on the robustness of h state feedback control to nonlinear perturbations abstract robustness of stability and h performance to nonlinear distortions of the central linear state feedback low are established under certain sector constraints the analysis is carried in a general setting of ltv systems

an optimal control approach to robust tracking of linear systems in our early work we show that one way to solve a robust control problem of an uncertain system is to translate the robust control problem into an optimal control problem if the system is linear then the optimal control problem becomes a linear quadratic regulator lqr problem which can be solved by solving an algebraic riccati equation in this article we extend the optimal control approach to robust tracking of linear systems we assume that the control objective is not simply to drive the state to zero but rather to track a non zero reference signal we assume that the reference signal to be tracked is a polynomial function of time we first investigated the tracking problem under the conditions that all state variables are available for feedback and show that the robust tracking problem can be solved by solving an algebraic riccati equation because the state feedback is not always available in practice we also investigated the output feedback we show that if we place the poles of the observer

a connection between state space and doubly coprime fractional representations explicit formulas for a doubly coprime fractional representation of the transfer function of a lumped lti system are given in terms of a stabilizable and detectable state space realization of the transfer function these formulas allow existing computational algorithms to be utilized for the purpose of computing doubly coprime factorizations several additional implications of this result are briefly discussed as are some immediate extensions and future research directions

robust stabilization of linear multivariable systems relations to approximation the stabilization of a linear system modelled as g a where g is a known rational transfer function and a is a perturbation satisfying w a2 g1 or equivalently there does not exist g with fewer poles in c than g such that w1 g g w2 a simple characterization of all robustly stabilizing controllers is then derived and state space formulae for maximally robust controllers are given finally reduced order controllers are considered

a state space formulation for optimal hankel norm approximations the optimal hankel norm approximation problem studied in 1 is reformulated in a state space setting the crucial extension theorem is reestablished in this framework the minimal degree optimal approximation is then derived in terms of state space parameters

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

on the dynamics of navier stokes and euler equations this is a detailed study on certain dynamics of navier stokes and euler equations via a combination of analysis and numerics we focus upon two main aspects a zero viscosity limit of the spectra of linear navier stokes operator b heteroclinics conjecture for euler equation its numerical verification melnikov integral and simulation and control of chaos due to the difficulty of the problem for the full navier stokes and euler equations we also propose and study two simpler models of them

l optimization and hankel approximation an important l infty optimization problem arising in feedback system design is shown to be equivalent to an optimal hankel norm approximation problem this enables the problem to be solved using state space methods based on balanced realizations

robust stabilization of linear time invariant systems the problem of robustly stabilizing families of linear time variant plants consisting of a nominal plant and its frequency domain neighborhood is discussed necessary and sufficient conditions are obtained for robust stabilizability of such families of plants the set of robustly stabilizing controllers is parameterized and the largest achievable stability margin is characterized for a linear time invariant plant

the category of f modules has finite global dimension abstract let r be a regular ring of characteristic p 0 in 4 hochster showed that the category of lyubeznik s f r modules has enough injectives so that every f r module has an injective resolution in this category we show in this paper that under mild conditions on r for example when r is essentially of finite type over an f finite regular local ring the category of f modules has finite global dimension d 1 where d dim r in 4 hochster also showed that when m and n are f r finite f r modules hom f r m n is finite we show that in general ext f r 1 m n is not necessarily finite

necessary and sufficient conditions for quadratic stability and stabilizability of uncertain linear time varying systems in this paper we consider linear time varying systems subject to norm bounded one block structured uncertainties first given an unforced system we find necessary and sufficient conditions for quadratic stability then considered a controlled system we provide necessary and sufficient conditions for quadratic stabilizability both for the state and the output feedback case

robust stabilization of discrete generalized systems the robust stabilization problem for generalized discrete time systems described by polynomial or improper transfer function matrices subject to perturbations acting on the normalized coprime factors is solved the maximum achievable stability margin and the robust stabilizing controller are given in terms of realizations and solutions to appropriate riccati equations

a computational algorithm for the super optimal solution of the model matching problem abstract in this paper we develop a computational algorithm to implement the strengthened minimization technique 1 in the model matching problem which arises in robust controller design and weighted sensitivity minimization a general algorithm is proposed for constructing unitary matrices and the extremal functions for the h optimization problem are diagonalized using the constructed unitary matrices we impose stronger minimization conditions to force the solution of the optimization problem to be unique and explain their physical significance

estimating generalized lyapunov exponents for products of random matrices we discuss several techniques for the evaluation of the generalized lyapunov exponents which characterize the growth of products of random matrices in the large deviation regime a monte carlo algorithm that performs importance sampling using a simple random resampling step is proposed as a general purpose numerical method which is both efficient and easy to implement alternative techniques complementing this method are presented these include the computation of the generalized lyapunov exponents by solving numerically an eigenvalue problem and some asymptotic results corresponding to high order moments of the matrix products taken together the techniques discussed in this paper provide a suite of methods which should prove useful for the evaluation of the generalized lyapunov exponents in a broad range of applications their usefulness is demonstrated on particular products of random matrices arising in the study of scalar mixing by complex fluid flows

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

feedback minimax sensitivity and optimal robustness in this paper we look for feedbacks that minimize the sensitivity function of a linear single variable feedback system represented by its frequency responses sensitivity to disturbances and robustness under plant perturbations are measured in a weighted h infty norm in an earlier paper zames proposed an approach to feedback design involving the measurement of sensitivity by multiplicative seminorms which have certain advantages over the widely used quadratic norm in problems where there is plant uncertainty or where signal power spectra are not fixed but belong to sets the problem was studied in a general setting and some h infty examples were solved here a detailed study of the single variable case is undertaken the results are extended to unstable plants and explicit formulas for the general situation of a finite number of right half plane rhp plant zeros or poles are provided the q or approximate inverse parametrization of feedbacks that maintain closed loop stability is extended to the ease of unstable plants the h infty and wiener hopf approaches are compared

finite dimensional controller design for infinite dimensional systems the circle criterion approach abstract the hilbert space generalization of the circle criterion is used for the finite dimensional compensator design for unstable infinite dimensional systems a design algorithm which generalizes the small gain based algorithm in 5 is presented

duality and de rham cohomology for graded d modules abstract we consider the graded matlis dual d m of a graded d module m over the polynomial ring r k x 1 x n k is a field of characteristic zero and show that it can be given a structure of d module in such a way that whenever dim k h d r i m is finite then h d r i m is k dual to h d r n i d m as a consequence we show that if m is a graded d module such that h d r n m is a finite dimensional k space then dim k h d r n m is the maximal integer s for which there exists a surjective d linear homomorphism m e s where e is the top local cohomology module h x 1 x n n r this extends a recent result of hartshorne and polini on formal power series rings to the case of polynomial rings we also apply the same circle of ideas to provide an alternate proof of their result when m is a finitely generated graded d module such that dim k h d r i m is finite we generalize the above result further showing that h d r n i m is k dual to ext d i m e

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

bass numbers of local cohomology modules let a be a regular local ring of positive characteristic this paper is concerned with the local cohomology modules of a itself but with respect to an arbitrary ideal of a the results include that all the bass numbers of all such local cohomology modules are finite that each such local cohomology module has finite set of associated prime ideals and that whenever such a local cohomology module is artinian then it must be injective this last result had been proved earlier by hartshorne and speiser under the additional assumptions that a is complete and contains its residue field which is perfect the paper ends with some low dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic 0 are true

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

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

maximally robust controllers for multivariable systems the set of all optimal controllers which maximize a robust stability radius for unstructured additive perturbations may be obtained using standard hankel norm approximation methods these controllers guarantee robust stability for all perturbations which lie inside an open ball in the uncertainty space say of radius r 1 necessary and sufficient conditions are obtained for a perturbation lying on the boundary of this ball to be destabilizing for all maximally robust controllers it is thus shown that a worst case direction exists along which all boundary perturbations are destabilizing by imposing a parametric constraint such that the permissible perturbations cannot have a projection of magnitude larger than 1 delta r 1 0 delta leq1 in the most critical direction the uncertainty region guaranteed to be stabilized by a subset of all maximally robust controllers can be extended beyond the ball of radius r1 the choice of the best maximally robust controller in the sense that the uncertainty region guaranteed to be stabilized becomes as large as possible is associated with the solution of a superoptimal approximation problem expressions for the improved stability radius are obtained and some interesting links with mu analysis are pursued

robust stabilizability for a class of transfer functions this paper is concerned with the robust stabilizability for single input single output plants robust stabilizability means that a fixed controller can stabilize simultaneously all the plants in a given class which is characterized by a frequency dependent uncertainty band function around the transfer function of a nominal model a necessary and sufficient condition for robust stabilizability is derived based on the well known nevanlinna pick theory in classical analysis it is shown that the values of the uncertainty band function should be restricted within a certain range at the unstable poles of the nominal model in order for the class to be robustly stabilizable a procedure of synthesizing a robust stabilizer is given and the parametrization of all the robust stabilizers is also shown

robust controllers for uncertain linear multivariable systems abstract this paper is addressed to three distinct yet related topics in the design of controllers for imprecisely known linear multivariable systems in the first part it is supposed that the plant to be stabilized is subject to additive or multiplicative uncertainties and necessary and sufficient conditions are derived for the existence of a controller that stabilizes all plants within this band of uncertainty in the second part in contrast with the first part it is supposed that the number of unstable poles of the plant to be stabilized is not precisely known the type of plant uncertainty is the so called stable factor uncertainty and necessary and sufficient conditions are given for robust stabilization in the third part the model of uncertainty is a ball in the space of rational matrices metrized by the so called graph metric and sufficient conditions for robust stabilization are derived

robust control of flexible structures a case study a comparison is made between three partial differential equation models for a flexible beam with different types of damping and varying parameter values robust controllers can be designed to stabilize all linear systems whose transfer functions lie within a ball in the l norm given a nominal model for our flexible beam we calculate which sets of models can be stabilized by the same finite dimensional robust controller by stabilize we mean here that the poles of the system are moved to the left of the line re s 0

input output norms in general linear systems upper bounds on the induced norm of the input output operators in linear systems with l2 input and output signals are provided the main idea is to transform the original problem into a parametized set of indefinite lq optimization problems using the classical maximum principle the existence of solutions to the latter is equivalent to the solvability of a boundary value hamiltonian system or equivalenty of an indefinite riccati equation our time domain approach enables the treatment of a very general linear set up including time varying infinite dimensional finite and infinite horizon systems

vorticity generation by instabilities of chaotic fluid flows abstract we show that at high reynolds number smooth lagrangian chaotic flows are typically linearly unstable and that the perturbed vorticity tends to concentrate on a fractal numerical integration of the relevant linear partial differential equations with reynolds number up to r 10 6 shows that the wavenumber power spectrum of the perturbed vorticity has a powerlaw behavior and that the magnitude of the perturbed vorticity is a multifractal it is then shown that a wavepacket picture whereby vorticity wavepackets are evolved according to ordinary differential equations yields scaling results and appropriately describes the small wavelength features of both the power spectrum and the multifractal dimension spectrum analytical results are derived establishing the validity of the latter approach as well as an equivalence principle for lyapunov partition functions these results are verified by numerical simulations involving a computationally efficient class of chaotic flows

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

fluctuating hydrodynamics and turbulence in a rotating fluid universal properties we analyze the statistical properties of three dimensional 3d turbulence in a rotating fluid to this end we introduce a generating functional to study the statistical properties of the velocity field mathbit v we obtain the master equation from the navier stokes equation in a rotating frame and thence a set of exact hierarchical equations for the velocity structure functions for arbitrary angular velocity mathbit ensuremath omega in particular we obtain the differential forms for the analogs of the well known von karman howarth relation for 3d fluid turbulence we examine their behavior in the limit of large rotation our results clearly suggest dissimilar statistical behavior and scaling along directions parallel and perpendicular to mathbit ensuremath omega the hierarchical relations yield strong evidence that the nature of the flows for large rotation is not identical to pure two dimensional flows to complement these results by using an effective model in the small ensuremath omega limit within a one loop approximation we show that the equal time correlation of the velocity components parallel to mathbit ensuremath omega displays kolmogorov scaling q ensuremath 5 3 wherein as for all other components the equal time correlators scale as q ensuremath 3 in the inertial range where mathbit q is a wave vector in 3d our results are generally testable in experiments and or direct numerical simulations of the navier stokes equation in a rotating frame

remarks on computing coprime factors for distributed parameter systems presents an indirect method for computing coprime factors for a class of linear time invariant distributed parameter systems the method overcomes the numerical instability of an existing direct decomposition method it is shown that coprime factorization can be reduced to constructing simultaneously stabilizing finite dimensional controllers for which an efficient iterative procedure is developed

robustness results in linear quadratic gaussian based multivariable control designs the robustness of control systems with respect to model uncertainty is considered using simple frequency domain criteria available and new results are derived under a common framework in which the minimum singular value of the return differences transfer matrix is the key quantity in particular robustness results associated with multivariable control systems designed on the basis of linear quadratic lq and the linear quadratic gaussian lqg design methodologies are presented

on the robust stabilizability of uncertain linear time invariant plants using nonlinear time varying controllers abstract we consider general families of linear time invariant plants described by a nominal plant model with additive or multiplicative unstructured modeling uncertainty we show that such families of plants are robustly uniformly stabilizable by nonlinear time varying controllers if and only if they are robustly stabilizable by linear time invariant ones some auxiliary results on hankel and l norm approximation are also obtained

nonlinear state observation using h sub spl infin filtering riccati design the authors propose an observer for continuous time nonlinear systems the observer gain is computed by a riccati differential equation similar to the extended kalman filter they prove that under certain conditions the proposed observer is an exponential observer by choosing an appropriate lyapunov function furthermore the authors explore some important relations of the proposed observer to robust control theory and h sub spl infin filtering to examine the practical usefulness of the proposed observer they applied it to an induction motor for the estimation of the rotor flux and the angular velocity

algebraic k theory of virtually free groups we provide a general procedure for computing the algebraic k theory of finitely generated virtually free groups the procedure describes these groups in terms of the algebraic k theory of various finite subgroups and various farrell nil groups we illustrate this process by carrying out the computation for several interesting classes of examples the first two classes serve as a check on the method and show that our algorithm recovers results that already exist in the literature the last two classes of examples yield new computations

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

uncertain feedback loops and robustness in general linear systems abstract our main results are a riccati equation based robust stability criterion and a robust stabilization scheme in systems with uncertain loop gain these issues are considered in the general framework of time varying and possibly infinite dimensional linear systems the current analysis is consequential to the recent development of a time domain variational approach to h control the report includes also remarks on the trade off between the type of uniform stability required lyapunov vs exponential and the type of perturbations considered real vs complex and on the small gain criterion in time varying systems

quadratic stabilization with an h performance bound of uncertain linear time varying systems in this paper we deal with linear time varying systems subject to uncertain parameters in the so called one block norm bounded form we consider the problem of designing a controller which guarantees the quadratic stabilization of the closed loop system while keeping the norm of the operator mapping the exogenus input to the controlled output under a prespecified level necessary and sufficient conditions for the existence of such a controller are given

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

multifractal structure of chaotically advected chemical fields the structure of the concentration field of a decaying substance produced by chemical sources and advected by a smooth incompressible two dimensional flow is investigated we focus our attention on the nonuniformities of the h older exponent of the resulting filamental chemical field they appear most evidently in the case of open flows where irregularities of the field exhibit strong spatial intermittency as they are restricted to a fractal manifold nonuniformities of the h older exponent of the chemical field in closed flows appears as a consequence of the nonuniform stretching of the fluid elements we study how this affects the scaling exponents of the structure functions displaying anomalous scaling and relate the scaling exponents to the distribution of local lyapunov exponents of the advection dynamics theoretical predictions are compared with numerical experiments

cartier modules finiteness results on a locally noetherian scheme x over a field of positive characteristic p we study the category of coherent ox modules m equipped with a pe linear map i e an additive map c ox ox satisfying rc m c r p e m for all m a m r a ox the notion of nilpotence meaning that some power of the map c is zero is used to rigidify this cate gory the resulting quotient category called cartier crystals satisfies some strong finiteness conditions the main result in this paper states that if the frobenius morphism on x is a finite map i e if x is f finite then all cartier crystals have finite length we further show how this and related results can be used to recover and generalize other finiteness results of hartshorne speiser 19 lyubeznik 27 sharp 30 enescu hochster 14 and hochster 20 about the structure of modules with a left action of the frobenius for example we show that over any regular f finite scheme x lyubeznik s f finite modules 27 have finite length

smooth and filamental structures in chaotically advected chemical fields this paper studies the spatial structure of decaying chemical fields generated by a chaotic advection flow and maintained by a spatially smooth chemical source previous work showed that in a regime where diffusion can be neglected large peclet number the structures are filamental or smooth depending on the relative strength of the chemical dynamics and the stirring induced by the flow the scaling exponent gamma q of the qth order structure function depends at leading order linearly on the ratio of the rate of decay of the chemical processes alpha and the average rate of divergence of neighboring fluid parcel trajectories lyapunov exponent h under a homogeneous stretching approximation gamma q q max alpha h 1 which implies that a well defined filamental smooth transition occurs at alpha h this approximation has been improved by using the distribution of finite time lyapunov exponents to characterize the inhomogeneous stretching of the flow however previous work focused more on the behavior of the exponents as q varies and less on the effects of alpha and hence the implications for the filamental smooth transition here we set out the precise relation between the stretching rate statistics and the scaling exponents and emphasize that the latter are determined by the distribution of the finite size rather than finite time lyapunov exponents we clarify the relation between the two distributions we show that the corrected exponents symbol see text depend nonlinearly on alpha with formula see text the magnitude of the correction to the homogeneous stretching approximation formula see text grows as alpha increases reaching a maximum when the leading order transition is reached alpha h the implication of these results is that there is no well defined bulk filamental smooth transition instead it is the case that the chemical field is unambiguously smooth for alpha h max where h max denotes the maximum finite time lyapunov exponent and unambiguously filamental for alpha h with an intermediate character for alpha between these two values theoretical predictions are confirmed by numerical results obtained for a linearly decaying chemistry coupled to a renewing type of flow together with careful calculations of the cramer function

test module filtrations for unit f modules abstract we extend the notion of test module filtration introduced by blickle for cartier modules we then show that this naturally defines a filtration on unit f modules and prove that this filtration coincides with the notion of v filtration introduced by stadnik in the cases where he proved existence of his filtration we also show that these filtrations do not coincide in general moreover we show that for a smooth morphism f x y test modules are preserved under f we also give examples to show that this is not the case if f is finite flat and tamely ramified along a smooth divisor

h2 optimal control of linear systems with tracking disturbance rejection constraints this paper considers the design of two degree of freedom linear multivariable controllers for four block matrix transfer functions with objectives pertaining to both random and deterministic signals the former are taken into account in the cost functional whereas the latter entail that the controller is constrained to solve a servomechanism problem involving a class of reference signals and two classes of disturbances the cost functional is set up by adding quadratic terms to the usual lqg wiener hopf cost in such a way as to ensure the existence of unique solutions even when the signals to be tracked rejected as well as models have poles on the imaginary axis the solutions of the servomechanism problem are explicitly parametrized by free stable rational and proper parameters under two sets of assumptions one which is necessary and sufficient for the existence of solutions and another stronger set which avoids kronecker products and the associated higher dimensional factorizations involved in

pointwise stabilizability of families of linear time invariant plants the topic of robust pointwise stabilizability of families of linear time invariant lti plants arising from unstructured modeling uncertainty is addressed the results presented extend some earlier work and show that there is no advantage in using general nonlinear time varying nltv controllers over lti ones for certain problems of robust pointwise stabilization

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

directional interpolation approach to h optimization and robust stabilization it is shown that under a mild condition h infty optimization and robust stabilization are equivalent to a directional interpolation problem which is a matrix extension of the classical pick nevanlinna interpolation problem a classical iterative method which is an extension of the schur nevanlinna algorithm is given for solving the problem this method does not require the inner outer factorization nor the balanced realization of the original plant a circuit theoretical parameterization of all solutions is derived that is expected to enhance the physical insight to the h infty optimal control and robust stabilization this parameterization has the degree much less than the one obtained previously

stability radius for structured perturbations and the algebraic riccati equation abstract in this paper we generalize the notion of stability radius introduced in 1 to allow for structured perturbations we then relate the stability radius to the existence of hermitian solutions of an algebraic riccati equation and give some applications of this result

f singularities via alterations we give characterizations of test ideals and f rational singularities via regular alterations formally the descriptions are analogous to standard characterizations of multiplier ideals and rational singularities in characteristic zero via log resolutions lastly we establish nadel type vanishing theorems up to finite maps for test ideals and further demonstrate how these vanishing theorems may be used to extend sections

d module and f module length of local cohomology modules let r be a polynomial or power series ring over a field k we study the length of local cohomology modules hji r in the category of d modules and fmodules we show that the d module length of hji r is bounded by a polynomial in the degree of the generators of i in characteristic p 0 we obtain upper and lower bounds on the f module length in terms of the dimensions of frobenius stable parts and the number of special primes of local cohomology modules of r i the obtained upper bound is sharp if r i is an isolated singularity and the lower bound is sharp when r i is gorenstein and f pure we also give an example of a local cohomology module that has different d module and f module lengths

generic vanishing in characteristic p 0 and the characterization of ordinary abelian varieties we prove a generic vanishing type statement in positive characteristic and apply it to prove positive characteristic versions of kawamata s theorems a characterization of smooth varieties birational to ordinary abelian varieties and the surjectivity of the albanese map when the frobenius stable kodaira dimension is zero

the robust optimal control of uncertain systems state space method the robust optimal control of linear uncertain systems with l sub 2 bounded uncertainties is addressed the optimal state feedback control law in the so called worst disturbance case is derived the lql method minimax state estimation maximin state feedback closed loop control scheme lql means linear quadratic l sub 2 bounded uncertainties is proposed in a sense the lql method uses the idea of the linear quadratic gaussian lqg method to treat h sub infinity optimizing design problems

intermittency in two dimensional turbulence with drag we consider the enstrophy cascade in forced two dimensional turbulence with a linear drag force in the presence of linear drag the energy wave number spectrum drops with a power law faster than in the case without drag and the vorticity field becomes intermittent as shown by the anomalous scaling of the vorticity structure functions using previous theory we compare numerical simulation results with predictions for the power law exponent of the energy wave number spectrum and the scaling exponents of the vorticity structure functions ensuremath zeta 2q we also study both by numerical experiment and theoretical analysis the multifractal structure of the viscous enstrophy dissipation in terms of its r enyi dimension spectrum d q we derive a relation between d q and ensuremath zeta 2q and discuss its relevance to a version of the refined similarity hypothesis in addition we obtain and compare theoretically and numerically derived results for the dependence on separation r of the probability distribution of ensuremath delta stackrel p vec r ensuremath omega the difference between the vorticity at two points separated by a distance r our numerical simulations are done on a 4096 ifmmode times else texttimes fi 4096 grid

a canonical linear system associated to adjoint divisors in characteristic p 0 suppose that x is a projective variety over an algebraically closed field of characteristic p 0 further suppose that l is an ample or more generally in some sense positive divisor we study a natural linear system in h 0 x ox kx l we further generalize this to incorporate a boundary divisor we show that these subsystems behave like the global sections associated to multiplier ideals h 0 x j x l in characteristic zero in particular we show that these systems are in many cases base point free while the original proof utilized kawamata viehweg vanishing and variants of multiplier ideals our proof uses test ideals

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

big tight closure test elements for some non reduced excellent rings abstract this paper is concerned with existence of big tight closure test elements for a commutative noetherian ring r of prime characteristic p let r denote the complement in r of the union of the minimal prime ideals of r a big test element for r is an element of r which can be used in every tight closure membership test for every r module and not just the finitely generated ones the main results of the paper are that if r is excellent and satisfies condition r 0 and c r is such that r c is gorenstein and weakly f regular then some power of c is a big test element for r if i r is a homomorphic image of an excellent regular ring of characteristic p for which the frobenius homomorphism is intersection flat or ii r is f pure or iii r is local the gamma construction is not used

graded annihilators of modules over the frobenius skew polynomial ring and tight closure this paper is concerned with the tight closure of an ideal a in a commutative noetherian local ring r of prime characteristic p several authors including r fedder k i watanabe k e smith n hara and f enescu have used the natural frobenius action on the top local cohomology module of such an r to good effect in the study of tight closure and this paper uses that device the main part of the paper develops a theory of what are here called special annihilator submodules of a left module over the frobenius skew polynomial ring associated to r this theory is then applied in the later sections of the paper to the top local cohomology module of r and used to show that if r is cohen macaulay then it must have a weak parameter test element even if it is not excellent

frobenius and cartier algebras of stanley reisner rings we study the generation of the frobenius algebra of the injective hull of a complete stanley reisner ring over a field with positive characteristic in particular by extending the ideas used by m katzman to give a counterexample to a question raised by g lyubeznik and k e smith about the finite generation of frobenius algebras we prove that the frobenius algebra of the injective hull of a complete stanley reisner ring can be only principally generated or infinitely generated also by using our explicit description of the generators of such algebra and applying the recent work by m blickle about cartier algebras and generalized test ideals we are able to show that the set of f jumping numbers of generalized test ideals associated to complete stanley reisner rings form a discrete subset inside the non negative real numbers