a restart procedure for the finite mass method the finite mass method is a gridless lagrangian method to simulate compressible flows that has been developed by gauger leinen and yserentant siam j numer anal 37 2000 pp 1768 1799 it is based on a discretization of mass not of space as with classical discretization schemes mass is subdivided into little mass packets of finite extension each of which is equipped with finitely many internal degrees of freedom these mass packets move under the influence of internal and external forces and the laws of thermodynamics and can change their size orientation and shape to follow the motion of the fluid the ability of the mass packets to deform with the flow is the reason for the high accuracy of the finite mass method but also requires that the computation is newly set up from time to time such restart procedures are described in this article

necessary and sufficient conditions under which an h 2 optimal control problem has a unique solution a set of necessary and sufficient conditions under which a general h 2 optimal control problem has a unique solution is derived it is shown that the solution for an h 2 optimal control problem if it exists is unique if and only if i the transfer function from the control input to the controlled output is left invertible and ii the transfer function from the disturbance to the measurement output is right invertible

rate of quantum ergodicity in euclidean billiards for a large class of quantized ergodic flows the quantum ergodicity theorem states that almost all eigenfunctions become equidistributed in the semiclassical limit in this work we give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the wigner function in the case of ergodic systems of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value this is studied numerically for three euclidean billiards stadium cosine and cardioid billiard using up to 6000 eigenfunctions we find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions such as bouncing ball modes or scarred eigenfunctions we give a detailed discussion and explanation of these effects using a simple but powerful model for the rate of quantum ergodicity in momentum space we observe a slower decay we also study the suitably normalized fluctuations of the expectation values around their mean and find good agreement with a gaussian distribution

continuity properties of the cheap control problem without stability abstract an open problem concerning the convergence of the optimal cost for the cheap control problem without stability is solved it turns out that the definition of a new type of linear quadratic optimal control problems is necessary this new problem requires the infimization of the cost functional under the constraint that the state trajectory modulo a certain subspace vanishes as time goes to infinity the associated optimal cost turns out to be the limit of the optimal cost for the cheap control problem without stability moreover for left invertible systems the optimal control state and output for the perturbed problem tend to the optimal control state and output for the new problem also a characterization of the optimal cost for the latter problem is given in terms of the dissipation inequality

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

h 2 optimal control with an h constraint the state feedback case in this paper we consider a mixed hz optimal control problem it is assumed that the plant as well as the feedback controller are finite dimensional and linear time invariant and that the plant state is available for feedback more specifically among all the state feedback controllers that minimize the he norm of a closed loop transfer matrix we give necessary and sufficient conditions for the existence of a controller that also satisfies a prescribed h norm bound on some other closed loop transfer matrix when these conditions are met the solution to the above problem is also a global solution to the contrained optimization problem of minimizing an he norm performance measure subject to an h norm constraint we also give state space formulae for computing the solutions some easily checkable sufficient conditions for the existence of solutions are given finally we give an example in which all solutions to the constrained optimization problem are necessarily dynamic i e there is no static gain solution even though plant state is available for feedback

separating the regular and irregular energy levels and their statistics in a hamiltonian system with mixed classical dynamics we look at the high lying eigenstates from the 10001 to the 13000 in the robnik billiard defined as a quadratic conformal map of the unit disk with the shape parameter lambda 0 15 all the 3000 eigenstates have been numerically calculated and examined in the configuration space and in the phase space which in comparison with the classical phase space enabled a clear cut classification of energy levels into regular and irregular this is the first successful separation of energy levels based on purely dynamical rather than special geometrical symmetry properties we calculate the fractional measure of regular levels as rho 1 0 365 or 0 01 which is in remarkable agreement with the classical estimate rho 1 0 360 or 0 001 this finding confirms the percival s 1973 classification scheme the assumption in berry robnik 1984 theory and the rigorous result by lazutkin 1981 1991 the regular levels obey the poissonian statistics quite well whereas the irregular sequence exhibits the fractional power law level repulsion and globally brody like statistics with beta 0 286 or 0 001 this is due to the strong localization of irregular eigenstates in the classically chaotic regions therefore in the entire spectrum we see that the berry robnik regime is not yet fully established so that the level spacing distribution is correctly captured by the berry robnik brody distribution

ergodicite et limite semi classique consider a self adjoint quantic hamiltonian p h p x hdx whereh 0 is the planck s constant andp some smooth classical observable on the phase space r2n when the classical flow on a compact energy shell p is ergodic we prove that in the limith 0 almost all the eigenfunctions ofp h whose energy is near of are distributed according to the liouville measure on p

lur e equations and even matrix pencils abstract in this work we consider the so called lur e matrix equations that arise e g in model reduction and linear quadratic infinite time horizon optimal control we characterize the set of solutions in terms of deflating subspaces of even matrix pencils in particular it is shown that there exist solutions which are extremal in terms of definiteness it is shown how these special solutions can be constructed via deflating subspaces of even matrix pencils

classical dynamics of a family of billiards with analytic boundaries the classical dynamics of a billiard which is a quadratic conformal image of the unit disc is investigated the author gives the stability analysis of major periodic orbits present the poincare maps demonstrate the mixing properties by following the evolution of a small element in phase space show the existence of homoclinic points and calculate the lyapunov exponent and the kolmogorov entropy h it turns out that the system becomes strongly chaotic positive h for sufficiently large deformations of the unit disc the system shows a generic stochastic transition the computations suggest that the system is mixing if the boundary is not convex

the generalized continuous algebraic riccati equation and impulse free continuous time lq optimal control abstract the purpose of this paper is to investigate the role that the so called constrained generalized riccati equation plays within the context of continuous time singular linear quadratic lq optimal control this equation has been defined following the analogy with the discrete time setting however while in the discrete time case the connections between this equation and the linear quadratic optimal control problem has been thoroughly investigated to date very little is known on these connections in the continuous time setting this note addresses this point we show in particular that when the continuous time constrained generalized riccati equation admits a solution the corresponding linear quadratic problem admits an impulse free optimal control we also address the corresponding infinite horizon lq problem for which we establish a similar result under the additional constraint that there exists a control input for which the cost index is finite

the quadratic matrix inequality in singular h control with state feedback in this paper the standard h infty control problem using state feedback is considered given a linear time invariant finite dimensional system this problem consists of finding a static state feedback such that the resulting closed loop transfer matrix has h infty norm smaller than some a priori given upper bound in addition it is required that the closed loop system is internally stable conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality reminiscent of the dissipation inequality of singular linear quadratic optimal control where the direct feedthrough matrix of the control input is injective the results presented here specialize to known results in terms of solvability of a certain indefinite algebraic riccati equation

the singular h 2 control problem in this paper we discuss the standard h2 control problem for linear finite dimensional time invariant systems without any assumptions on the system parameters we give an explicit formula for the infimum over all internally stabilizing strictly proper compensators and give a characterization when the infimum is attained

squaring down lti systems a geometric approach in this paper the problem of reducing a given lti system into a left or right invertible one is addressed and solved with the standard tools of the geometric control theory first it will be shown how an lti system can be turned into a left invertible system thus preserving key system properties like stabilizability phase minimality right invertibility relative degree and infinite zero structure moreover the additional invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane by duality the scheme of a right inverter will be derived straightforwardly moreover the squaring down problem will be addressed in fact when the left and right reduction procedures are applied together a system with an unequal number of inputs and outputs is turned into a square and invertible system furthermore as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility of the plant in many optimization problems

on general ℋ2 control from frequency to time domain this article presents a general formula for discrete time 2 control it works with the regular and singular case of 2 control i e in the case of possibly non left invertible matrices g 12 and non right invertible matrices g 21 with possible unit circle invariant zeros in the generic case it can be simplified and adapted to work with the plant transfer matrix directly without invoking the matrices of the parameterisation of stabilising controllers a further result of this article is the presented necessary and sufficient conditions for state space 2 control under only stabilisability and detectability assumptions if the conditions are satisfied an observer based 2 controller is constructed the corresponding numerical algorithm consists of solving two discrete time algebraic riccati systems darss and two eigen problems

a new class of particle methods particle methods are numerical methods designed to solve problems in fluid mechanics and related problems in continuum mechanics a general approach to the construction of such particle methods is presented in this article the particles are no mass points but possess a finite extension they can rotate in space and have a spin the conservation of mass is automatically guaranteed by the ansatz the forces of interaction between the particles are derived in a canonical way from the force laws of continuum mechanics and are directly based on a regularized stress tensor in the absence of external forces and of heat sources and sinks momentum angular momentum and energy are conserved as in the continuum case

entropy generation and shock resolution in the particle model of compressible fluids the examination of the particle model of compressible fluids that has been developed by the author numer math 1997 76 111 142 and that has recently been extended to particles of variable size numer math 1999 82 143 159 is continued it is shown that in the limit of particle sizes tending to zero both the mass density and the mass flux density and the entropy density and the entropy flux density converge in the weak sense and satisfy the corresponding conservation laws to incorporate entropy generation in shocks a new kind of viscous force is introduced

singular lq control optimal pd controller and inadmissible initial conditions we consider a classical control problem the infinite horizon singular lq problem i e some inputs are unpenalized in the quadratic performance index in this case it is known that the slow dynamics is constrained to be in a proper subspace of the state space with the optimal input for the slow dynamics implementable by feedback in this technical note we show that both the fast dynamics and the slow dynamics can be implemented by a feedback controller moreover we show that the feedback controller cannot be a static feedback controller but can be pd i e proportional differentiate exactly once in the state we show that the closed loop system is a singular descriptor state space system and we also characterize the conditions on the system performance index for existence of inadmissible initial conditions i e initial conditions that cause impulsive solutions there are no inadmissible initial conditions in the controlled system if and only if in the strictly proper transfer matrix from the unpenalized inputs to the penalized states there exists at least one maximal minor of relative degree equal to the number of unpenalized inputs in addition to the above we prove solvability of the infinite horizon singular lq problem under milder assumptions than in the literature

computation of generalized h 2 optimal controllers the generalized h 2 optimal control problem for a linear time invariant system is one in which the conventional h 2 norm is replaced by an operator norm the closed loop system is described in terms of a mapping between the space of time domain input disturbances in l 2 and the space of time domain regulated outputs in l a minimum of this norm is then sought over all stabilizing controllers it is shown that optimal controllers for such problems have the structure of a kalman filter with estimated state feedback where the feedback gains are obtained from the solution to a weighted lqr problem a computational algorithm is presented to determine the weights in this lqr problem and examples are given which demonstrate various problems which may arise in obtaining the optimal weights in particular it is shown that the generalized h 2 problem may involve the solution to a singular lqr problem

the h2 control problem a general transfer function solution a general solution of the h2 control problem is presented for linear systems described by rational transfer matrices not necessarily proper or stable the control system is considered in the standard configuration which includes the synthesis model of the plant and the controller the h2 control problem consists of internally stabilizing the control system while minimizing the h2 norm of its transfer function the notion of internal stability is based on bounded input bounded output stability and means that subsystems defined by any pair of input and output signals within the control system all are bounded input bounded output stable in this manner the optimal control system is devoid of impulsive as well as non decaying exponential modes the solution proceeds in three steps first the set of all controllers that internally stabilize the control system is parameterized then the subset of the stabilizing controllers that achieve a finite value of the h2 norm of the control system transfer matrix is

survey of the eigenfunctions of a billiard system between integrability and chaos we study numerically the eigenfunctions and their wigner phase space distributions of the two dimensional billiard system defined by the quadratic conformal image of the unit disk as introduced by robnik 1983 this system is a generic kam system and displays a transition from integrability to almost ergodicity as the billiard shape changes we clearly identify two classes of states the regular ones associated with integrable regions and the irregular states supported on classically chaotic regions whilst the mixed type states were not found in support of percival s conjecture 1973 we confirm the existence of extremely intense scars in the classically chaotic regions and demonstrate their association with classical periodic orbits three classes of scars are revealed one orbit scars many orbit one family scars of statistically similar orbits in the homoclinic neighbourhood and many orbit many family scars we argue that it is impossible to find an a priori semiclassical theory of individual eigenstates but do not deny the usefulness of general semiclassical arguments in analysing the collective and statistical properties of eigenstates

the role of the dissipation matrix in singular optimal control the optimal cost in the regular stationary linear quadratic optimal control problem is given by the maximal solution of an algebraic riccati equation this solution minimizes the rank of a certain matrix which we call the dissipation matrix the rank minimization problem is also meaningful in the singular case does it provide the optimal cost for the singular control problem this question was posed by j c willems in 1971 the answer is yes

simplified formula for a controller in optimal control problems we derive a simplified frequency domain formula for a controller transfer matrix for lq optimal output feedback control of stochastic systems for this purpose we apply a generalization of the wiener hopf method the generalization is characterized by the following three properties 1 we generalize the usual operator cdot on rational matrices in the traditional wiener hopf approach the same algorithms as for computation of cdot are applicable to compute the new operator 2 the matrices of the youla kuccera parametrization do not appear in the optimal controller transfer matrix bf c they appear only in its derivation even if the plant is unstable 3 unlike the traditional wiener hopf method d c youla h a jabr and j j bongiorno jr ieee trans automat control ac 21 1976 pp 319 338 where the spectral factors in the two spectral factorizations are both stable and minimum phase our spectral factors need to be minimum phase only finally three state space applications of the formula are presented

h2 optimal control for generalized discrete time systems abstract for a generalized discrete time system whose transfer function matrix may be improper or polynomial we give realization based formulas for the h 2 optimal output feedback controller the formulas bear essentially the same elegant simplicity of the proper case

amplitude distribution of eigenfunctions in mixed systems we study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space for an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and a good agreement with numerical computations for the family of limacon billiards is found the natural extension of our result to more general systems e g with a potential is also discussed

principles for the design of billiards with nonvanishing lyapunov exponents we introduce a large class of billiards with convex pieces of the boundary which have nonvanishing lyapunov exponents

construction and parameterization of all static and dynamic h sub 2 optimal state feedback solutions optimal fixed modes and fixed decoupling zeros a problem of h sub 2 optimization via state feedback is considered the problems dealt with are of the general singular type with a left invertible transfer matrix function from the control input to the controlled output all the static and dynamic h sub 2 optimal state feedback solutions are constructed and parameterized and all the eigenvalues of an optimal closed loop system are characterized all optimal closed loop systems share a set of eigenvalues which are termed the optimal fixed modes which must be assigned among the closed loop eigenvalues this set includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and location of pole zero cancellations present in any h sub 2 optimal design it is shown that both the sets of optimal fixed modes and optimal fixed decoupling zeros do not vary

energy level statistics in the transition region between integrability and chaos a generic one parameter family of billiards discovered and introduced by robnik 1983 is used to study the spectral properties of corresponding quantum systems when the parameter is varied a smooth transition from an integrable system over a typical kam system to an almost ergodic system can be observed the authors calculate up to 7600 lowest reliable energy levels a detailed analysis of the numerical data shows significant deviation from the semiclassical berry robnik formulae for the nearest neighbour level spacing distribution p s except for large level spacings s 1 which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct at small s the power law level repulsion is clearly observed and a fit by the phenomenological formula by izrailev 1988 1989 is statistically significant

the propagation of sound in particle models of compressible fluids the propagation of sound in compressible fluids is described by the acoustic equations that result from the linearization of the euler equations around a state of constant mass density and velocity zero in this article it is shown that a stable and convergent discretization of the acoustic wave equation for the velocity field can be recovered from the particle model of compressible fluids recently developed by the author in numer math 1997 76 111 142 by linearizing the equations of motion for the particles for particles of proper shape this discretization is second order accurate and with an obvious modification of the basic particle model one can even reach an arbitrarily high order of convergence

semiclassical spatial correlations in chaotic wave functions we study the spatial autocorrelation of energy eigenfunctions psi n q corresponding to classically chaotic systems in the semiclassical regime our analysis is based on the weyl wigner formalism for the spectral average c epsilon q q e of psi n q psi n q defined as the average over eigenstates within an energy window epsilon centered at e in this framework c epsilon is the fourier transform in the momentum space of the spectral wigner function w x e epsilon our study reveals the chord structure that c epsilon inherits from the spectral wigner function showing the interplay between the size of the spectral average window and the spatial separation scale we discuss under which conditions is it possible to define a local system independent regime for c epsilon in doing so we derive an expression that bridges the existing formulas in the literature and find expressions for c epsilon q q e valid for any separation size q q

a geometric approach to the singular filtering problem we consider the least squares filtering problem for a stationary gaussian process when the observation is not fully corrupted by white noise the so called singular case an optimal estimator is constructed consisting of an integrating part which is as in the regular case computed from a spectral factorization or an equivalent matrix problem and a differentiating part whose parameters are computed from a single matrix equation this improves on older results which either work under restrictive assumptions or describe the solution only as the result of some nested algorithm

lmi solution to general suboptimal control problems abstract in this paper we derive an lmi solution to general h 2 suboptimal control problems without any unnecessary assumptions necessary and sufficient conditions for the h 2 norm of the closed loop transfer matrix to be bounded by a given number is given by 4 coupled lmis a suboptimal controller is also obtained the obtained solution applies to both nonsingular and singular cases

quantising a generic family of billiards with analytic boundaries a generic family of plane billiards has been discovered recently the shape of the boundary is given by the quadratic conformal image of the unit circle and is thus real analytic for small deformations of the unit disc the billiard is a typical kam system but becomes ergodic or even mixing when the curvature of the boundary vanishes at some point the kolmogorov entropy has been calculated and it increases with the deformation of the boundary the author studies aspects of the quantum chaos for this billiard he solves numerically the eigenvalue problem for the laplace operator with dirichlet s boundary condition he examines the spectrum and inspects the avoided crossings at which mixing of nearby states occurs the variation of the nodal structure and of the localisation properties of the eigenfunctions is studied in analysing the level spacing distribution he finds a continuous transition from the poisson distribution towards the wigner distribution the exponent in the level repulsion law varies continuously along with a generic perturbation for small perturbations it seems to be proportional to the square root of the perturbation parameter

the finite mass method on domains with boundary the finite mass method is a gridless lagrangian method to simulate compressible flows that has been introduced in a recent paper from gauger leinen and yserentant siam j numer anal 37 2000 pp 1768 1799 it is based on a discretization of mass not of space as with classical discretization schemes mass is subdivided into little mass packets of finite extension each of which is equipped with finitely many internal degrees of freedom these mass packets move under the influence of internal and external forces and the laws of thermodynamics and can change their shape to follow the motion of the fluid only free flows in vacuum have been considered so far in this article a concept is presented to extend the method to flows in domains having boundaries it maintains the second order accuracy of the basic method and can be implemented along the same lines

on the k property of some planar hyperbolic billiards thek property is demonstrated for a class of planar billiards satisfying wojtkowski s principles their boundary may consist of convex scattering concave and linear pieces earlier wojtkowski showed that these billards had non zero lyapunov exponents

performance with regulation constraints we study the problem where we have a regulation asymptotic tracking requirement together with a performance requirement for continuous time systems typically we measure performance by the h 2 or h norm of a chosen transfer function matrix although any other norm such as the l 1 norm could be used in the case when the performance is measured by the h 2 norm there is no loss in performance due to the regulation constraint on the other hand when the performance is measured by the h norm there exists in general certain loss of performance due to the regulation constraint and we explicitly characterize such a loss determining suitable controllers for a given system with both regulation and performance constraints requires solving only an unconstrained optimization problem for an auxiliary system which is explicitly constructed here

random matrix elements and eigenfunctions in chaotic systems the expected root mean square value of a matrix element a ensuremath alpha ensuremath beta in a classically chaotic system where a is a smooth ensuremath elzxh independent function of the coordinates and momenta and ensuremath alpha and ensuremath beta label different energy eigenstates has been evaluated in the literature in two different ways by treating the energy eigenfunctions as gaussian random variables and averaging a ensuremath alpha ensuremath beta 2 over them and by relating a ensuremath alpha ensuremath beta 2 to the classical time correlation function of a we show that these two methods give the same answer only if berry s formula for the spatial correlations in the energy eigenfunctions which is based on a microcanonical density in phase space is modified at large separations in a manner that we previously proposed

the singular control problem with dynamic measurement feedback this paper is concerned with the h infty problem with measurement feedback the problem is to find a dynamic feedback from the measured output to the control input such that the closed loop system has an h infty norm strictly less than some a priori given bound gamma and such that the closed loop system is internally stable necessary and sufficient conditions are given under which such a feedback exists the only assumption that must be made is that there are no invariant zeros on the imaginary axis for two subsystems contrary to recent publications no assumptions are made on the direct feedthrough matrices of the plant it turns out that this problem can be reduced to an almost disturbance decoupling problem with measurement feedback and internal stability i e the problem in which we can make the h infty norm arbitrarily small

study of regular and irregular states in generic systems in this work we present the results of a numerical and semiclassical analysis of high lying states in a hamiltonian system whose classical mechanics is of a generic mixed type where the energy surface is split into regions of regular and chaotic motion as predicted by the principle of uniform semiclassical condensation when the effective tends to zero each state can be classified as regular or irregular we were able to semiclassically reproduce individual regular states by the einstein brillouin keller torus quantization for which we devise a new approach while for the irregular ones we found the semiclassical prediction of their autocorrelation function in a good agreement with numerics we also looked at the low lying states to get a better understanding of the onset of semiclassical behaviour

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

singular optimal control a geometric approach linear quadratic singular optimal control problem is solved for nonminimum phase and noninvertible systems a state space decomposition is obtained and a reduced order nonsingular subproblem is solved the optimal stabilizing input of the singular problem has been found when there are no transmission zeros on the imaginary axis

autocorrelation function of eigenstates in chaotic and mixed systems we study the autocorrelation function of different types of eigenfunctions in quantum mechanical systems with either chaotic or mixed classical limits we obtain an expansion of the autocorrelation function in terms of the correlation distance for localized states in billiards like bouncing ball modes or states living on tori a simple model using only classical input gives good agreement with the exact result in particular a prediction for irregular eigenfunctions in mixed systems is derived and tested for chaotic systems the expansion of the autocorrelation function can be used to test quantum ergodicity on different length scales

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

particles of variable size a problem of all particle methods is that they produce large vacuum regions when they are applied to a free gas flow for example with the approach recently proposed by the author numer math 1997 76 111 142 this difficulty can be avoided one can let the particles adapt their size to the local state of the fluid how is described in the present article the diameter as an additional degree of freedom strongly improves the performance of the numerical methods based on this particle model

brief paper a new approach to the cheap lq regulator exploiting the geometric properties of the hamiltonian system the cheap lq regulator is reinterpreted as an output nulling problem which is a basic problem of the geometric control theory in fact solving the lq regulator problem is equivalent to keep the output of the related hamiltonian system identically zero the solution lies on a controlled invariant subspace whose dimension is characterized in terms of the minimal conditioned invariant of the original system and the optimal feedback gain is computed as the friend matrix of the resolving subspace this study yields a new computational framework for the cheap lq regulator relying only on the very basic and simple tools of the geometric approach namely the algorithms for controlled and conditioned invariant subspaces and invariant zeros

statistical properties of high lying chaotic eigenstates we study the statistical properties of the high lying chaotic eigenstates 200000 and above which are deep within the semiclassical regime the system we are analysing is the billiard system inside the region defined by the quadratic complex conformal map of the unit disk as introduced by robnik 1983 we are using heller s method of plane wave decomposition of the numerical eigenfunctions and perform extensive statistical analysis with the following conclusions i the local average probability density is in excellent agreement with the microcanonical assumption and all statistical properties are also in excellent agreement with the gaussian random model ii the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with berry s 1977 prediction iii although the scars of unstable classical periodic orbits in such an ergodic regime are expected to exist so far we have not found any around the 200000th state other than a scar like feature resembling the whispering gallery modes

brief paper generalized asymptotic regulation with guaranteed h2 performance an lmi solution a multi objective controller synthesis problem is considered in which a set of generalized asymptotic regulation constraints are to be satisfied while also guaranteeing a desired level of performance measured in terms of the asymptotic variance of a performance output in response to a white noise random process with zero mean and identity covariance matrix the generalized regulation constraints are expressed as bounds on the state steady peak in response to disturbances generated by an anti stable autonomous exogenous system with nonzero initial states the controllers that guarantee these constraints can be realized by replicating the dynamics of the exogenous system within a certain structure formed by gain matrices which are subject to convex constraints and an accompanying controller with which the feedback loop is to be stabilized formulating the design of the accompanying controller in a way to guarantee the additional performance objective the overall design is rendered tractable over a set of free variables in terms of which the synthesis of a suitable controller is also outlined the order of the controller is equal to the order of the plant plus the order of the exogenous system by an adaptation of the developed techniques a solution is also provided for the problem of generalized asymptotic regulation with suboptimal transient response

all optimal controls for the singular linear quadratic problem without stability a new interpretation of the optimal cost abstract the singular linear quadratic control problem without stability is solved by means of a generalized dual structure algorithm in order to generate all optimal inputs furthermore it is shown that the optimal cost can be interpreted as the smallest nonnegative rank minimizing solution of a certain matrix inequality the so called dissipation inequality

gaussian random eigenfunctions and spatial correlations in quantum dots we show that spatial correlations in wave functions of quantum dots obtained earlier by averaging over a random potential via supermatrix techniques can be computed with much less effort by making use of berry s conjecture that the energy eigenfunctions in a quantized chaotic system are gaussian random variables furthermore in the case of a time reversal invariant system we find a greatly simplified though equivalent formula for these correlations textcopyright 1996 the american physical society

how chaotic is the stadium billiard a semiclassical analysis the impression gained from the literature published to date is that the spectrum of the stadium billiard can be adequately described semiclassically by the gutzwiller periodic orbit trace formula together with a modified treatment of the marginally stable family of bouncing ball orbits i show that this belief is erroneous the gutzwiller trace formula is not applicable for the phase space dynamics near the bouncing ball orbits unstable periodic orbits close to the marginally stable family in phase space cannot be treated as isolated stationary phase points when approximating the trace of the green s function semiclassical contributions to the trace show an dependent transition from hard chaos to integrable behaviour for trajectories approaching the bouncing ball orbits a whole region in phase space surrounding the marginal stable family acts semiclassically like a stable island with boundaries being explicitly dependent the localized bouncing ball states found in the billiard derive from this semiclassically stable island the bouncing ball orbits themselves however do not contribute to individual eigenvalues in the spectrum an ebk like quantization of the regular bouncing ball eigenstates in the stadium can be derived the stadium billiard is thus an ideal model for studying the influence of almost regular dynamics near marginally stable boundaries on quantum mechanics the behaviour is generically found at the border of classically stable islands in systems with a mixed phase space structure

families of linear quadratic problems continuity properties in this paper we investigate for given one parameter families of linear time invariant finite dimensional systems the parameter dependence of the linear quadratic optimal cost optimal control inputs optimal state trajectories and optimal outputs it is shown that results that have been obtained in the past in the context of the problem of cheap control can in fact be generalized to a much broader class of parameter dependent cost functionals including cost functionals in which for every parameter value the weighting matrix of the control inputs is singular essentially only two assumptions on the parameter dependence of the cost functionals are required in order to have continuity of the optimal cost and optimal control inputs with respect to the underlying parameter one assumption is concerned with the continuity of the weighting matrices with respect to this parameter the other with the monotonicy of the weighting matrices with respect to the parameter instrumental in our development is a characterization of the linear quadratic optimal cost in terms of the so called dissipation inequality the results obtained are applied to the problem of cheap control and to a problem of priority control the latter provides an example of a family of quadratic cosl functionals with a polynomial parameter dependence

a particle model of compressible fluids fluid mechanics describes the motion of mass in space under the influence of internal and external forces the particle model presented in this article is based on this fact the fluid is subdivided into a finite number of small mass packets the particles these mass packets have a finite extension and share all properties with the fluid except for the restriction that they cannot get deformed and can perform only rigid body motions the forces acting upon the particles are identical to those acting on a part of a fluid the exact conservation of mass and for the case of adiabatic flows also of entropy is automatically guaranteed by the approach when the particle size tends to zero the mean local displacement of the particles converges in the weak sense in the inviscid case the resulting flows can be regarded as solutions of the euler equations

full and reduced order observer based controller design for h_2 optimization in this paper the most general h2 control problem is considered we derive necessary and sufficient conditions when the infimum is attained by state feedback we do the same for the measurement feedback case where we derive necessary and sufficient conditions when the infimum is attained by proper dynamic compensators we also investigate reduced order compensators if some states are observable without noise we discuss for all of these cases the freedom that the non uniqueness of optimal compensators gives us in assigning the closed loop eigenvalues the second half of this paper investigates the case when the infimum cannot be attained we give a constructive algorithm to find a minimizing sequence of stabilizing controllers and discuss the freedom in the asymptotic locations of the closed loop eigenvalues again we do the above for three different cases static state feedback full order measurement feedback and reduced order measurement feedback

the finite mass method the finite mass method a new lagrangian method for the numerical simulation of gas flows is presented and analyzed in contrast to the finite volume and the finite element method the finite mass method is founded on a discretization of mass not of space mass is subdivided into small mass packets of finite extension each of which is equipped with finitely many internal degrees of freedom these mass packets move under the influence of internal and external forces and the laws of thermodynamics and can undergo arbitrary linear deformations the method is based on an approach recently developed by yserentant and can attain a very high accuracy

h2 optimal controllers with measurement feedback for continuous time systems flexibility in closed loop pole placement for a general h 2 optimal control problem at first all h 2 optimal measurement feedback controllers are characterized and parameterized and then attention is focused on controllers with observer based architecture both full order as well as reduced order observer based h 2 optimal controllers are characterized and parameterized also systematic methods of designing them are presented an important problem coined as an h 2 optimal control problem with simultaneous pole placement is formulated and solved that is since in general there exist many h 2 optimal measurement feedback controllers utilizing such flexibility and freedom we can solve the problem of simultaneously placing the closed loop poles at desirable locations whenever possible while still preserving h 2 optimality all the design algorithms developed here are easily computer implementable