pareto optimal strategy for stochastic weakly coupled large scale systems with state dependent system noise this note is concerned with the decentralized infinite horizon stochastic pareto optimal static output feedback strategy for a class of weakly coupled systems with state dependent noise first pareto optimal control problems are formulated using a static output feedback strategy the necessary conditions are given by the cross coupled stochastic algebraic riccati type equations csares for proving the existence of the static output feedback strategy that minimizes the quadratic cost function after determining the asymptotic structure of the solutions of the csares a new sequential numerical algorithm and newton s method for solving the csares are described the resulting numerical solution is used to develop the pareto optimal strategy finally the efficiency of the proposed algorithm is demonstrated by solving a numerical example for a practical megawatt frequency control problem

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

control effort considerations in the stabilization of uncertain dynamical systems abstract this paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system we focus attention on the amplitude norm of the controller which is used to achieve this end the uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded a lyapunov function is used to establish the stability of the closed loop system the paper gives necessary and sufficient concitions for the uncertain system to be stabilizable with a given lyapunov function furthermore a procedure is indicated for the construction of the desired feedback control law

sets of efficient points in a normed space abstract sets of efficient points in a normed space with respect to the distances to the points of a given compact set are geometrically characterized hull and closure properties are obtained these results are relevant to geometry of normed spaces and are mostly useful in the context of location theory

on the robustness of lq regulators it is shown that in spite of its large gain and phase margins the linear quadratic state feedback regulator may suffer from poor robustness where small changes in the parameters of the system may lead to fast unstable closed loop modes

robust discrete controllers guaranteeing l sub 2 and l sub infinity performances an algorithm is given for designing discrete time controllers which are robust with respect to parameter variations and which also guarantee specified l sub 2 performance constraints for stochastic inputs and l sub infinity performance constraints for deterministic inputs this is not an analysis but a design method guaranteeing the specified robust performances also an upper bound on the covariance matrix is provided

lqg control with an h sup infinity performance bound a riccati equation approach an lqg linear quadratic gaussian control design problem involving a constraint on h sup infinity disturbance attenuation is considered the h sup infinity performance constraint is embedded within the optimization process by replacing the covariance lyapunov equation by a riccati equation whose solution leads to an upper bound on l sub 2 performance in contrast to the pair of separated riccati equations of standard lqg theory the h sup infinity constrained gains are given by a coupled system of three modified riccati equations the coupling illustrates the breakdown of the separation principle for the h sup infinity constrained problem both full and reduced order design problems are considered with an h sup infinity attenuation constraint involving both state and control variables an algorithm is developed for the full order design problem and illustrative numerical results are given

positive proper efficient points and related cone results in vector optimization theory positive proper efficient points are defined as solutions of appropriate linear scalar optimization problems a geometric characterization of positive proper efficient points is given as well as conditions under which the set of positive proper efficient points is dense in the set of all efficient points it is shown that these results are applicable in the normed vector lattices c a b l p and l p for 1 leqq p leqq infty and that previous related results which required the ordering cone to have a compact or weak compact base are not applicable in many normed vector lattices including c a b l p and l p for 1 leqq p leqq infty

a double commutator guaranteed cost bound for robust stability and performance in this paper we present a new guaranteed cost bound motivated by the maximum entropy equation this bound is of interest due to its sensitivity to skew symmetric structured uncertainty which is difficult to address by means of conventional guaranteed cost bounds

non vanishing of high derivatives of automorphic l functions at the center of the critical strip keywords modular l functions mollification nonvanishing results central value of high derivatives analytic rank jacobian variety reference tan article 2000 004doi 10 1515 crll 2000 074 record created on 2008 11 14 modified on 2017 05 12

state feedback control of systems with multiplicative noise via linear matrix inequalities we consider lti systems perturbed by parametric uncertainties modeled as white noise disturbances we show how to maximize via state feedback control the smallest norm of the noise intensity vector producing instability in the mean square sense using convex optimization over linear matrix inequalities we also show how to maximize performance robustness where performance is measured by expected output energy with either bounded initial conditions and zero inputs classical lqg cost or zero initial conditions and deterministic inputs of bounded energy a generalization of the h norm

weighted averages of modular l values using an explicit relative trace formula on gl 2 we derive a formula for averages of modular l values in the critical strip weighting by fourier coefficients hecke eigenvalues and petersson norms as an application we show that a grh holds for these averages as the weight or the level goes to we also use the formula to give explicit zero free regions of the form im s 0 for some particular modular l functions

on central l derivative values of automorphic forms we study the average and nonvanishing of the central l derivative values of l s f and l s f k scriptscriptstyle d for f in an orthogonal hecke eigenbasis mathcal h 2k of weight 2k cusp forms of level 1 for large odd k here f k scriptscriptstyle d is the base change of f to an imaginary quadratic field k scriptscriptstyle d mathbb q sqrt d with fundamental discriminant d we prove asymptotic formulas for the first and second moments of l frac 1 2 f as well as the first moment of l frac 1 2 f k scriptscriptstyle d over mathcal h 2k as odd k rightarrow infty further we employ mollifiers to establish that for sufficiently large k there are positive proportion of hecke eigenforms f in mathcal h 2k with l frac 1 2 f ne 0 we also give applications of our results to heegner cycles of high weights of the modular curve x 0 1

on the central value of symmetric square l functions let s k n be the space of cusp forms of weight k level n and character for f in s k n chi let l s sym2 f be the symmetric square l function and l s f otimes f be the rankin selberg square attached to f for fixed k 2 n prime and real primitive asymptotic formulas for the first and second moment of the central value of l s sym2 f and l s f otimes f over a basis of s k n are given as n as an application it is shown that a positive proportion of the central values l 1 2 sym2 f does not vanish

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions

inclusion theorems for non explosive and strongly exposed cones in normed spaces abstract this paper is composed of five sections in the first one we classify cones into pseudonuclear intermediate and explosive the notion of strongly exposed cone and the equivalence between pseudonuclearity and strong exposition for cones in a normed space are recalled in the second one it is shown that several properties for a cone in a normed space exist each one being equivalent to strong exposition of the cone furthermore 2 5 an estimate by chung is improved in the third one through projections of rank one some cones that are an immediate extension of elliptic cones in elementary geometry are defined and it is established 3 3 that a cone is strongly exposed if and only if it can be contained in one of that type in the fourth one it is shown 4 2 that also the non explosivity of a cone is expressible in terms of metric properties more precisely a cone is non explosive if and only if it can be contained in a balanced cone of elliptic kind but in which the axis becomes a finite dimensional subspace it may be interesting to observe that the result does not depend on particular hypotheses e g reflexivity on the space in the fifth one it is put in evidence a property of a cone in a reflexive normed space which excludes that the cone is intermediate an orientative essential bibliography concludes the paper

maximum entropy type lyapunov functions for robust stability and performance analysis we present two lyapunov functions that ensure the unconditional stability and robust performance of a modal system with uncertain damped natural frequency each lyapunov function involves the sum of two matrices the first being the solution to the so called maximum entropy equation and the second being a constant auxiliary portion the significant feature of these lyapunov functions is that the guaranteed robust stability region is independent of the weighting matrix while the performance bounds are relatively tight compared to alternative approaches thus these lyapunov functions are less conservative than standard bounds that tend to be highly sensitive to the choice of state space basis kevwords maximum entropy function robust stability robust performance

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

appendix an effective zero free region all the notation in this appendix will be as in the preceding paper let f be a maass form which is a newform for fo n with eigenvalue a and central character x normalized so that f f 1 we have seen that the size of p l the first fourier coefficient of f is intimately related to the behavior of l s f near s 1 here l s f is the l series of f the adjoint square lift of f to gl 3 and the crucial question is whether or not l s f vanishes when s is real and close to 1 it was shown in theorem 0 1 that if l s f is nonzero in a sufficiently wide neighborhood of 1 then ip 1 12 0 where the implied constant depends on e and is ineffective about a year after the preceding paper was first circulated it developed from conversations involving the above three authors that by slightly modifying the techniques introduced in that paper the possibility of a siegel zero could be completely eliminated in many cases in particular theorem 0 1 is now true unconditionally in the generic situation when f is not a lift from gl 1 that is to say when the l series of f is not equal to a hecke l series defined over a quadratic field if we include all cusp forms we can still obtain the theorem in the a aspect but must restrict ourselves to either an ineffective constant or a weaker effective constant in the n aspect we have

controller design of systems with multiplicative noise general linear continuous stochastic systems are considered with multiplicative noise in the control and state channels and stabilizing optimal inputs are synthesized for both time varying and time invariant situations in the nonstationary case a set of nonlinear matrix differential equations has to be solved in the stationary problem a set of nonlinear matrix algebraic equations provides the solution the separation principle is not valid and both the state filtering and the control synthesis problem should be treated simultaneously

design of robust state feedback laws existence criteria and computational methods are presented for finding stabilizing state feedback laws for systems with uncertain state or control matrices the uncertainties are modelled as sector bounded non linearities and feedback laws ensuring global asymptotic closed loop stability are obtained using lyapunov e direct method the algorithms essentially amount to repeatedly solving a parameter dependent riccati equation until the maximal solution becomes positive definite

robust stabilization of uncertain systems in this paper we consider the systems described by dx axdt budt sum i sigma i f i xd beta i qquad text or qquad dot x ax bu sum i b i f i x t c i x and we will derive conditions under which there exists a feedback control law u kx such that the closed loop system is stable for all sigma i or smooth nonlinearities f i the nonlinearities f i and the noisy gains sigma i d beta i are unknown uncertainties in the system and the problem considered is to obtain a control law which is robust against these uncertainties as far as stability is concerned

stochastic h sub 2 h sub spl infin control with state dependent noise this paper discusses the stochastic h sub 2 h sub spl infin control problem with state dependent noise by means of the stabilization exact observability and stochastic detectability of stochastic systems the infinite horizon stochastic h sub 2 h sub spl infin control design is developed for the finite horizon h sub 2 h sub spl infin control problem our results generalize the corresponding deterministic ones to the stochastic models finally the observer based suboptimal stochastic h sub 2 h sub spl infin control is discussed in which the state variables cannot be measured directly and a feasible design algorithm is proposed

stabilization of linear systems by noise it is proved that the biggest lyapunov number lambda max of the system dot x a f t x where a is a fixed d times d matrix and f t is a zero mean strictly stationary matrix valued stochastic process satisfies 1 d trace a leqq lambda max on the other hand for each varepsilon 0 there is a process f t for which lambda max leqq 1 d trace a varepsilon in particular the system dot x ax can be stabilized by zero mean stationary parameter noise if and only if trace a 0 the stabilization can be accomplished by a one dimensional noise source the results carry over to the case where a is a stationary process they are also true for f t white noise

a riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems this paper presents a procedure for designing a full state observer and feedback control law which will stabilize a given uncertain linear system the uncertain linear systems under consideration are described by state equations which depend on uncertain parameters these uncertain parameters may be time varying their values however are constrained to lie within known compact bounding sets the design procedure involves solving two algebraic riccati equations a feature of the design procedure presented is the fact that it reduces to the standard lqg design procedure if the system contains no uncertain parameters

the constrained lyapunov problem and its application to robust output feedback stabilization given a dynamical system whose description includes time varying uncertain parameters it is often desirable to design an output feedback controller leading to asymptotic stability of a given equilibrium point when designing such a controller one may consider static i e memoryless or dynamic compensation in this paper we show that solvability of various output feedback design problems is implied by existence of a solution to a certain constrained lyapunov problem clp the clp can be stated in purely algebraic terms once the clp is described we provide necessary and sufficient conditions for its solution to exist subsequently we consider application of the clp to a number of robust stabilization problems involving static output feedback and observer based feedback

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

brief paper optimal reduced order compensation of time varying discrete time systems with deterministic and white parameters the finite horizon optimal compensation problem is considered in the case of linear time varying discrete time systems with deterministic and white stochastic parameters and quadratic criteria the dimensions of the compensator are a priori fixed and may be time varying also the dimensions of the system may be time varying strengthened discrete time optimal projection equations sdope are developed which within the class of minimal compensators are equivalent to the first order necessary optimality conditions based on the sdope and their associated boundary conditions two numerical algorithms are presented to solve the two point boundary value problem one is a homotopy algorithm while the second iterates the sdope repeatedly forward and backward in time the latter algorithm is much more efficient and constitutes a generalization of the single iteration of the control and estimation riccati equations associated with the full order problem for systems with deterministic parameters the algorithms are illustrated with a numerical example the case of systems with deterministic parameters will be treated as a special case of systems with white parameters

robust reduced order nonstrictly proper state estimation via the optimal projection equations with guaranteed cost bounds a state estimation design problem involving parametric plant uncertainties is considered an estimation error bound suggested by multiplicative white noise modeling is utilized for guaranteeing robust estimation over a specified range of parameter uncertainties necessary conditions that generalize the optimal projection equations for reduced order state estimation are used to characterize the estimator that minimizes the error bound the design equations thus effectively serve as sufficient conditions for synthesizing robust estimators additional features include the presence of a static estimation gain in conjunction with the dynamic kalman estimator to obtain a nonstrictly proper estimator

the optimal projection equations for reduced order discrete time state estimation for linear systems with multiplicative white noise the optimal projection equations obtained in 2 3 for reduced order discrete time state estimation are generalized to include the effects of state and measurement dependent noise to provide a model of parameter uncertainty in contrast to the single matrix riccati equation arising in the full order kalman filter case the optimal steady state reduced order discrete time estimator is characterized by three matrix equations one modified riccati equation and two modified lyapunov equations coupled by both an oblique projection and stochastic effects

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

paper linear ultimate boundedness control of uncertain dynamical systems given a state equation whose description includes uncertain parameters it is often desirable to design a feedback controller which renders all solutions uniformly and ultimately bounded when facing such a problem one often has available or assumes an upper bound for the excursions of the uncertain parameters within this framework the objective is to guarantee no matter how nature selects the uncertainty that the state will eventually end up and remain within some pre specified region when this region is a small neighborhood about the origin the concept of uniform ultimate boundedness is tantamount to practical asymptotic stability the objectives of this paper are twofold first and foremost our goal is to show that a class of ultimate boundedness problems which have been solved to date via nonlinear control can in point of fact be solved via linear control this is the main result and is given in theorem 4 1 secondly we show that some of the so called matching assumptions which have been required in the earlier literature can be weakened somewhat this enables one to design controllers for a larger class of dynamical systems than previously considered

contractibility of efficient point sets in normed spaces l e a topological vector space over reals c a convex cone and x a set in e c specifies a partial order in e as follows x y e x c y if x y c with this order one can define the set of efficient points of x with respect to c as the set min x c x x if x c y for some y x then y c x whenever the relative interior of c ric is nonempty the cone 0 u ric gives another order defining the set of weakly efficient points of x with respect to c denoted w min x c one of the most important problems of vector optimization is to investigate topological properties of the sets min x c and w min x c our aim is to develop a new method which makes it possible to prove most important properties of efficient point sets such as contractibility and connectedness in general normed spaces

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

the lqg ltr procedure for multivariable feedback control design this paper provides a tutorial overview of the lqg ltr design procedure for linear multivariable feedback systems lqg ltr is interpreted as the solution of a specific weighted h2 tradeoff between transfer functions in the frequency domain properties of this solution are examined for both minimum phase and nonminimum phase systems this leads to a formal weight augmentation procedure for the minimum phase case which permits essentially arbitrary specification of system sensitivity functions in terms of the weights while such arbitrary specifications are not possible for nonminimum phase problems a direct relationship between weights and sensitivities is developed for nonminimum phase siso and certain nonminimum phase mimo cases which guides the weight selection process

the dirichlet series of koecher and maas and modular forms of weight 3 2 the basis problem for modular forms is the question which modular forms can be expressed as linear combinations of theta series in order that this question makes sense one has of course to specify which type of theta series should be taken into account we want here to follow in a special case the approach taken e g in b61 w a l l which in its general form asks which cuspidal modular forms f of degree n and weight k for the group foc n can be expressed as linear combinations of the theta series of degree n of integral quadratic forms of rank m 2 k and the same level n moreover one wants to derive an explicit expression for such a linear combination in terms of the petersson inner products of f with the theta series involved the representation theoretic version of this question is to ask which cuspidat automorphic representations of the adelic symplectic or metaplectic group

two generalizations of a theorem of arrow barankin and blackwell in 1953 arrow barankin and blackwell proved that if r n is equipped with its natural ordering and if a is a closed convex subset of r n then the set of points in a that can be supported by strictly positive linear functionals is dense in the set of all efficient maximal points of a in this note two generalizations of this result are given the first of these is in the setting of a dual system and requires relatively weak assumptions on the ordering cone but a rather strong compactness assumption on the set a the second generalization which is in the setting of a locally convex space relaxes the compactness assumption on the set a but demands more stringent assumptions on the ordering cone this second result was recently obtained by petschke for normed spaces m petschke on a theorem of arrow barankin and blackwell siam j control optim 28 1990 pp 395 401 the proof given here is substantially different from that given by petschke

pareto optimization in topological vector spaces to identify the efficient point of a set is an important problem in vector optimization the existence of efficient points in infinite dimensional space has been studied by many authors such as 2 10 12 15 17 19 21 23 25 etc to study pareto efficiency in locally convex topological vector space in 1983 the second author isac 14 has introduced an important notion of nuclear cone which has also other applications as for example in the fixed point theory in the best approximation theory in the study of vector optimization for multivalued functions in the study of conically bounded sets and in the study of nuclearity of topological vector space about these one can refer to hyers isac rassias book 11 and references therein in 1990 pontini 20 defined a more general notion of pseudonuclear in general topological vector space may not be locally convex pontini remarked that in a normed vector space a cone is pseudo nuclear if and only if it is nuclear but in a general topological vector space this result is not true in 11 13 isac hyers and rassias discussed the existence of efficient points in locally convex topological space ordered by pointed nuclear cone and got some general results in book 11 and paper 13 they also posed the following open problem

robust optimal parametric lq control with a guaranteed cost bound and applications for an optimal parametric linear quadratic lq control problem a design objective is to determine a controller of constrained structure such that the closed loop system is asymptotically stable and an associated performance measure is optimized in the presence of system uncertainty the system via a parametric lq design is further required to be robust in terms of maintaining the closed loop stability with a guaranteed cost bound this problem is referred to as robust optimal parametric lq control with a guaranteed cost bound and is addressed in this work a new design method is proposed to find an optimal controller for simultaneously guaranteeing robust stability and performance over a specified range of parameter variations the results presented generalize some previous work in this area a versatile numerical algorithm is also given for computing the robust optimal gains the usefulness of the design method is demonstrated by numerical examples and a design of the robust control of a vtol helico

the majorant lyapunov equation a nonnegative matrix equation for robust stability and performance of large scale systems a new robust stability and performance analysis technique is developed the approach involves replacing the state covariance by its block norm matrix i e the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics a bound i e majorant for the block norm matrix is given by the majorant lyapunov equation a lyapunov type nonnegative matrix equation existence uniqueness and computational tractability of solutions to the majorant lyapunov equation are shown to be completely characterized in terms of m matrices two examples are considered for a damped simple harmonic oscillator with uncertain but constant natural frequency the majorant lyapunov equation predicts unconditional stability and for a pair of nominally uncoupled oscillators with uncertain coupling the majorant lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators a result not predictable from quadratic or vector lyapunov functions

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

the optimal projection equations for fixed order dynamic compensation first order necessary conditions for quadratically optimal steady state fixed order dynamic compensation of a linear time invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form in contrast to the pair of matrix riccati equations for the full order lqg case the optimal steady state fixed order dynamic compensator is characterized by four matrix equations two modified riccati equations and two modified lyapunov equations coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains the coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced order controller case

robust controller synthesis using the maximum entropy design equations this note presents an application of the optimality conditions obtained in 1 for dynamic compensation in the presence of state control and measurement dependent noise by solving these equations which represent a fundamental generalization of standard steady state lqg theory a series of increasingly robust control designs is obtained for the example considered in 2

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

symmetry transitions in random matrix theory l functions we compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices defined by averages with respect to haar measure on so 2n and usp 2n to leading order as n on the unit circle as functions of the angle measured from one of the two symmetry points in the eigenvalue spectrum exp pm i theta n 1 le n le n our results extend previous formulae that relate just to the symmetry points i e to 0 local spectral statistics are expected to converge to those of random unitary matrices in the limit as n when is fixed and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing if y n they become functions of y in the limit when n we verify that this is true for the spectral two point correlation function but show that it is not true for the moments of the characteristic polynomials for which the leading order asymptotic approximation is a function of rather than y symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing we also investigate the moments of the logarithms of the characteristic polynomials in the same context the moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of l functions previously moments at the symmetry point 0 have been related to the moments of families of l functions evaluated at the centre of the critical strip our results motivate general conjectures for the moments of orthogonal and symplectic families of l functions evaluated at a fixed height t up the critical line these conjectures suggest that the symmetry of the non trivial zeros of the l functions influences the moments asymptotically far on the scale of the mean zero spacing from the centre of the critical strip we verify that the second moments of real quadratic dirichlet l functions and a family of automorphic l functions are consistent with our conjectures

on the stability robustness of the continuous time lqg optimal control the robustness properties of the lqg optimal controller that is obtained by the asymptotic recovery method are discussed it is shown that in spite of the good recovered stability margins the resulting closed loop design is inherently unrobust in the sense that small parameter variations in the plant and or small realization errors in the controller may lead to an instantaneous instability of the design

averages of central l values of hilbert modular forms with an application to subconvexity we use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change l functions averaged over hilbert modular forms of a fixed weight and level we apply these formulas to the subconvexity problem for these l functions we also establish an equidistribution result for the hecke eigenvalues weighted by these l values

brief paper robust multi objective feedback design with linear guaranteed cost bounds this paper considers the problem of designing a state feedback control law which minimizes multiple quadratic performance objectives in the presence of uncertain linear plant dynamics vector optimization techniques are used to minimize upper bounds on each performance objective guaranteed cost by exploiting the linear uncertainty bound introduced by bernstein 1987 the vector optimization problem is scalarized and pareto optimal solutions are computed the theory is applied to a numerical example with two quadratic performance measures

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice

linear time varying system control based on the inversion transformation this paper proposes a new state feedback control design for uniformly controllable linear time varying systems the new design is based on a nonlinear state transformation that converts the stabilization problem of the system state into a destabilization problem of the transformed state such a transformation results in a new control that has two advantages over previous controls 1 in some of the previous control designs forecast of future information of the time varying system matrices over some time interval t t is required to calculate the state feedback gain at any time instant t while in the new control design no such forecast is needed 2 in other previous control designs consecutive time differentiation on the time varying system matrices is required while in the new control design time integration instead of differentiation is applied to the system matrices the new control can therefore be applied to systems with time varying parameters that are piecewise continuous in time

robust static and dynamic output feedback stabilization deterministic and stochastic perspectives three parallel gaps in robust feedback control theory are examined sufficiency versus necessity deterministic versus stochastic uncertainty modeling and stability versus performance deterministic and stochastic output feedback control problems are considered with both static and dynamic controllers the static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time varying parameter variations while the static and dynamic stochastic optimal control problems feature state control and measurement dependent white noise general sufficiency conditions for the deterministic problems are obtained using lyapunov s direct method while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion the sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems as an additional application of the deterministic result the modified riccati equation approach of petersen and hollot is generalized in the static case and extended to dynamic compensation

robust stability and performance via fixed order dynamic compensation the discrete time case a discrete time feedback control design problem involving parametric uncertainty is considered a quadratic bound suggested by recent work on discrete time state space h sub infinity theory is utilized in conjunction with the guaranteed cost approach to guarantee robust stability with a robust performance bound the principal result involves sufficient conditions for characterizing robust full and reduced order controllers with a worst case h sub 1 performance bound