stabilization and regulation of nonlinear singularly perturbed systems composite control for a general class of nonlinear singularly perturbed systems a two time scale analysis and design procedure for stabilization and regulation is presented it is shown that under a set of conditions the composite control proposed by chow and kokotovic 5 asymptotically stabilizes the desired equilibrium and produces a finite cost j uc which tends to the cost of the slow regulator as the perturbation parameter e tends to zero thus the well posedness of the full regulator problem is established furthermore under different stability requirements on the fast subsystem explicit upper and lower bounds on j uc are obtained finally for each result in this paper an upper bound on the perturbation parameter e is provided under which the result is valid

on the mean field and classical limits of quantum mechanics the main result in this paper is a new inequality bearing on solutions of the n body linear schrodinger equation and of the mean field hartree equation this inequality implies that the mean field limit of the quantum mechanics of n identical particles is uniform in the classical limit and provides a quantitative estimate of the quality of the approximation this result applies to the case of c1 1 interaction potentials the quantity measuring the approximation of the n body quantum dynamics by its mean field limit is analogous to the monge kantorovich or wasserstein distance with exponent 2 the inequality satisfied by this quantity is reminiscent of the work of dobrushin on the mean field limit in classical mechanics func anal appl 13 115 123 1979 our approach to this problem is based on a direct analysis of the n particle liouville equation and avoids using techniques based on the bbgky hierarchy or on second quantization

balancing for a class of non linear singularly perturbed systems in this work a balancing method is investigated for a class of non linear singularly perturbed systems the main result presented here shows that the well known two stages strategy used with singular perturbations in control theory can be extended to compute a balancing form of non linear singularly perturbed systems so an approximate balancing form is derived from the balancing forms of the slow and fast subsystems both computed separately this two stage method avoids the difficult task of solving high dimensional and ill conditioned hamilton jacobi equations due to the presence of the small singular perturbation parameter

ricci curvature of finite markov chains via convexity of the entropy we study a new notion of ricci curvature that applies to markov chains on discrete spaces this notion relies on geodesic convexity of the entropy and is analogous to the one introduced by lott sturm and villani for geodesic measure spaces in order to apply to the discrete setting the role of the wasserstein metric is taken over by a different metric having the property that continuous time markov chains are gradient flows of the entropy using this notion of ricci curvature we prove discrete analogues of fundamental results by bakry emery and otto villani further we show that ricci curvature bounds are preserved under tensorisation as a special case we obtain the sharp ricci curvature lower bound for the discrete hypercube

convexity of quasi entropy type functions lieb s and ando s convexity theorems revisited given a positive function f on 0 and a non zero real parameter we consider a function if a b x tr x f larb 1 rb x in three matrices a b 0 and x this generalizes the notion of monotone metrics on positive definite matrices and in the literature 1 has been typical we investigate how operator monotony of f is sufficient and or necessary for joint convexity concavity of if a b x similar discussions are given for quasi entropies and quantum skew informations

concentration of quantum states from quantum functional and transportation cost inequalities quantum functional inequalities e g the logarithmic sobolev and poincare inequalities have found widespread application in the study of the behavior of primitive quantum markov semigroups the classical counterparts of these inequalities are related to each other via a so called transportation cost inequality of order 2 tc2 the latter inequality relies on the notion of a metric on the set of probability distributions called the wasserstein distance of order 2 tc2 in turn implies a transportation cost inequality of order 1 tc1 in this paper we introduce quantum generalizations of the inequalities tc1 and tc2 making use of appropriate quantum versions of the wasserstein distances one recently defined by carlen and maas and the other defined by us we establish that these inequalities are related to each other and to the quantum modified logarithmic sobolev and poincare inequalities as in the classical case we also show that these inequalities imply certain concentration type results

a simple criterion for global stabilizability of a class of nonlinear singularly perturbed systems in this paper a simple criterion is proposed for the global stabilizability of a class of nonlinear singularly perturbed systems our approach is based on the global nonlinear controller canonical transformation and the integral manifold method a globally stabilizing feedback control is also proposed such that the pre specified design manifold becomes an exact slow integral manifold and the trajectories of the closed loop system starting from any initial states are steered along the integral manifold to the origin

non commutative calculus optimal transport and functional inequalities in dissipative quantum systems we study dynamical optimal transport metrics between density matrices associated to symmetric dirichlet forms on finite dimensional c algebras our setting covers arbitrary skew derivations and it provides a unified framework that simultaneously generalizes recently constructed transport metrics for markov chains lindblad equations and the fermi ornstein uhlenbeck semigroup we develop a non nommutative differential calculus that allows us to obtain non commutative ricci curvature bounds logarithmic sobolev inequalities transport entropy inequalities and spectral gap estimates

linear control of nearly singularly perturbed hydropower plants abstract in this paper we consider a special class of linear control systems represented by the standard singularly perturbed system matrix and with the control input matrix having three different nonstandard forms many real systems such as hydropower plants systems with only few actuators possess the control structure studied in this paper the obtained results are quite simplified comparing to the standard singularly perturbed control systems and in one case the optimal solution of the algebraic riccati equation is completely determined in terms of the reduced order algebraic lyapunov equations the proposed method is successfully applied to the reduced order design of optimal controllers for the real hydropower plant of the serbian power system

petrov galerkin finite element model for compressible flows a petrov galerkin method for the solution of the compressible euler and navier stokes equations is presented it is based on the introduction of an anisotropic blancing diffusion in the direction of the local direction of propagation of the scalar variables the local direction in which the anisotropic diffusion is introduced is uniquely determined and the magnitude of the balancing diffusion is automatically calculated locally using a criterion that is optimal for one dimensional transport equations the algorithm has been implemented using four noded bilinear elements with forward euler and second order runge kutta methods of integration in time several applications are presented and show the stability and approximation properties of the method

contractivity of positive and trace preserving maps under lp norms we provide a complete picture of contractivity of trace preserving positive maps with respect to p norms we show that for p 1 contractivity holds in general if and only if the map is unital when the domain is restricted to the traceless subspace of hermitian matrices then contractivity is shown to hold in the case of qubits for arbitrary p 1 and in the case of qutrits if and only if p 1 in all noncontractive cases best possible bounds on the p norms are derived

quantum logarithmic sobolev inequalities and rapid mixing a family of logarithmic sobolev inequalities on finite dimensional quantum state spaces is introduced the framework of non commutative lp spaces is reviewed and the relationship between quantum logarithmic sobolev inequalities and the hypercontractivity of quantum semigroups is discussed this relationship is central for the derivation of lower bounds for the logarithmic sobolev ls constants essential results for the family of inequalities are proved and we show an upper bound to the generalized ls constant in terms of the spectral gap of the generator of the semigroup these inequalities provide a framework for the derivation of improved bounds on the convergence time of quantum dynamical semigroups when the ls constant and the spectral gap are of the same order convergence bounds on finite dimensional state spaces are particularly relevant for the field of quantum information theory we provide a number of examples where improved bounds on the mixing time of several semigroups are obtained incl

matrix optimal mass transport a quantum mechanical approach in this paper we describe a possible generalization of the wasserstein 2 metric originally defined on the space of scalar probability densities to the space of hermitian matrices with trace one and to the space of matrix valued probability densities our approach follows a control theoretic optimization formulation of the wasserstein 2 metric having its roots in fluid dynamics and utilizes certain results from the quantum mechanics of open systems in particular the lindblad equation it allows determining the gradient flow for the quantum entropy relative to this matricial wasserstein metric

singular perturbations and input to state stability this paper establishes a type of total stability for the input to state stability property with respect to singular perturbations in particular if the boundary layer system is uniformly globally asymptotically stable and the reduced system is input to state stable with respect to disturbances then these properties continue to hold up to an arbitrarily small offset for initial conditions disturbances and their derivatives in an arbitrarily large compact set as long as the singular perturbation parameter is sufficiently small

a corrective feedback design for nonlinear systems with fast actuators recent two time scale results can be derived from a geometric framework which allows further extensions and computational improvements in this note the two time scale behavior of singularly perturbed systems is exploited to design slow and fast controls and to combine them into a composite control as an illustration we present a corrective design to compensate for fast actuator dynamics modeled as singular perturbations

an efficient algorithm for finding the d stability bound of discrete singularly perturbed systems with multiple time delays in this paper we present an original work on the d stabilization problem of discrete singularly perturbed systems with multiple time delays a new robust d stability criterion in terms of stability radius is first derived to guarantee that all poles of the discrete multiple time delay systems remain inside the specific disk d alpha r uncertainties then by using the technique of time scale separation we derive the corresponding slow and fast subsystems of a discrete multiple time delay singularly perturbed system the state feedback controls for the d stabilization of the slow and the fast subsystems are separately designed and a composite state feedback control for the original system is subsequently synthesized from these state feedback controls thereafter we derive a frequency domain epsilon dependent dstability criterion for the original discrete multiple time delay singularly perturbed system under the composite state feedback control if any one of the conditions of this criterion is fulfilled

composite feedback control for a class of non linear singularly perturbed systems with fast actuators a design methodology for a class of non linear singularly perturbed systems with fast actuators is proposed using the composite control technique the origin of the nominally unforced unperturbed system is assumed to be asymptotically stable and a lyapunov function for the nominal system is available then by appropriate choice of the design manifold a composite feedback control can be constructed to guarantee the asymptotic stability of the origin of the closed loop system

a finite element model for double diffusive convection presentation d un algorithme aux elements finis pour les equations determinant la dynamique de la convection a double diffusion exemples d applications numeriques et comparaison avec des resultats experimentaux discussion des extensions possibles a d autres problemes

a novel approach to control of nonminimum phase nonlinear systems we present an approach to control of nonminimum phase multiple input multiple output mimo nonlinear systems based on a singular perturbation like technique first we show that a mimo nonlinear system can be converted into a singularly perturbed system through magnitude and time scaling transformation in particular the proposed transformation technique can make the hidden or internal dynamics behave like a fast subsystem however the fast subsystem takes a weakly controllable form which implies that a small positive low gain parameter multiplies the control input in the fast subsystem for this reason the well known integral manifold approach is then utilized to decompose the resulting singularly perturbed system into two subsystems in separate timescales each of which is of a lower dimension and more importantly controllable in the sense that the control input will explicitly appear in the boundary layer system the proposed method is particularly useful to make the input output dynamic characteristics of a decouplable nonminimum phase nonlinear system decoupled and linear

the adjoint method for an inverse design problem in the directional solidification of binary alloys this paper presents a systematic procedure based on the adjoint method for solving a class of inverse directional alloy solidification design problems in which a desired growth velocityvfis achieved under stable growth conditions to the best of our knowledge this is the first time that a continuum adjoint formulation is proposed for the solution of an inverse problem with simultaneous heat and mass transfer thermo solutal convection and phase change in this paper the interfacial stability is considered to imply a sharp solid liquid freezing interface this condition is enforced using the constitutional undercooling criterion in the form of an inequality constraint between the thermal and solute concentration gradients gandgc respectively at the freezing front the main unknowns of the design problem are the heating and or cooling boundary conditions on the mold walls the inverse design problem is formulated as a functional optimization problem the cost functional is defined by the square of thel2norm of the deviation of the freezing interface temperature from the temperature corresponding to thermodynamic equilibrium a continuum adjoint system is derived to calculate the adjoint temperature concentration and velocity fields such that the gradient of the cost functional can be expressed analytically the cost functional minimization process is realized by the conjugate gradient method via the finite element method solutions of the continuum direct sensitivity and adjoint problems the developed formulation is demonstrated with an example of designing the directional solidification of a binary aqueous solution in a rectangular mold such that a stable vertical interface advances from left to right with a desired growth velocity

exact design manifold control of a class of nonlinear singularly perturbed systems the integral manifold approach captures from a geometric point of view the intrinsic two time scale behavior of singularly perturbed systems an important class of nonlinear singularly perturbed systems considered in this note are fast actuator type systems for a class of fast actuator type systems which includes many physical systems an explicit corrected composite control the sum of a slow control and a fast control is derived this corrected control will steer the system exactly to a required design manifold

composite controls for singularly perturbed systems the controllability and the stabilizability invariance of the reduced system slow subsystem of the nonlinear singularly perturbed system due to a class of fast feedback controls is shown the general type composite control for the nonlinear singularly perturbed system is proposed and its application for the synthesis problem is considered

operator lipschitz functions in schatten von neumann classes this paper resolves a number of problems in the perturbation theory of linear operators linked with the 45 year old conjecure of m g kre n in particular we prove that every lipschitz function is operator lipschitz in the schatten von neumann ideals s 1 0 such that left f a f b right alpha leqslant c alpha left f right text lip 1 left a b right alpha where f is a lipschitz function with left f right text lip 1 mathop sup limits lambda ne mu lambda mu in mathbb r left frac f left lambda right f left mu right lambda mu right infty left cdot right alpha is the norm is s and a and b are self adjoint linear operators such that a b in s alpha

a simple criterion for global exponential stabilizability of nonlinear multiple time delay singularly perturbed systems a simple criterion for global exponential stability of a class of nonlinear multiple time delay singularly perturbed systems without the so called interconnection conditions is proposed in this paper a globally exponentially stabilizing composite feedback control is also proposed such that the chosen design manifold becomes an exact integral manifold and the trajectories of the closed loop systems starting from any initial states are steered along the integral manifold to the origin exponentially for all sufficiently small singular perturbation parameters e the composite control technique is adopted in this paper

brief a descriptor system approach to nonlinear singularly perturbed optimal control problem this paper considers the infinite horizon nonlinear quadratic optimal control problem for a singularly perturbed system which is nonlinear in both the slow and the fast variables the relationship between this problem and the analogous one for a descriptor system is investigated parameter independent controllers are constructed that solve the problem for the descriptor system and lead the full order system to the near optimal performance estimates on the closeness of the cost under near optimal controllers to the optimal one are obtained

gradient flow and entropy inequalities for quantum markov semigroups with detailed balance abstract we study a class of ergodic quantum markov semigroups on finite dimensional unital c algebras these semigroups have a unique stationary state and we are concerned with those that satisfy a quantum detailed balance condition with respect to we show that the evolution on the set of states that is given by such a quantum markov semigroup is gradient flow for the relative entropy with respect to in a particular riemannian metric on the set of states this metric is a non commutative analog of the 2 wasserstein metric and in several interesting cases we are able to show in analogy with work of otto on gradient flows with respect to the classical 2 wasserstein metric that the relative entropy is strictly and uniformly convex with respect to the riemannian metric introduced here as a consequence we obtain a number of new inequalities for the decay of relative entropy for ergodic quantum markov semigroups with detailed balance

gradient flows of the entropy for finite markov chains let k be an irreducible and reversible markov kernel on a finite set x we construct a metric w on the set of probability measures on x and show that with respect to this metric the law of the continuous time markov chain evolves as the gradient flow of the entropy this result is a discrete counterpart of the wasserstein gradient flow interpretation of the heat flow in rn by jordan kinderlehrer and otto 1998 the metric w is similar to but different from the l2 wasserstein metric and is defined via a discrete variant of the benamou brenier formula

a geometric approach to nonlinear singularly perturbed control systems abstract applications of two time scale singular perturbation methods have been limited to the class of models appearing in a standard form many nonlinear control systems such as models of aircraft and robotic manipulators with flexible joints are two time scale systems but do not appear in the standard form the main result of this paper is a coordinate free characterization of time scales in terms of invariant manifolds which express conservation and equilibrium properties of the control system a procedure for finding slow and fast states is given previously developed slow fast composite control designs are thus made applicable to a wider class of nonlinear systems

derivations as square roots of dirichlet forms abstract the aim of this work is to analyze the structure of a tracially symmetric dirichlet form on a c algebra in terms of a killing weight and a closable derivation taking values in a hilbert space with a bimodule structure it is shown that the generator of the associate markovian semigroup always appears in a natural way as the divergence of a closable derivation applications are shown to the decomposition of dirichlet forms and to the construction of differential calculus on metric spaces

optimal hypercontractivity for fermi fields and related non commutative integration inequalities optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained these are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by nelson in the process several results of independent interest in the theory of non commutative integration are established

an analog of the 2 wasserstein metric in non commutative probability under which the fermionic fokker planck equation is gradient flow for the entropy let mathfrak c denote the clifford algebra over mathbb r n which is the von neumann algebra generated by n self adjoint operators qj j 1 n satisfying the canonical anticommutation relations qiqj qjqi 2 iji and let denote the normalized trace on mathfrak c this algebra arises in quantum mechanics as the algebra of observables generated by n fermionic degrees of freedom let mathfrak p denote the set of all positive operators rho in mathfrak c such that 1 these are the non commutative analogs of probability densities in the non commutative probability space mathfrak c tau the fermionic fokker planck equation is a quantum mechanical analog of the classical fokker planck equation with which it has much in common such as the same optimal hypercontractivity properties in this paper we construct a riemannian metric on mathfrak p that we show to be a natural analog of the classical 2 wasserstein metric and we show that in analogy with the classical case the fermionic fokker planck equation is gradient flow in this metric for the relative entropy with respect to the ground state we derive a number of consequences of this such as a sharp talagrand inequality for this metric and we prove a number of results pertaining to this metric several open problems are raised

poincare modified logarithmic sobolev and isoperimetric inequalities for markov chains with non negative ricci curvature abstract we study functional inequalities for markov chains on discrete spaces with entropic ricci curvature bounded from below our main results are that when curvature is non negative but not necessarily positive the spectral gap the cheeger isoperimetric constant and the modified logarithmic sobolev constant of the chain can be bounded from below by a constant that only depends on the diameter of the space with respect to a suitable metric these estimates are discrete analogues of classical results of riemannian geometry obtained by li and yau buser and wang

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100