structure and properties of hughston s stochastic extension of the schrodinger equation hughston has recently proposed a stochastic extension of the schrodinger equation expressed as a stochastic differential equation on projective hilbert space we derive new projective hilbert space identities which we use to give a general proof that hughston s equation leads to state vector collapse to energy eigenstates with collapse probabilities given by the quantum mechanical probabilities computed from the initial state we discuss the relation of hughston s equation to earlier work on norm preserving stochastic equations and show that hughston s equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix we discuss the behavior of systems constructed as direct products of independent subsystems and briefly address the question of whether an energy based approach such as hughston s suffices to give an objective interpretation of the measurement process in quantum mechanics

characterization of the solution to the optimal h model reduction problem abstract in this paper the h norm approximation of a given stable proper rational transfer function by a lower order stable proper rational transfer function is studied the solution method is based on the authors observation that the h model reduction problem can be converted into a hankel norm model reduction problem a simple characterization of the solution to the optimal h model reduction problem is developed

on the precise connection between the grw master equation and master equations for the description of decoherence we point out that the celebrated grw master equation is invariant under translations reflecting the homogeneity of space thus providing a particular realization of a general class of translation covariant markovian master equations such master equations are typically used for the description of decoherence due to momentum transfers between the system and environment building on this analogy we show the exact relationship between the grw master equation and decoherence master equations further providing a collisional decoherence model formally equivalent to the grw master equation this allows for a direct comparison of order of magnitudes of relevant parameters this formal analogy should not lead to confusion on the utterly different spirit of the two research fields in particular it has to be stressed that the decoherence approach does not lead to a solution of the measurement problem building on this analogy however the feasibility of the extension of spontaneous localization models in order to avoid the infinite energy growth is discussed apart from a particular case considered in the paper it appears that the amplification mechanism is generally spoiled by such modifications

a variational formulation of anisotropic geometric evolution equations in higher dimensions we present a novel variational formulation of fully anisotropic motion by surface diffusion and mean curvature flow in mathbb r d d 2 this new formulation leads to an unconditionally stable fully discrete parametric finite element approximation in the case d 2 or 3 the resulting scheme has very good properties with respect to the distribution of mesh points and if applicable volume conservation this is demonstrated by several numerical experiments for d 3 including regularized crystalline mean curvature flow and regularized crystalline surface diffusion

quantum stochastic processes as models for state vector reduction an elementary introduction of quantum state valued markovian stochastic processes qsp for n state quantum systems is given it is pointed out that a so called master constraint must be fulfilled for a given master equation a continuous and as a new alternative possibility a discontinuous qsp are derived both are discussed as possible models for state reduction during measurement

generation motion and thickness of transition layers for a nonlocal allen cahn equation abstract we investigate the behavior as e 0 of the nonlocal allen cahn equation u t u 1 e 2 f u e u where f u 0 is of the bistable type given a rather general initial datum u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of the interface and obtain a new estimate for its thickness more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the limit problem within an error margin of order e

martingale models for quantum state reduction stochastic models for quantum state reduction give rise to statistical laws that are in most respects in agreement with those of quantum measurement theory here we examine the correspondence of the two theories in detail making a systematic use of the methods of martingale theory an analysis is carried out to determine the magnitude of the fluctuations experienced by the expectation of the observable during the course of the reduction process and an upper bound is established for the ensemble average of the greatest fluctuations incurred we consider the general projection postulate of l uders applicable in the case of a possibly degenerate eigenvalue spectrum and derive this result rigorously from the underlying stochastic dynamics for state reduction in the case of both a pure and a mixed initial state we also analyse the associated lindblad equation for the evolution of the density matrix and obtain an exact time dependent solution for the state reduction that explicitly exhibits the transition from a general initial density matrix to the l uders density matrix finally we apply girsanov s theorem to derive a set of simple formulae for the dynamics of the state in terms of a family of geometric brownian motions thereby constructing an explicit unravelling of the lindblad equation

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

structured coprime factor model reduction based on lmis in this paper we discuss dynamic model reduction methods which preserve a certain structure in the underlying system specifically we consider the situation where the reduction must be consistent with a partition of the system states this is motivated for instance in situations where state variables are associated with the topology of a networked system and the reduction should preserve this we build on the observation that imposing block structure to generalized controllability and observability gramians automatically yields such state partitioned model reduction the difficulty lies in ensuring feasibility of the resulting lyapunov inequalities which is in general very restrictive to overcome this we consider coprime factor model reduction we derive an lmi characterization of expansive and contractive coprime factorizations that preserve structure and use this to build a more flexible method for structured model reduction an example is given to illustrate the method

coprime factors reduction methods for linear parameter varying and uncertain systems abstract we present a generalization of the coprime factors model reduction method of meyer and propose a balanced truncation reduction algorithm for a class of systems containing linear parameter varying and uncertain system models a complete derivation of coprime factorizations for this class of systems is also given the reduction method proposed is thus applicable to linear parameter varying and uncertain system realizations that do not satisfy the structured l 2 induced stability constraint required in the standard nonfactored case reduction error bounds in the l 2 induced norm of the factorized mapping are given

frequency dependent error bounds for uncertain linear models we study frequency dependent error bounds for approximation and truncation of linear dynamic models with uncertainty a possible application is in model simplification relevant for control design the uncertainty is described by quadratic constraints and the error bounds are calculated based on solutions to linear matrix inequalities

balanced truncation of linear time varying systems in this paper balanced truncation of linear time varying systems is studied in discrete and continuous time based on relatively basic calculations with time varying lyapunov equations inequalities we are able to derive both upper and lower error bounds for the truncated models these results generalize well known time invariant formulas the case of time varying state dimension is considered input output stability of all truncated balanced realizations is also proven the method is finally successfully applied to a high order model

continuous quantum jumps and infinite dimensional stochastic equations from a mathematical point of view a class of infinite dimensional stochastic differential equations describing continuous spontaneous localization in quantum dynamics will be studied existence and uniqueness of weak and strong solutions of respective equations are proven via cameron martin girsanov transformation the case of gaussian initial states is explicitly solved

the singular limit of the allen cahn equation and the fitzhugh nagumo system abstract we consider an allen cahn type equation of the form u t u e 2 f e x t u where e is a small parameter and f e x t u f u e g e x t u a bistable nonlinearity associated with a double well potential whose well depths can be slightly unbalanced given a rather general initial data u 0 that is independent of e we perform a rigorous analysis of both the generation and the motion of interface more precisely we show that the solution develops a steep transition layer within the time scale of order e 2 ln e and that the layer obeys the law of motion that coincides with the formal asymptotic limit within an error margin of order e this is an optimal estimate that has not been known before for solutions with general initial data even in the case where g e 0 next we consider systems of reaction diffusion equations of the form u t u e 2 f e u v v t d v h u v which include the fitzhugh nagumo system as a special case given a rather general initial data u 0 v 0 we show that the component u develops a steep transition layer and that all the above mentioned results remain true for the u component of these systems

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

relaxation of quantum states under energy perturbations the energy based stochastic extension of the schrodinger equation is perhaps the simplest mathematically rigorous and physically plausible model for the reduction of the wave function in this article we apply a new simulation methodology for the stochastic framework to analyse the dynamics of a particle confined to a square well potential we consider the situation when the width of the well is expanded instantaneously through this example we are able to illustrate in detail how a quantum system responds to an energy perturbation and the mechanism according to the stochastic evolutionary law by which the system relaxes spontaneously into one of the stable eigenstates of the hamiltonian we examine in particular how the expectation value of the hamiltonian and the probability distribution for the position of the particle change in time an analytic expression for the typical time scale of relaxation is derived we also consider the small perturbation limit and discuss the relation between the stochastic framework and the quantum adiabatic theorem

efficient simulation of quantum state reduction the energy based stochastic extension of the schrodinger equation is a rather special nonlinear stochastic differential equation on hilbert space involving a single free parameter that has been shown to be very useful for modeling the phenomenon of quantum state reduction here we construct a general closed form solution to this equation for any given initial condition in terms of a random variable representing the terminal value of the energy and an independent brownian motion the solution is essentially algebraic in character involving no integration and is thus suitable as a basis for efficient simulation studies of state reduction in complex systems

development of interfaces in ℝ n consider the reaction diffusion equation in n u t h 2 u u 0 where is the derivative of a bistable even potential and h is a small parameter if the initial data have a smooth noncritical zero set we prove that an interface appears in time o log h 1 and that the solution stays close to it for at least time o 1 h

signal processing with levy information levy processes which have stationary independent increments are ideal for modelling the various types of noise that can arise in communication channels if a levy process admits exponential moments then there exists a parametric family of measure changes called esscher transformations if the parameter is replaced with an independent random variable the true value of which represents a message then under the transformed measure the original levy process takes on the character of an information process in this paper we develop a theory of such levy information processes the underlying levy process which we call the fiducial process represents the noise type each such noise type is capable of carrying a message of a certain specification a number of examples are worked out in detail including information processes of the brownian poisson gamma variance gamma negative binomial inverse gaussian and normal inverse gaussian type although in general there is no additive decomposition of information into signal and noise one is led nevertheless for each noise type to a well defined scheme for signal detection and enhancement relevant to a variety of practical situations

geometry of stochastic state vector reduction the state space of a quantum mechanical system is a complex projective space the space of rays in the associated hilbert space the state space comes equipped with a natural riemannian metric the fubini study metric and a compatible symplectic structure the operations of ordinary quantum mechanics can thus be reinterpreted in the language of differential geometry it is interesting in this spirit to scrutinize the probabilistic assumptions that are brought in at various stages in the analy sis of quantum dynamics particularly in connection with state vector reduction a promising approach to understanding reduction studied recently by a number of authors involves the use of nonlinear stochastic dynamics to modify the ordinary linear schrodinger evolution here we use stochastic differential geometry to give a systematic geometric formulation for such stochastic models of state vector collapse in this picture the conventional schrodinger evolution which corresponds to the unitary flow associated with a killing vector of the fubini study metric is replaced by a more general stochastic flow on the state manifold in the simplest example of such a flow the volatility term in the stochastic differential equation for the state trajectory is proportional to the gradient of the expectation of the hamiltonian the conservation of energy is represented by the requirement that the actual process followed by the expectation of the hamiltonian as the state evolves should be a martingale this requirement implies the existence of a nonlinear term in the drift vector of the state process which is always oriented opposite to the direction of in creasing energy uncertainty as a consequence the state vector necessarily collapses to an energy eigenstate and a martingale argument can be used to show that the probability of collapse to a given eigenstate from any particular initial state is in fact given by precisely the usual quantum mechanical probability

modeling and control of quantum systems an introduction the scope of this work is to provide a self contained introduction to a selection of basic theoretical aspects in the modeling and control of quantum mechanical systems as well as a brief survey on the main approaches to control synthesis while part of the existing theory especially in the open loop setting stems directly from classical control theory most notably geometric control and optimal control a number of tools specifically tailored for quantum systems have been developed since the 1980s in order to take into account their distinctive features the probabilistic nature of atomic scale physical systems the effect of dissipation and the irreversible character of the measurements have all proved to be critical in feedback design problems the relevant dynamical models for both closed and open quantum systems are presented along with the main results on their controllability and stability a brief review of several currently available control design methods is meant to provide the interested reader with a roadmap for further studies

model reduction of multidimensional and uncertain systems model reduction methods are presented for systems represented by a linear fractional transformation on a repeated scalar uncertainty structure these methods involve a complete generalization of balanced realizations balanced gramians and balanced truncation model reduction with guaranteed error bounds based on solutions to a pair of linear matrix inequalities which generalize lyapunov equations the resulting reduction methods immediately apply to uncertainty simplification and state order reduction in the case of uncertain systems but also may be interpreted as state order reduction for multidimensional systems

is measurement based feedback still better for quantum control systems in this paper we put forward a fundamental question concerning feedback control of quantum systems is measurement based feedback control still better than open loop control in contrast to the classical control theory the answer is far from obvious this is because measurement based feedback needs measurement to reduce the system uncertainty whereas the measurement on a quantum system will inevitably increase the system uncertainty in turn in fact there is a complicated tradeoff between the uncertainty introduced and the information gained by the measurement on a quantum system to investigate this fundamental problem we will only focus on a typical model of coherent control mode with and without the decoherence term in the paper by establishing some fundamental limits on the performances of both the open loop and measurement based feedback controls we will demonstrate via simulation that the measurement based feedback control of quantum systems is still superior to the open loop control in some sense for the typical model under consideration

singular limit of a reaction diffusion equation with a spatially inhomogeneous reaction term we study reaction diffusion equations with a spatially inhomogeneous reaction term if the coefficient of these reaction term is much larger than the diffusion coefficient a sharp interface appears between two different phases we show that the equation of motion of such an interface involves a drift term despite the absence of drift in the original diffusion equations in particular we show that the same rich spatial patterns observed for a chemotaxis growth model can be realized by a system without a drift term

the structure of characteristic values for classes of linear time invariant systems abstractin this paper a particular structure of the characteristic values found in some classes of systems is dealt with in particular for the linear time invariant systems in minimal form a b c d with transfer function matrix g s for which if i is a characteristic value then also 1 i is consequently when the system order is odd a singular value is equal to one the classes of systems for which this property holds are characterised by the existence of specific state space representations moreover the same structure of the characteristic values can be also retrieved for the class of even g s gt s and odd g s gt s systems several examples related to multi input multi output mimo systems are reported also referring to real physical systems

error bounds for balanced model reduction of linear time varying systems error bounds are developed for balanced truncation of linear time varying systems leading to an extension of the twice the sum of the tail formulas well known in the time invariant case the approach relies on an operator theoretic framework for analysis of linear time varying systems this provides a multivariable notion of frequency for such systems which are thus characterized by rational functions of many complex variables allowing the problem to be formulated in the linear fractional framework using a time varying version of standard necessary conditions for reduced order modeling based on convex operator inequalities we show that these error bounds for balanced truncation are related to the closest possible reduced order modeling error in a sense which parallels the time invariant case

an improved error estimate for reduced order models of discrete time systems the authors derive under weaker conditions a discrete time counterpart of k glover s 1984 error estimate for reduced order models following the same lines a simple proof of glover s result for continuous time systems can be given

a gramian based approach to model reduction for uncertain systems the technical note considers a problem of model reduction for a class of uncertain systems with structured norm bounded uncertainty the technical note introduces controllability and observability gramians in terms of certain parameterized algebraic riccati inequalities based on these gramians three model reduction approaches are investigated for the underlying uncertain systems

motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic allen cahn equation we consider the spatially inhomogeneous and anisotropic reaction diffusion equation ut m x 1 div m x ap x u e 2 f u involving a small parameter e 0 and a bistable nonlinear term whose stable equilibria are 0 and 1 we use a finsler metric related to the anisotropic diffusion term and work in relative geometry we prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces more precisely we show that within the time scale of order e2 ln e the unique weak solution u e develops a steep transition layer that separates the regions u e 0 and u e 1 then on a much slower time scale the layer starts to propagate consequently as e 0 the solution u e converges almost everywhere a e to 0 in t and 1 in t where t and t are sub domains of separated by an interface t whose motion is driven by its anisotropic mean curvature we also prove that the thickness of the transition layer is of order e

notions of observability for uncertain linear systems with structured uncertainty this paper introduces a notion of observability for a class of uncertain linear systems with structured uncertainty in the uncertain systems under consideration the uncertainty is described by averaged integral quadratic constraints in order to define a notion of observability for uncertain linear systems the paper introduces a robust observability function which extends the usual definition of the observability gramian to the case of uncertain systems using this observability function a corresponding unobservable cone is defined and an uncertain system is said to be robustly observable if this cone contains only the origin the paper presents an algorithm for finding the robust observability function and corresponding unobservable cone this algorithm involves solving a parameterized riccati differential equation

finite time stochastic reduction models a new energy based stochastic extension of the schrodinger equation for which the wave function collapses after the passage of a finite amount of time is proposed an exact closed form solution to the dynamical equation valid for all finite dimensional quantum systems is presented and used to verify explicitly that reduction of the state vector to an energy eigenstate occurs a time change technique is introduced to construct a clock variable that relates the asymptotic and the finite time collapse models by means of a nonlinear transformation

a new set of invariants for linear systems application to reduced order compensator design a new set of invariants for linear systems weighting the contribution of each state component to the inherent closed loop lqg behavior of the system is presented together with applications to model order reduction and reduced order compensator design

stochastic schrodinger evolution over piecewise enlarged filtrations this paper constructs a nonlinear filtering framework that admits appearances of new information processes at random times by introducing piecewise enlargements of filtrations and proposes a new energy based schrodinger evolution expressed as a stochastic differential equation on a complex hilbert space each information process is modeled as the sum of a random variable taking the eigenvalues of a hamiltonian and an independent brownian bridge noise it is shown that under a piecewise enlarged filtration the wave function is a jump diffusion process until it collapses at some terminal time in between discontinuities the dynamics of the state vector are governed by different wiener processes and diffusion coefficients this motivates the introduction of an inclusive chain of kolmogorov probability spaces or a isomorphic chain of commutative von neumann probability spaces on which the quantum system evolves differently based on the number of active information processes the expectation of the hamilt

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

a convex characterization of gain scheduled h sub spl infin controllers an important class of linear time varying systems consists of plants where the state space matrices are fixed functions of some time varying physical parameters spl theta small gain techniques can be applied to such systems to derive robust time invariant controllers yet this approach is often overly conservative when the parameters spl theta undergo large variations during system operation in general higher performance can be achieved by control laws that incorporate available measurements of spl theta and therefore adjust to the current plant dynamics this paper discusses extensions of h sub spl infin synthesis techniques to allow for controller dependence on time varying but measured parameters when this dependence is linear fractional the existence of such gain scheduled h sub spl infin controllers is fully characterized in terms of linear matrix inequalities the underlying synthesis problem is therefore a convex program for which efficient optimization techniques are available the formalism and derivation techniques developed here apply to both the continuous and discrete time problems existence conditions for robust time invariant controllers are recovered as a special case and extensions to gain scheduling in the face of parametric uncertainty are discussed in particular simple heuristics are proposed to compute such controllers

control of quantum systems despite feedback delay feedback control based on the quantum continuous measurement of quantum systems inevitably suffers from estimation delays in this technical note we give a delay dependent stability criterion for a wide class of nonlinear stochastic systems including quantum spin systems we utilize a semi algebraic problem approach to incorporate the structure of density matrices to show the effectiveness of the result we derive a globally stabilizing control law for a quantum spin 1 2 systems in the face of feedback delays

induced l 2 norm model reduction of polytopic uncertain linear systems abstract in this paper we study an induced l 2 norm model reduction problem for polytopic uncertain linear systems the polytopic system has its state space data contained in a convex polytope a situation which often arises the problem we address is to approximate this type of system by a lower order polytopic uncertain linear system with a guaranteed induced l 2 norm error a sufficient solvability condition is provided in terms of lmis with one extra coupling rank constraint which generally leads to a nonconvex feasibility problem we use an improved computational scheme based on the alternating projection method to obtain solutions to the problem but only local convergence is guaranteed the reduced order uncertain system with its associated polytope is given explicitly by the solution of the feasibility problem

collapse models with non white noises we set up a general formalism for models of spontaneous wavefunction collapse with dynamics represented by a stochastic differential equation driven by general gaussian noises not necessarily white in time in particular we show that the non schrodinger terms of the equation induce the collapse of the wavefunction to one of the common eigenstates of the collapsing operators and that the collapse occurs with the correct quantum probabilities we also develop a perturbation expansion of the solution of the equation with respect to the parameter which sets the strength of the collapse process such an approximation allows one to compute the leading order terms for the deviations of the predictions of collapse models with respect to those of standard quantum mechanics this analysis shows that to leading order the imaginary noise trick can be used for non white gaussian noise