uniform convergence to equilibrium for granular media we study the long time asymptotics of a nonlinear nonlocal equation used in the modelling of granular media we prove a uniform exponential convergence to equilibrium for degenerately convex and non convex interaction or confinement potentials improving in particular results by j a carrillo r j mccann and c villani the method is based on studying the dissipation of the wasserstein distance between a solution and the steady state

discrete optimal control with eigenvalue assigned inside a circular region a discrete time optimal control that guarantees that all the closed loop poles will lie inside a circle centered at beta 0 with radius is formulated it is shown how the exposed problem can be reduced to a standard discrete time linear quadratic regulator problem furthermore a quantitative measure of the robustness of linear quadratic state feedback design in the presence of a perturbation is obtained bounds are derived for allowable nonlinear perturbations such that the resultant closed loop is stable

entropic multipliers method for langevin diffusion and weighted log sobolev inequalities in his work about hypocercivity villani 18 considers in particular convergence to equilibrium for the kinetic langevin process while his convergence results in l 2 are given in a quite general setting convergence in entropy requires some boundedness condition on the hessian of the hamiltonian we will show here how to get rid of this assumption in the study of the hypocoercive entropic relaxation to equilibrium for the langevin diffusion our method relies on a generalization to entropy of the multipliers method and an adequate functional inequality as a byproduct we also give tractable conditions for this functional inequality which is a particular instance of a weighted logarithmic sobolev inequality to hold

pole assignment robustness in a specified disk abstract a sufficient condition is proposed to guarantee robust pole location within a specified circular region for a linear uncertain system the explicit bounds on linear time invariant perturbations with highly structured information are obtained under these allowable highly structured perturbations both stability robustness and certain performance robustness will thus be ensured the merit of the proposed new approach is demonstrated by a given example where the results achieved are much better than the ones reported recently

controllability observability and duality in singular systems the concepts of controllability and observability for systems of the form e dot x ax bu y cx e singular are considered a theory is presented which unifies the three main approaches to this topic already existing in the literature the development includes a generalization of the duality theorem from state space theory

convergence to equilibrium in wasserstein distance for fokker planck equations we describe conditions on non gradient drift diffusion fokker planck equations for its solutions to converge to equilibrium with a uniform exponential rate in wasserstein distance this asymptotic behaviour is related to a functional inequality which links the distance with its dissipation and ensures a spectral gap in wasserstein distance we give practical criteria for this inequality and compare it to classical ones the key point is to quantify the contribution of the diffusion term to the rate of convergence which to our knowledge is a novelty

on the monotonicity principle of optimal skorokhod embedding problem this is a continuation of our accompanying paper siam j control optim 54 2016 pp 2174 2201 we provide an alternative proof of the monotonicity principle for the optimal skorokhod embedding problem established in beiglbock cox and huesmann optimal transport and skorokhod embedding preprint 2013 our proof is based on the adaptation of the monge kantorovich duality in our context a delicate application of the optional cross section theorem and a clever conditioning argument introduced in beiglbock cox and huesmann optimal transport and skorokhod embedding preprint 2013

a lyapunov approach to analysis of discrete singular systems in this paper a new type generalized lyapunov equation for discrete singular systems is proposed then it is applied to study problems such as pole clustering controllability and observability for discrete singular systems first some necessary and sufficient conditions for pole clustering are derived via the solution of this new type lyapunov equation further the relationship between the solution of the lyapunov equation and structure properties of discrete singular systems will be investigated based on these results finally a type of generalized riccati equation is proposed and its solution is used to design state feedback law for discrete singular systems such that all the finite poles of the closed loop systems are clustered into a specified disk

pole placement for singular systems abstract in this paper we present a pole placement design for singular systems in polynomial fraction form the design can also be used for the role placement of singular systems in generalized state space form and requires no decomposition

complete duality for martingale optimal transport on the line we study the optimal transport between two probability measures on the real line where the transport plans are laws of one step martingales a quasi sure formulation of the dual problem is introduced and shown to yield a complete duality theory for general marginals and measurable reward cost functions absence of a duality gap and existence of dual optimizers both properties are shown to fail in the classical formulation as a consequence of the duality result we obtain a general principle of cyclical monotonicity describing the geometry of optimal transports

a perturbation theory for ergodic markov chains and application to numerical approximations perturbations to markov chains and markov processes are considered the unperturbed problem is assumed to be geometrically ergodic in the sense usually established through the use of foster lyapunov drift conditions the perturbations are assumed to be uniform in a weak sense on bounded time intervals the long time behavior of the perturbed chain is studied applications are given to numerical approximations of a randomly impulsed ode an ito stochastic differential equation sde and a parabolic stochastic partial differential equation spde subject to space time brownian noise existing perturbation theories for geometrically ergodic markov chains are not readily applicable to these situations since they require very stringent hypotheses on the perturbations

pole placement and compensator design of generalized systems abstract the system considered is a generalized system e ax bu y cx with matrix e singular when the impulsive modes of the system are both controllable and observable it is shown in this paper that these modes can be eliminated by almost any constant output feedback for a strongly controllable and observable generalized system after its impulsive modes are eliminated the poles of the system can be assigned arbitrarily by state feedback even if the states are not available the paper adopts an efficient method to design a compensator for the system

graph theoretic approach to symbolic analysis of linear descriptor systems abstract continuous descriptor systems e x ax bu y cx where e is a possibly singular matrix are symbolically analyzed by means of digraphs starting with four different digraph characterizations of square matrices and determinants the author favors the cauchy coates interpretation then an appropriate digraph representation of the matrix pencil se a is given which is followed by a digraph interpretation of det se a and the transfer function matrix c se a 1 b next a graph theoretic procedure is derived to reveal a possibly hidden factorizability of the determinant det se a this is very important for large scale systems finally as an application of the derived results an electrical network is analyzed symbolically

homogeneous indices feedback invariants and control structure theorem for generalized linear systems we define a new set of indices for a generalized linear system these indices referred to as the homogeneous indices are a natural generalization of the minimal column indices kronecker indices of an ordinary state space system we prove that the homogeneous indices are a complete set of invariants for the action of a natural group of feedback transformations on generalized linear systems we also show that the homogeneous indices determine exactly which closed loop invariant polynomials can be assigned by feedback thereby generalizing the control structure theorem of rosenbrock

robustness of regional pole placement for uncertain continuous time implicit systems in this note under assumptions that the nominal linear implicit system is regular impulse free with all its generalized eigenvalues lying inside some specified regions sufficient conditions are proposed to preserve the assumed properties when structured perturbations are added into the nominal system explicit bounds on perturbations with different structures are given in terms of the spectral radii of some nonnegative matrices

kantorovich duality for general transport costs and applications we introduce a general notion of transport cost that encompasses many costs used in the literature including the classical one and weak transport costs introduced by talagrand and marton in the 90 s and prove a kantorovich type duality theorem as a by product we obtain various applications in different directions we give a short proof of a result by strassen on the existence of a martingale with given marginals we characterize the associated transport entropy inequalities together with the log sobolev inequality restricted to convex concave functions some explicit examples of discrete measures satisfying weak transport entropy inequalities are also given

quantitative harris type theorems for diffusions and mckean vlasov processes we consider d valued diffusion processes of type dxt b xt dt dbt assuming a geometric drift condition we establish contractions of the transitions kernels in kantorovich l1 wasserstein distances with explicit constants our results are in the spirit of hairer and mattingly s extension of harris theorem in particular they do not rely on a small set condition instead we combine lyapunov functions with reflection coupling and concave distance functions we retrieve constants that are explicit in parameters which can be computed with little effort from one sided lipschitz conditions for the drift coefficient and the growth of a chosen lyapunov function consequences include exponential convergence in weighted total variation norms gradient bounds bounds for ergodic averages and kantorovich contractions for nonlinear mckean vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities we also establish quantitative bounds for sub geometric ergodicity assuming a sub geometric drift condition

linear quadratic regulators with eigenvalue placement in a specified region a linear optimal quadratic regulator is developed for optimally placing the closed loop poles of multivariable continuous time systems within the common region of an open sector bounded by lines inclined at 2k k 2 or 3 from the negative real axis with a sector angle 2 and the left hand side of a line parallel to the imaginary axis in the complex s plane also a shifted sector method is presented to optimally place the closed loop poles of a system in any general sector having a sector angle between 2 and the optimal pole placement is achieved without explicitly utilizing the eigenvalues of the open loop system the design method is mainly based on the solution of a linear matrix lyapunov equation and the resultant closed loop system with its eigenvalues in the desired region is optimal with respect to a quadratic performance index

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

fundamental theorem of state feedback for singular systems abstract the limits of state feedback in altering the dynamics of singular systems are studied a necessary and sufficient condition is given for a list of polynomials to be invariant polynomials of a proper system obtained by state feedback from the singular system the condition consists of inequalities which involve the controllability indices of the singular system and the degrees of the invariant polynomials the development is based on the properties of polynomial matrices a procedure is given for the calculation of a feedback gain which achieves the desired dynamics

robustness analysis of uncertain linear singular systems with output feedback control the robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered the problem is transformed into a robust nonsingularity problem based on the linear fractional transformation lft approach the robustness bounds to preserve regularity impulse immunity and stability are found in terms of the structured singular value spl mu with respect to parametric uncertainties the lft approach provides a unified framework for robustness analysis of both uncertain linear continuous discrete time singular systems

curvature concentration and error estimates for markov chain monte carlo we provide explicit nonasymptotic estimates for the rate of convergence of empirical means of markov chains together with a gaussian or exponential control on the deviations of empirical means these estimates hold under a positive curvature assumption expressing a kind of metric ergodicity which generalizes the ricci curvature from differential geometry and on finite graphs amounts to contraction under path coupling

stability analysis and stabilization of 2 d singular roesser models abstract this paper addresses the problem of controller design for 2 d singular roesser models sufficient stability conditions are formulated in terms of strict linear matrix inequalities lmis for 2 d singular systems described by the roesser models then a set of sufficient conditions for existence of desired a state feedback controller are obtained numerical examples are provided to demonstrate the effectiveness of the proposed method

a canonical form for controllable singular systems abstract a new canonical form for the action of restricted system equivalence on controllable singular systems is given the construction of this form is based on the weierstrass decomposition of the singular system into a slow and a fast subsystem both subsystems are transformed into hermite canonical form the resulting hermite canonical form for singular systems has a particularly simple structure and is expected to be useful for e g identification purposes continuity properties of the hermite form are investigated and the nonexistence of a globally defined continuous canonical form for controllable singular systems is shown

robustness of discrete systems under structured uncertainties we give a sufficient condition for the robust stability of the discrete system x k 1 f e x k where the perturbation e is bounded e u a necessary and sufficient condition is derived in the case where f is a positive or a negative matrix and u f the stability of interval matrices is also considered numerical examples are given for illustration

optimal transportation under controlled stochastic dynamics we consider an extension of the monge kantorovitch optimal transportation problem the mass is transported along a continuous semimartingale and the cost of transportation depends on the drift and the diffusion coefficients of the continuous semimartingale the optimal transportation problem minimizes the cost among all continuous semimartingales with given initial and terminal distributions our first main result is an extension of the kantorovitch duality to this context we also suggest a finite difference scheme combined with the gradient projection algorithm to approximate the dual value we prove the convergence of the scheme and we derive a rate of convergence we finally provide an application in the context of financial mathematics which originally motivated our extension of the monge kantorovitch problem namely we implement our scheme to approximate no arbitrage bounds on the prices of exotic options given the implied volatility curve of some maturity

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

an lmi approach to d stability and d stabilization of linear discrete singular systems with state delay abstract the problems of d stability and d stabilization for linear discrete singular systems with state delay are addressed in this paper by using the state augmentation technique a necessary and sufficient d stability condition and a sufficient d stabilization condition for linear discrete singular time delay systems are firstly proposed then based on these conditions quadratic d stability and d stabilization conditions for linear discrete singular time delay systems with norm bounded uncertainties and interval uncertainties are obtained respectively both the d stability and d stabilization conditions are proposed in terms of linear matrix inequalities finally illustrative examples show that the proposed conditions are effective to check the d stability and to design the d stabilizing controller for linear discrete singular time delay systems

theoretical guarantees for approximate sampling from smooth and log concave densities sampling from various kinds of distributions is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators test procedures or confidence intervals in many situations the exact sampling from a given distribution is impossible or computationally expensive and therefore one needs to resort to approximate sampling strategies however there is no well developed theory providing meaningful nonasymptotic guarantees for the approximate sampling procedures especially in the high dimensional problems this paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log concave density defined on rp for some integer p 0 we establish nonasymptotic bounds for the error of approximating the true distribution by the one obtained by the langevin monte carlo method and its variants we illustrate the effectiveness of the established guarantees with various experiments underlying our analysis are insights from the theory of continuous time diffusion processes which may be of interest beyond the framework of distributions with log concave densities considered in the present work

two end points marginal problem by stochastic optimal transportation we give a sufficient condition under which the stochastic optimal transportation problem is finite which implies the existence of a semimartingale with given initial and terminal distributions the idea of the proof is to show the finiteness of the supremum in the duality theorem for the stochastic optimal transportation problem as a special case it also gives a new approach for the construction of the h path process with given initial and terminal distributions we also consider a problem similar to the above for a class of optimal control problems for a family of solutions to fokker planck equations

exponential mixing properties of stochastic pdes through asymptotic coupling we consider parabolic stochastic partial differential equations driven by white noise in time we prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions these conditions amount essentially to the fact that the equation transmits the noise to all its determining modes several examples are investigated including some where the noise does not act on every determining mode directly

reflection couplings and contraction rates for diffusions we consider contractivity for diffusion semigroups w r t kantorovich l 1 wasserstein distances based on appropriately chosen concave functions these distances are inbetween total variation and usual wasserstein distances it is shown that by appropriate explicit choices of the underlying distance contractivity with rates of close to optimal order can be obtained in several fundamental classes of examples where contractivity w r t standard wasserstein distances fails applications include overdamped langevin diffusions with locally non convex potentials products of these processes and systems of weakly interacting diffusions both of mean field and nearest neighbour type

contractions in the 2 wasserstein length space and thermalization of granular media an algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions this rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite dimensional manifold an abstract theory is developed for gradient flows in length spaces which shows how degenerate convexity or even non convexity if uniformly controlled will quantify contractivity limit expansivity of the flow

the stability robustness of generalized eigenvalues the concept of stability radius is generalized to matrix pairs a matrix pair is said to be stable if its generalized eigenvalues are located in the open left half of the complex plane the stability radius of a matrix pair a b is defined to be the norm of the smallest perturbation delta a such that a delta a b is unstable the purpose is to estimate the stability radius of a given matrix pair depending on whether the matrices under consideration are complex or real the problem can be classified into two cases the complex case is easy and a complete solution is provided the real case is more difficult and only a partial solution is given

transport inequalities gradient estimates entropy and ricci curvature we present various characterizations of uniform lower bounds for the ricci curvature of a smooth riemannian manifold m in terms of convexity properties of the entropy considered as a function on the space of probability measures on m as well as in terms of transportation inequalities for volume measures heat kernels and brownian motions and in terms of gradient estimates for the heat semigroup 2004 wiley periodicals inc

superreplication of financial derivatives via convex programming we give a method based on convex programming to calculate the optimal superreplicating and subreplicating prices and corresponding hedging portfolios of a financial derivative in terms of other financial derivatives in a discrete time setting our method produces a model that matches the superreplicating or subreplicating price within an arbitrary precision and is consistent with the other financial derivatives prices applications include robust replication in terms of call prices with various strikes and maturities of forward start options volatility and variance swaps and derivatives cliquet calls barrier options and lookback and asian options numerical examples show that in some cases the best superreplicating and or subreplicating prices are within 10 of the price obtained by a standard model but considerably differ from it in other cases our method can incorporate bid ask spreads interest rates and dividends and various limitations to the diffusion model the supplementary material is a

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

pole assignment in a specified disk the problem of assigning all poles of a closed loop system in a specified disk by state feedback is considered for both continuous and discrete systems a state feedback control law is determined by using a discrete riccati equation this kind of pole assignment problem is named d pole assignment and its relation to the optimal control problem and its robustness properties are discussed the gain and phase margins for all closed loop poles to stay inside the specified disk d are determined for the proposed control

probabilistic approach for granular media equations in the non uniformly convex case we use here a particle system to prove both a convergence result with convergence rate and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex the proof of convergence is simpler than the one in carrillo mccann villani rev mat iberoamericana 19 971 1018 2003 arch rat mech anal 179 217 263 2006 all the results complete former results of malrieu ann appl probab 13 540 560 2003 in the uniformly convex case the main tool is an uniform propagation of chaos property and a direct control in wasserstein distance of solutions starting with different initial measures the deviation inequality is obtained via a t 1 transportation cost inequality replacing the logarithmic sobolev inequality which is no more clearly dimension free

robust control analysis and design for discrete time singular systems abstract in this paper we propose a simple approach to analyse stability robustness of discrete time singular systems under structured perturbations the developed robustness criteria are then applied to solve robust regional pole assignment problems of singular systems a robust control design algorithm via state feedback is also given the robust stability problem of singular systems is more complicated than that of regular systems not only stability robustness but system regularity and impulse elimination should be considered simultaneously since the results of robust control and analysis for singular systems is not available in the literature as much as other fields the paper may be viewed as a complementary result in this field although only discrete time case is discussed several results can be directly applied to continuous time systems as well

asymptotic coupling and a general form of harris theorem with applications to stochastic delay equations there are many markov chains on infinite dimensional spaces whose one step transition kernels are mutually singular when starting from different initial conditions we give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of harris theorem the first uses the existence of couplings which draw the solutions together as time goes to infinity such asymptotic couplings were central to mattingly and sinai in comm math phys 219 3 523 565 2001 mattingly in comm math phys 230 3 461 462 2002 hairer in prob theory relat field 124 345 380 2002 bakhtin and mattingly in commun contemp math 7 553 582 2005 on which this work builds as in bakhtin and mattingly 2005 the emphasis here is on stochastic differential delay equations harris celebrated theorem states that if a markov chain admits a lyapunov function whose level sets are small in the sense that transition probabilities are uniformly bounded from below then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed this convergence takes place in a total variation norm weighted by the lyapunov function a second aim of this article is to replace the notion of a small set by the much weaker notion of a d small set which takes the topology of the underlying space into account via a distance like function d with this notion at hand we prove an analogue to harris theorem where the convergence takes place in a wasserstein like distance weighted again by the lyapunov function this abstract result is then applied to the framework of stochastic delay equations in this framework the usual theory of harris chains does not apply since there are natural examples for which there exist no small sets except for sets consisting of only one point this gives a solution to the long standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it

feedback canonical form for singular systems controllability and impulse controllability indices for singular systems are defined via the concept of minimal indices for modules over polynomial rings these indices are invariant under the transformations of restricted system equivalence and feedback equivalence they yield a characterization of controllability and impulse controllability of singular systems analogous to the state space situation and form a complete list of invariants for impulse controllable systems under feedback equivalence a feedback canonical form described by these invariants is constructed explicitly