quadratic stabilizability of uncertain systems a two level optimization setup abstract the problem of stabilizing linear systems subject to possibly fast time varying uncertainties is investigated necessary and sufficient conditions of quadratic stabilizability are discussed the design process is formulated as a two level optimization process which can be simplified if the uncertainty is bounded by a hyperpolyhedron

a general theory of canonical induction formulae abstract so far there exist three independent constructions of two different canonical versions of brauer s induction theorem for complex characters due to v snaith p symonds and the author canonical in this context means functorial with respect to restrictions along group homomorphisms in this article we axiomatize the situation in which the above canonical induction formulae are constructed mackey functors and related structures arise in this way naturally as a convenient language this approach allows us to construct canonical induction formulae for arbitrary mackey functors in particular we obtain canonical induction formulae for the brauer character ring the group of projective characters the ring of trivial source modules and the ring of linear source modules in most cases it is not difficult to construct such formulae over the rational numbers a much more subtle question is whether the constructed formula comes from a canonical induction formula defined over the integers we give a sufficient condition in the general framework of mackey functors for a canonical induction formula to be integral as an application we show how canonical induction formulae allow the construction of functorial maps on representation rings in terms of functorial maps on subrings as for example the span of linear characters in the case of the canonical brauer induction formula this will be used in a subsequent article in the case of adams operations and chern classes

robust stabilization of discrete time linear systems with norm bounded time varying uncertainty abstract this paper presents an algorithm for the stabilization of a class of discrete time uncertain systems suffering from uncertainty of the norm bounded time varying type the stabilizing control gain matrix is obtained by solving a certain discrete riccati equation the solution of this riccati equation is also a lyapunov matrix which is used to establish the stability of closed loop system the results presented in this paper generalize those obtained in previous works for continuous time uncertain systems

monomial resolutions of trivial source modules abstract for trivial source modules admitting a filtration related to their generalized brauer constructions a resolution in terms of monomial modules is given which provides a categorical interpretation for the canonical induction formula applied to these modules

robust stability of interval matrices a computational approach a simple computational procedure is given to determine the robust stability of matrices containing interval parameter uncertainty the computation consists of determining the maximal phase difference of the characteristic polynomial evaluated over the vertices of the uncertain parameter set at each point of the stability boundary and verifying that this maximal phase difference is less than radians this follows as a direct application of the mapping theorem which allows us effectively to treat rank one perturbations of matrices we discuss the conservatism of the method and show how the solution may be improved to any desired level of accuracy we give several numerical examples of systems with state space perturbations and compare our results with those obtained by other methods

fibred permutation sets and the idempotents and units of monomial burnside rings we study the units of monomial burnside rings and the idempotents of monomial burnside algebras introducing a tenduction map we realise the unit group and the torsion unit group as a mackey functor

on the cauchy problem and initial traces for a degenerate parabolic equation the authors study the cauchy problem for the degenerate parabolic equation ut div du p 2 du p 2 and find sufficient conditions on the initial trace u0 and in particular on its behaviour as x for existence of a solution in some strip rn 0 t using a harnack type inequality they show that these conditions are optimal in the case of nonnegative solutions uniqueness of solutions is shown if u0 belongs to l1loc rn but is left open in the case that u0 is merely a locally bounded measure the results are closely related to papers by aronson caffarelli benilan crandall pierre and dahlberg kenig about the porous medium equation ut um the proofs are different and allow one to generalize some of the above results to equations with variable coefficients

quadratic stability and stabilization of linear systems with frobenius norm bounded uncertainties in this paper a quadratic stability condition and a quadratic stabilizer are proposed for linear systems with frobenius norm bounded uncertainties the necessary and sufficient condition for the quadratic stability of the uncertain system is given by a riccati inequality the proposed state feedback quadratic stabilizer can be obtained by solving a riccati equation

remarks on a nonlinear parabolic equation the equation ut u u r is studied in rn and in the periodic case it is shown that the equation is well posed in l1 and possesses regularizing properties for nonnegative initial data and 0 the solution decays in l1 rn as t in the periodic case it tends uniformly to a limit a consistent difference scheme is presented and proved to be stable and convergent

robust stability for linear discrete time systems with structured perturbations the robust stability problems of linear discrete time systems with structured perturbations are considered the necessary and sufficient condition of interval stability is given and the maximal bound of robust stability is presented for systems with a non negative or non positive nominal system matrix further it is shown that the given necessary and sufficient condition and the given maximal robust bound are still valid for time varying perturbations and nonlinear perturbations

robust stability of linear systems with delayed perturbations we consider the robust stability of linear systems with the time varying perturbations in the time delayed state and present sufficient conditions for guaranteeing the asymptotic stability the obtained results are analytically shown to be less conservative than those reported in the literature a numerical example is given to show the usefulness of the results

a generalized brauer construction and linear source modules for a complete discrete valuation ring 0 with residue field f a subgroup h of a finite group g and a homomorphism p h ox we define a functor v 4 v h p from the category of og modules to the category of fng h p modules and investigate its behaviour with respect to linear source modules

parametric quadratic stabilizability of uncertain nonlinear systems abstract the objective of this paper is to introduce the notion of parametric quadratic stabilizability pq stabilizability of nonlinear control systems x a a p x b p u when we consider nonlinear systems reference inputs and disturbances may alter dynamics in an essential way by moving the equilibrium to a new location or destroying it altogether for this reason when we consider quadratic stabilizability q stabilizability of nonlinear systems we need to combine the two concepts of parametric stability and q stabilizability and consider pq stabilizability when nonlinear function belongs to a subclass of passive functions it is shown that the necessary and sufficient condition for quadratic stabilizability q stabilizability of linear systems via dynamic state feedback which was derived by zhou and khargonekar is also a necessary and sufficient condition for that of nonlinear systems considered in this paper moreover when nonlinear function belongs to a subclass of incresing functions and when b 0 it shown that the necessary and sufficient condition for q stabilizability of linear systems via static state feedback which was derrived by petersen is also a necessary and sufficient condition for that of nonlinear systems considered in this paper

on the cauchy problem for the evolution p laplacian equations with gradient term and source abstract in this paper the authors study the equation u t div d u p 2 d u u q 1 u d u l in r n with p 2 we first prove that for 1 l p 1 the solution exists at least for a short time then for p 2 l p 1 the existence and nonexistence of global in time solutions are studied in various situations

improved measures of stability robustness for linear state space models in this paper the aspect of stability robustness of linear systems is analyzed in the time domain a bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a lyapunov matrix equation solution the resulting bound is shown to be an improved bound over the ones recently reported in the literature also special cases of the nominal system matrix are considered for which the bound is given in terms of the nominal matrix thereby avoiding the solution of the lyapunov matrix equation examples given include comparison of the proposed approach with the recently reported results

reduced conservatism in stability robustness bounds by state transformation this note addresses the issue of conservatism in the time domain stability robustness bounds obtained by the lyapunov approach a state transformation is employed to improve the upper bounds on the linear time varying perturbation of an asymptotically stable linear time invariant system for robust stability this improvement is due to the variance of the conservatism of the lyapunov approach with respect to the basis of the vector space in which the lyapunov function is constructed improved bounds are obtained using a transformation on elemental and vector norms of perturbations i e structured perturbations as well as on a matrix norm of perturbations i e unstructured perturbations for the case of a diagonal transformation an algorithm is proposed to find the optimal transformation several examples are presented to illustrate the proposed analysis

robust stability of discrete time systems under parametric perturbations stability robustness analysis of a system under parametric perturbations is concerned with characterizing a region in the parameter space in which the system remains stable in this paper two methods are presented to estimate the stability robustness region of a linear time invariant discrete time system under multiparameter additive perturbations an inherent difficulty which originates from the nonlinear appearance of the perturbation parameters in the inequalities defining the robustness region is resolved by transforming the problem to stability of a higher order continuous time system this allows for application of the available results on stability robustness of continuous time systems to discrete time systems the results are also applied to stability analysis of discrete time interconnected systems where the interconnections are treated as perturbations on decoupled stable subsystems

robustness analysis of interval matrices based on kharitonov s theorem this paper deals with the robustness analysis problem of interval matrices expansion of det a b is used to get characteristic polynomials of two corresponding systems kharitonov s theorem is applied to test hurwitz properties of the two polynomials a new sufficient condition for all eigenvalues of the interval matrix to lie in a damping cone on complex plan is derived illustrative examples are given

real representation spheres and the real monomial burnside ring abstract we introduce a restriction morphism called the boltje morphism from a given ordinary representation functor to a given monomial burnside functor in the case of a sufficiently large fibre group this is robert boltje s splitting of the linearization morphism by considering a monomial lefschetz invariant associated with real representation spheres we show that in the case of the real representation ring and the fibre group 1 the image of a modulo 2 reduction of the boltje morphism is contained in a group of units associated with the idempotents of the 2 local burnside ring we deduce a relation on the dimensions of the subgroup fixed subspaces of a real representation

technical communique on the robustness of linear systems with nonlinear uncertain parameters the present paper studies the stability robustness problem of an uncertain state space system we propose a new uncertain state space model which allows second order uncertain parameters the uncertainty in the model appears in the form of a combination of additive perturbation and multiplicative perturbation it is shown that the model has a strong practical background a robustness criterion based on the model is derived an example shows advantages of the proposed approach

stabilization of uncertain systems with norm bounded uncertainty a control a robust stabilization problem in a state space setting is treated it is assumed that the states are available for feedback using a fixed lyapunov function approach quadratic stability it is shown that an open loop stabilizability condition is equivalent to the existence of a stabilizing memoryless linear state feedback controller as a consequence it is shown that the existence of a quadratically stabilizing nonlinear time varying dynamic state feedback controller implies the existence of a quadratically stabilizing memoryless linear time invariant state feedback compensator

stabilization of an uncertain linear system in which uncertain parameters enter into the input matrix this paper presents a procedure for stabilizing a class of uncertain linear systems the uncertain systems under consideration are described by state equations in which the input matrix depends on a matrix of uncertain parameters this matrix of uncertain parameters may be time varying however it is constrained by a bound on its induced norm the stabilization procedure presented involves the repeated solution of an algebraic riccati equation when a positive definite solution to this riccati equation is obtained this solution is used to construct a stabilizing linear control law another important result contained in this paper can be stated roughly as follows for the class of uncertain systems under consideration if a system can be stabilized via nonlinear control then it is also possible to stabilize the system via linear control

the and constructions for biset functors abstract in this article we define the construction and the construction that was crucial in the theory of canonical induction formulas see 3 in the setting of biset functors thus providing the necessary framework to define and construct canonical induction formulas for representation rings that are most naturally viewed as biset functors additionally this provides a unified approach to the study of a class of functors including the burnside ring the monomial burnside ring and global representation ring

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

the blow up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source abstract the cauchy problem for a degenerate parabolic equation with a source and variable coefficient of the form u t d i v x u m 1 d u 1 d u u p is studied global in time existence and nonexistence conditions are found for a solution to the cauchy problem exact estimates of a solution are obtained in the case of global solvability a sharp universal i e independent of the initial function estimate of a solution near the blow up time is obtained

new robustness bounds for discrete systems with random perturbations some new stochastic stability robustness bounds for state space models are reported nominally exponentially stable discrete time systems are assumed to be subject to random parameter perturbations and novel bounds are obtained on the maximum variances of these random perturbations to maintain stability robustness the methods employed in the analysis are the existing ones used in the deterministic framework after transforming the stochastic robustness problem to a deterministic one the results are compared with each other and with the exact stability region in an example

on the cauchy problem for the singular parabolic equations with gradient term abstract the cauchy problem for a singular parabolic equation with gradient term of the form u t div d u p 2 d u d u q is studied this paper both extends the existence of solutions of the porous media equation with the same gradient term to this cauchy problem for the case q p 1 and obtains the existence of solutions on the assumptions that measures as initial data for the case 0 q p 1 however as q p 1 the optimality of the stipulated assumptions to existence of solutions is still open

generalized burnside rings and group cohomology we define the cohomological burnside ring b n g m of a finite group g with coefficients in a zgmodule m as the grothendieck ring of the isomorphism classes of pairs x u where x is a g set and u is a cohomology class in a cohomology group h n x g m the cohomology groups h x g m are defined in such a way that h x g m i h hi m when x is the disjoint union of transitive g sets g hi i fa is an abelian group with trivial action then b 1 g a is the same as the monomial burnside ring over a and when m is taken as a g monoid then b 0 g m is equal to the crossed burnside ring b c g m we discuss the generalizations of the ghost ring and the mark homomorphism and prove the fundamental theorem for cohomological burnside rings we also give an interpretation of b 2 g m in terms of twisted

multiple burnside rings and brauer induction formulae abstract let g be a finite group and suppose that g is an operator group of a finite group a define s g a h h s g and z 1 h a where s g is the set of subgroups of g and z 1 h a is the set of crossed homomorphisms from h to a we view g as an operator group of the opposite group a of a and make s g a into a left a g set the ring g a is defined to be a commutative ring consisting of all formal z linear combinations of a g orbits in s g a idempotent formulae for q z g a not only imply a generalization of dress induction theorem but bring in the case where z 1 g a is the set of linear c characters of g boltje s explicit formula for brauer s induction theorem and its hyperelementary version

a lyapunov robustness bound for linear systems with periodic uncertainties abstract for linear systems involving uncertain parameters with known constant nominal values and uncertain perturbations that vary sinusoidally with time lyapunov robustness anslysis is used to determine a stability bound or margin for the amplitudes of the parameter perturbations this bound is the size of a hypercube in parameter space for which asymptotic stability is guaranteed the bound which is based on a quadratic lyapunov function that depends linearly on parameter perturbations varies with the frequency of the uncertain parameter perturbations the bound is asymptotically proportional to the square root of this frequency as it becomes large

stability robustness bounds for linear state space models with structured uncertainty in this note we consider the robust stability analysis problem in linear state space models we consider systems with structured uncertainty some lower bounds on allowable perturbations which maintain the stability of a nominally stable system are derived these bounds are shown to be less conservative than the existing ones

cauchy problem for nonlinear parabolic equations with a gradient term we consider the cauchy problem of the nonlinear diffusive hamilton jacobi equation ut um duq where m 0 the existence and nonexistence of local and global solutions are studied by a priori estimates and compactness methods we prove that the cauchy problem of this equation has no nontrivial nonnegative global solutions if 0 moreover we obtain the global existence of solutions in the limiting case and the exact l estimates for local and global solutions

degenerate parabolic equations with initial data measures we address the problem of existence of solutions to degenerate and nondegenerate parabolic equations under optimal assumptions on the initial data which are assumed to be measures the requirements imposed on the initial data are connected both with the degeneracy of the principal part of the equation and with the form of the nonlinear forcing term the latter depends on the space gradient of a power of the solution applications to related problems are also outlined

multiplicative induction and units for the ring of monomial representations abstract let g be a finite group and let a be a finite abelian g group for each subgroup h of g h a denotes the ring of monomial representations of h with coefficients in a which is a generalization of the burnside ring h of h we research the multiplicative induction map h a g a derived from the tensor induction map h g and also research the unit group of g a the results are explained in terms of the first cohomology groups h 1 k a for k g we see that tensor induction for 1 cocycles plays a crucial role in a description of multiplicative induction the unit group of g a is identified as a finitely generated abelian group we especially study the group of torsion units of g a and study the unit group of g as well

holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations abstract we prove interior and boundary holder continuity for bounded weak soluttions of a class of quasilinear parabolic equations whose prototype is u t div u m 1 du p 2 du

the cauchy problem for ut δu u q with q a positive real number the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces in that context it is known as the generalized deterministic kpz equation the paper is concerned with the initial value problem for the equation under the assumption that the initial data function is bounded and continuous results on the existence uniqueness and regularity of solutions are obtained

robust stability analysis of linear continuous discrete time systems with output feedback controllers the robust stability of linear systems with output feedback controllers and time varying uncertain parameters is considered the robust stability bounds for time varying uncertain parameters are given using the lyapunov method when there are no uncertain parameters in the input and or output matrices it is shown that the result for continuous time systems is the same as that presented by k m zhou and p p khargonekar 1987 and the result for discrete time systems is better than that of s r kolla 1989 for the same example

on fibred biset functors with fibres of order prime and four abstract this note has two purposes first to present a counterexample to a conjecture parametrizing the simple modules over green biset functors appearing in an author s previous article this parametrization fails for the monomial burnside ring over a cyclic group of order four second to classify the simple modules for the monomial burnside ring over a group of prime order for which the above mentioned parametrization holds

fibered biset functors abstract the theory of biset functors introduced by serge bouc gives a unified treatment of operations in representation theory that are induced by permutation bimodules in this paper by considering fibered bisets we introduce and describe the basic theory of fibered biset functors which is a natural framework for operations induced by monomial bimodules the main result of this paper is the classification of simple fibered biset functors

a note on robust stability bounds the authors comment on stability robustness bounds for linear systems with multiple uncertain parameters which were obtained by means of lyapunov functions results obtained with quadratic and with vector lyapunov functions are compared simple examples are used to illustrate the effects of the choice of functions corresponding majorizations and system decompositions

unit groups of representation rings and their ghost rings as inflation functors abstract the theory of biset functors developed by serge bouc has been instrumental in the study of the unit group of the burnside ring of a finite group in particular for the case of p groups the ghost ring of the burnside ring defines an inflation functor and becomes a useful tool in studying the burnside ring functor itself we are interested in studying the unit group of another representation ring the trivial source ring of a finite group in this article we show how the unit groups of the trivial source ring and its associated ghost ring define inflation functors since the trivial source ring is often seen as connecting the burnside ring to the character ring and brauer character ring of a finite group we study all these representation rings at the same time we point out that restricting all of these representation rings unit groups to their torsion subgroups also give inflation functors which we can completely determine in the case of the character ring and brauer character ring

cauchy problem for doubly degenerate parabolic equation with gradient source abstract we consider the cauchy problem for a doubly degenerate parabolic equation with a source term depending solely on the gradient a major result of this paper is to establish the local existence of solutions for this equation under optimal stipulated assumptions on the initial data

on solutions of some doubly nonlinear degenerate parabolic equations with absorption abstract the existence uniqueness regularity and behavior of solutions to the initial boundary value problem for the equation u t i 1 n x i u x i p 2 u x i u 0 with zero dirichlet boundary condition and with nonnegative initial condition are studied in a bounded domain in r n where 0 and p 1

robust stabilizability for linear systems with both parameter variation and unstructured uncertainty the authors consider the problem of finding a single compensator which stabilizes a plant having both parameter variation and high frequency unstructured uncertainty an uncertain system model is proposed to characterize plants containing those two different uncertainties closed loop stability criteria are then proposed for the model robust stabilizability for single input single output siso systems is studied a number of assumptions describing the set of admissable siso plants are imposed the satisfaction of these assumptions guarantees the existence of a proper stable compensator c s achieving robust stabilization the algorithm used for controller design is recursive in nature and allows the designer to select one compensator coefficient at a time also the minimum order of the desired stabilizer c s depends only on the largest relative degree of the nominal plants which often leads to a lower order compensator

a robust stability bound of linear systems with structured uncertainty the robust stability of linear uncertain systems with multiple uncertain parameters is considered a sphere norm bound condition on the uncertainties which assures stability of the systems is derived this bound is shown to be less conservative than existing ones a necessary condition to maximize this bound is also given

robust h control for linear time invariant systems with norm bounded uncertainty in the input matrix abstract this paper focuses on the problem of robust h control design for a class of linear time invariant systems with uncertainty in the state space model we consider uncertain systems with norm bounded parameter uncertainty in the input matrix the paper presents a state feedback control design which stabilizes the plant for all admissible uncertainties and also guarantees an h norm bound constraint on disturbance attenuation paralleling to the theory of robust control the robust h control problem is solved via the notion of quadratic stabilization with an h norm bound constraint necessary and sufficient conditions for quadratic stabilization with an h norm bound are derived it is shown that the solution to this problem involves solving a parameter dependent algebraic riccati equation the results can be regarded as extensions of existing results on robust stabilization of linear uncertain systems and h optimal control

an induction theorem for the unit groups of burnside rings of 2 groups abstract let g be a 2 group and b g denote the group of units of the burnside ring of g for each subquotient h k of g there is a generalized induction map from b h k to b g defined as the composition of inflation and multiplicative induction maps we prove that the product of generalized induction maps b h k b g is surjective when the product is taken over the set of all subquotients that are isomorphic to the trivial group or a dihedral 2 group of order 2 n with n 4 as an application we give an algebraic proof for a theorem by tornehave the unit group for the burnside ring of a 2 group aarhus universitet preprint series 1983 84 41 may 1984 which states that tom dieck s exponential map from the real representation ring of g to b g is surjective we also give a sufficient condition for the surjectivity of the exponential map from the burnside ring of g to b g

on the cauchy problem for the evolution p laplacian equations with gradient term and source and measures as initial data abstract in this paper we study the cauchy problem for the evolution p laplacian equations with gradient term and source on the assumption that measures as initial conditions

monomial g posets and their lefschetz invariants abstract let g be a finite group and c be an abelian group we introduce the notions of c monomial g sets and c monomial g posets and state some of their categorical properties this gives in particular a new description of the c monomial burnside ring b c g we also introduce lefschetz invariants of c monomial g posets which are elements of b c g these invariants allow for a definition of a generalized tensor induction multiplicative map t u b c g b c h associated to any c monomial g h biset u which in turn gives a group homomorphism b c g b c h between the unit groups of c monomial burnside rings

lyapunov stability robust analysis and robustness design for linear continuous time systems the linear continuous time systems to be discussed are described by state space models with structured time varying uncertainties first the explicit maximal perturbation bound for maintaining quadratic lyapunov stability of the closed loop systems is presented then a robust design method is proposed given an open loop nominal system with information of the uncertainty structure the proposed robust design method produces a linear constant output feedback control law which maximizes the perturbation bound for maintaining quadratic lyapunov stability both the optimal results for robust analysis and robustness design can be obtained by using an efficient numerical algorithm e g minimax of matlab with analytic gradients given in this paper examples are investigated in some detail to show the improvements and advantages of the proposed methods over previous ones

new robust stability bounds of linear discrete time systems with time varying uncertainties this paper presents new uncertain parameter variation bounds for linear discrete time systems to preserve asymptotic stability the lyapunov method is utilized to treat both structured and unstructured uncertainties and the results are optimized with respect to a parameter in the inequality used when applied to examples considered by previous authors our results give less conservative bounds