k is algebraically closed

a convex characterization of multidimensional linear systems subject to sqi constraints this technical note proposes a convex characterization of the set of all stable closed loop linear systems that are obtained from a given plant which may be multidimensional by interconnecting it in feedback with controllers that satisfy a certain pre selected constraint we take an approach that is particularly useful when a doubly coprime factorization of the plant is difficult to obtain and a stable stabilizing controller may not exist the plant is not strongly stabilizable or when one may be difficult to find most related work requires one of these we adopt the so called coordinate free approach which unlike youla s parametrization does not rely on a doubly coprime factorization of the plant we show that if constraints which satisfy a condition called strong quadratic invariance sqi are imposed on the controllers then the set of all stable closed loop multidimensional linear systems has a convex representation and norm optimal control problems can be cast in convex form although the sqi condition is in general slightly stronger than quadratic invariance qi which was developed in related work they are equivalent for common classes of problems arising in decentralized control

on the automorphism group of inversive planes of odd order abstract the structure of the fix bundle free automorphism groups of inversive planes of odd order is determined as a special case of our main result the automorphism groups with a transitive action on the points of an inversive plane of odd order are essentially determined and the plane is shown to be miquelian when these have no non trivial normal subgroups of odd order

stability and stabilization of multidimensional input output systems in this paper we discuss stability and stabilization of continuous and discrete multidimensional input output io behaviors of dimension r which are described by linear systems of complex partial differential resp difference equations with constant coefficients where the signals are taken from various function spaces in particular from those of polynomial exponential functions stability is defined with respect to a disjoint decomposition of the r dimensional complex space into a stable and an unstable region with the standard stable region in the one dimensional continuous case being the set of complex numbers with negative real part a rational function is called stable if it has no poles in the unstable region an io behavior is called stable if the characteristic variety of its autonomous part has no points in the unstable region this is equivalent to the stability of its transfer matrix and an additional condition the system is called stabilizable if there is a compensator io system such that the output feedback system is well posed and stable we characterize stability and stabilizability and construct all stabilizing compensators of a stabilizable io system parametrization the theorems and proofs are new but essentially inspired and influenced by and related to the stabilization theorems concerning multidimensional io maps as developed for instance by bose guiver shankar sule xu lin ying zerz and quadrat and of course the seminal papers of vidyasagar youla and others in the one dimensional case in contrast to the existing literature the theorems and proofs of this paper do not need or employ the so called fractional representation approach i e various matrix fraction descriptions of the transfer matrix thus avoiding the often lengthy matrix computations and seeming to be of interest even for one dimensional systems at least to the author an important mathematical tool new in systems theory is gabriel s localization theory which only in the case of ideal convex shankar sule unstable regions coincides with the usual one algorithmic tests for stability stabilizability and ideal convexity and the algorithmic construction of stabilizing compensators are addressed but still encounter many difficulties see in particular the open problems listed by xu e

on the cohomology of algebraic and related finite groups in the above theorem h gl m denotes the rational cohomology of the algebraic group gl with coefficients in m the rational gl module obtained from the adjoint module m m through twisting by the r th power of the frobenius endomorphism on gl the algebra structure on the cohomology is induced from the associative algebra structure on m on the other hand we provide the following systematic determination of certain rational cohomology groups h g v r where v r is the rational gmodule obtained from the rational g module v by applying the r th frobenius twist see 1 2 this is a somewhat sharpened version of theorem 4 of 36

feedback stabilization of mimo 3 d linear systems the authors solve the open problem of the existence of double coprime factorizations for a large class of multi input multi output mimo three dimensional 3 d linear systems it is proven that if all the unstable zeros of the contents associated with left and right matrix fraction descriptions of a given feedback stabilizable causal mimo 3 d plant are simple then the plant has a double coprime factorization the authors then give a parameterization of all stabilizing compensators for a mimo 3 d system in this class the key result developed in the paper is a novel and constructive technique of replacing an unstable polynomial with a stable polynomial step by step an illustrative example is also provided

weil restriction and support varieties this paper extends the comparison results for cohomological support varieties for chevalley groups of the form g fp established by j carlson z lin and d nakano to chevalley groups of the form g fpd other topics considered are complexity and non maximal support varieties for modules techniques developed by j pevtsova and the author together with the use of weil restriction functors are employed

a generalization of serre s conjecture and some related issues abstract several topics concerned with multivariate polynomial matrices like unimodular matrix completion matrix determinantal or primitive factorization matrix greatest common factor existence and subsequent extraction along with relevant primeness and coprimeness issues are related to a conjecture which may be viewed as a type of generalization of the original serre problem conjecture solved nonconstructively in 1976 and constructively more recently this generalized serre conjecture is proved to be equivalent to several other unsettled conjectures and therfore all these conjectures constitute a complete set in the sense that solution to any one also solves all the remaining

on a generalization of the youla kucera parametrization part i the fractional ideal approach to siso systems abstract in this paper we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for siso systems within this mathematical framework we give necessary and sufficient conditions so that a plant is internally strongly bistably stabilizable or admits a weak coprime factorization moreover we show how to generalize the youla kucera parametrization of the stabilizing controllers to any stabilizable plant which does not necessarily admit a coprime factorization this parametrization is generally affine in two free parameters

elementary proof of controller parametrization without coprime factorizability recently a controller parametrization method without coprime factorizability was presented by mori in this note we present this method using elementary matrix operations unlike in the earlier paper good knowledge of commutative algebra and module theory is not required

relationship between standard control problem and model matching problem without coprime factorizability we present the equivalence between a standard control problem and a model matching problem without coprime factorizability of plants this equivalence is well known if every stabilizable plant admit a doubly coprime factorization the result of this note does not require the coprime factorizability of plants

on the elementary factors of linear systems over unique factorization domains abstract the concept of elementary factors introduced by sule siam j control optim 32 1994 1675 1695 plays an important role in the output feedback stabilizability and stabilization of linear systems over unique factorization domains ufds this paper gives a simple proof that the elementary factors of t and w are identical where t and w are matrices associated with a right and a left matrix fraction description of a linear system over a ufd relationship between the reduced minors and the elementary factors is established an efficient method for computing the elementary factors based on the reduced minors is also proposed

on generalized inverses of matrices over integral domains it is proved that a matrix a over an integral domain admits a 1 inverse if and only if a linear combination of all the r r minors of a is equal to one where r is the rank of a some results on the existence of moore penrose inverses are also obtained

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

support varieties for modules over chevalley groups and classical lie algebras let g be a connected reductive algebraic group over an algebraically closed field of characteristic p 0 g 1 be the first frobenius kernel and g f p be the corresponding finite chevalley group let m be a rational g module in this paper we relate the support variety of m over the first frobenius kernel with the support variety of m over the group algebra kg f p this provides an answer to a question of parshall applications of our new techniques are presented which allow us to extend results of alperin mason and janiszczak jantzen and to calculate the dimensions of support varieties for finite chevalley groups

on stabilization and the existence of coprime factorizations let k be an integral domain and f its quotient field we show there are plants over f which have no stable coprime factorizations but can be stabilized in a stable closed loop this answers a question posed by vidyasagar schneider and francis

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

parameterization of stabilizing controllers with either right or left coprime factorization a parameterization method of stabilizing controllers that requires only one of right left coprime factorizations is presented as an illustrative model we will consider the multidimensional system with structural stability for which it is not yet known in general whether or not a doubly coprime factorization always exists a parameterization of stabilizing controllers for the model is presented