a bogolyubov type theorem with a nonconvex constraint in banach spaces we prove an analogue of the classical bogolyubov theorem with a nonconvex constraint in the case we consider the constraint is the solution set of a cauchy problem for a differential inclusion with a nonconvex right hand side satisfying a lipschitz condition our approach is based on a relaxation argument as in the filippov wazewski theorem

polyhedral sweeping processes with unbounded nonconvex valued perturbation abstract a polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable hilbert space polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes we consider the polyhedral sweeping process as an evolution inclusion with subdifferential operators depending on time the widely used assumption of lipschitz continuity for the multivalued perturbation term is replaced by a weaker notion of h lipschitzness the existence of solutions is proved for this sweeping process

chern simons invariants in kk theory abstract for a unitary representation of the fundamental group of a compact smooth manifold atiyah patodi singer defined the so called invariant of using the chern simons invariants in this article using traces on c algebras we give an intrinsic definition of an element in kk with real coefficients theory whose pullback by the representation is the invariant

adaptive dynamic programming for a class of complex valued nonlinear systems in this brief an optimal control scheme based on adaptive dynamic programming adp is developed to solve infinite horizon optimal control problems of continuous time complex valued nonlinear systems a new performance index function is established on the basis of complex valued state and control using system transformations the complex valued system is transformed into a real valued one which overcomes cauchy riemann conditions effectively with the transformed system and the performance index function a new adp method is developed to obtain the optimal control law by using neural networks a compensation controller is developed to compensate the approximation errors of neural networks stability properties of the nonlinear system are analyzed and convergence properties of the weights for neural networks are presented finally simulation results demonstrate the performance of the developed optimal control scheme for complex valued nonlinear systems

stability and euler approximation of one sided lipschitz differential inclusions ordinary differential and functional differential inclusions with compact right hand sides are considered stability theorems of filippov s type in the convex and nonconvex case are proved under a one sided lipschitz condition which extends the notions of lipschitz continuity dissipativity and the uniform one sided lipschitz condition for set valued mappings the accuracy of approximation of the solution sets by means of the euler discretization scheme for both types of inclusions is estimated

equivariant novikov conjecture for groups acting on euclidean buildings we prove the equivariant novikov conjecture for groups acting on euclidean buildings by using an equivariant f tilsum skandalis methocl we also obtain an equivariant version of the connes gromov moscovic th eorem for almost flat c algebra bundles

unbounded pseudodifferential calculus on lie groupoids abstract we develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids also called lie groupoids with compact basis we analyze these operators as unbounded operators acting on hilbert modules over c g and we show in particular that elliptic operators are regular we construct a scale of sobolev modules which are the abstract analogues of the ordinary sobolev spaces and analyze their properties furthermore we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense

remarks on coarse embeddings of metric spaces into uniformly convex banach spaces we study the support and convergence conditions for a metric space to be coarsely embeddable into a uniformly convex banach space by using ultraproducts we also show that the coarse embeddability of a metric space into a uniformly convex banach space is determined by its finite subspaces

un renforcement de la propriete t nous proposons un renforcement de la propriete t de kazhdan en remplacant les representations unitaires par les representations dans des espaces de hilbert qui ne sont pas necessairement unitaires mais a croissance exponentielle suffisamment petite on sait qu un groupe localement compact a la propriete t si et seulement si il existe un idempotent p dans c max g tel que pour toute representation unitaire continue h de g p est le projecteur orthogonal sur le sous espace ferme de h forme des vecteurs ginvariants pour la propriete t renforcee nous posons une definition dans ce style on appelle longueur sur un groupe localement compact g une fonction continue g r verifiant g 1 g et g1g2 g1 g2 pour g g1 g2 g dans la suite pour tout groupe localement compact g on supposera choisie une mesure de haar a gauche qui munit cc g d une structure d algebre par convolution

the coarse baum connes conjecture for spaces which admit a uniform embedding into hilbert space corollary 1 2 let be a finitely generated group if as a metric space with a word length metric admits a uniform embedding into hilbert space and its classifying space b has the homotopy type of a finite cw complex then the strong novikov conjecture holds for i e the index map from k b to k c r is injective corollary 1 2 follows from theorem 1 1 and the descent principle 23 by index theory the strong novikov conjecture implies the novikov conjecture on the homotopy invariance of higher signatures cf 8 for an excellent

k theorie bivariante pour les algebres de banach et conjecture de baum connes je suis parvenu dans ma these a construire une k theorie bivariante pour les algebres de banach cela m a permis de demontrer la conjecture de baum connes pour les groupes de lie semi simples et les groupes reductifs p adiques ainsi que pour les sous groupes discrets de type fini de sl 3 f avec f une extension finie de q p et pour les sous groupes discrets cocompacts de sp n 1 sl 3 r et sl 3 c j ai aussi demontre une conjecture analogue a la conjecture de baum connes pour les algebres l 1 et plus generalement toutes les bonnes completions des sous groupes fermes des groupes de lie semi simples et des groupes reductifs p adiques

on banach spaces with kasparov and yu s property h abstract a new geometric property of banach spaces recently introduced by g kasparov and g yu called property h has important applications to the strong novikov conjecture and the coarse novikov conjecture yet is far from being well understood in this paper we investigate various uniformly continuous maps on unit spheres of banach spaces and prove that all separable banach lattices with nontrivial cotype have property h

fibred coarse embedding into non positively curved manifolds and higher index problem abstract let x n 1 x n be the coarse disjoint union of a sequence of finite metric spaces with uniform bounded geometry in this paper we show that the coarse novikov conjecture holds for x if x admits a fibred coarse embedding into a simply connected complete riemannian manifold of non positive sectional curvature this includes a large class of expander graphs with geometric property t

nonpositive curvature is not coarsely universal we prove that not every metric space embeds coarsely into an alexandrov space of nonpositive curvature this answers a question of gromov geometric group theory cambridge university press cambridge 1993 and is in contrast to the fact that any metric space embeds coarsely into an alexandrov space of nonnegative curvature as shown by andoni et al ann sci ec norm super 4 51 3 657 700 2018 we establish this statement by proving that a metric space which is q barycentric for some q in 1 infty has metric cotype q with sharp scaling parameter our proof utilizes nonlinear metric space valued martingale inequalities and yields sharp bounds even for some classical banach spaces this allows us to evaluate the bi lipschitz distortion of the ell infty grid m infty n 1 ldots m n vert cdot vert infty into ell q for all q in 2 infty from which we deduce the following discrete converse to the fact that ell infty n embeds with distortion o 1 into ell q for q o log n a rigidity theorem of ribe ark mat 14 2 237 244 1976 implies that for every n in mathbb n there exists m in mathbb n such that if m infty n embeds into ell q with distortion o 1 then q is necessarily at least a universal constant multiple of log n ribe s theorem does not give an explicit upper bound on this m but by the work of bourgain geometrical aspects of functional analysis 1985 86 springer berlin 1987 it suffices to take m n and this was the previously best known estimate for m we show that the above discretization statement actually holds when m is a universal constant

large riemannian manifolds which are flexible for each k z we construct a uniformly contractible metric on euclidean space which is not mod k hypereuclidean we also construct a pair of uniformly contractible riemannian metrics on r n n 11 so that the resulting manifolds z and z are bounded homotopy equivalent by a homotopy equivalence which is not boundedly close to a homeomorphism we show that for these spaces the c algebra assembly map k lf z k c z from locally finite k homology to the k theory of the bounded propagation algebra is not a monomorphism this shows that an integral version of the coarse novikov conjecture fails for real operator algebras if we allow a single cone like singularity a similar construction yields a counterexample for complex c algebras

adaptive optimal control of highly dissipative nonlinear spatially distributed processes with neuro dynamic programming highly dissipative nonlinear partial differential equations pdes are widely employed to describe the system dynamics of industrial spatially distributed processes sdps in this paper we consider the optimal control problem of the general highly dissipative sdps and propose an adaptive optimal control approach based on neuro dynamic programming ndp initially karhunen loeve decomposition is employed to compute empirical eigenfunctions eefs of the sdp based on the method of snapshots these eefs together with singular perturbation technique are then used to obtain a finite dimensional slow subsystem of ordinary differential equations that accurately describes the dominant dynamics of the pde system subsequently the optimal control problem is reformulated on the basis of the slow subsystem which is further converted to solve a hamilton jacobi bellman hjb equation hjb equation is a nonlinear pde that has proven to be impossible to solve analytically thus an adaptive optimal control method is developed via ndp that solves the hjb equation online using neural network nn for approximating the value function and an online nn weight tuning law is proposed without requiring an initial stabilizing control policy moreover by involving the nn estimation error we prove that the original closed loop pde system with the adaptive optimal control policy is semiglobally uniformly ultimately bounded finally the developed method is tested on a nonlinear diffusion convection reaction process and applied to a temperature cooling fin of high speed aerospace vehicle and the achieved results show its effectiveness

the coarse novikov conjecture and banach spaces with property h in this paper we prove the coarse novikov conjecture for metric spaces with bounded geometry which are coarsely embeddable into banach spaces with a geometric condition called property h introduced by g kasparov and g yu

discrete approximations and refined euler lagrange conditions fornonconvex differential inclusions this paper deals with the bolza problem p for differential inclusions subject to general endpoint constraints we pursue a twofold goal first we develop a finite difference method for studying p and construct a discrete approximation to p that ensures a strong convergence of optimal solutions second we use this direct method to obtain necessary optimality conditions in a refined euler lagrange form without standard convexity assumptions in general we prove necessary conditions for the so called intermediate relaxed local minimum that takes an intermediate place between the classical concepts of strong and weak minima in the case of a mayer cost functional or boundary solutions to differential inclusions this euler lagrange form holds without any relaxation the results obtained are expressed in terms of nonconvex valued generalized differentiation constructions for nonsmooth mappings and sets

an equivariant higher index theory and nonpositively curved manifolds in this paper we define an equivariant higher index map from k x to k c x if a torsion free discrete group acts on a manifold x properly where c x is the norm closure of all locally compact invariant operators with finite propagation when acts on x properly and cocompactly this equivariant higher index map coincides with the baum connes map p baum a connes k theory for discrete groups in d evens m takesaki eds operator algebras and applications cambridge univ press cambridge 1989 pp 1 20 p baum a connes n higson classifying space for proper actions and k theory of group c algebras in c algebras 1943 1993 san antonio tx 1993 in contemp math vol 167 amer math soc providence ri 1994 pp 240 291 when is trivial this equivariant higher index map is the coarse baum connes map j roe coarse cohomology and index theory on complete riemannian manifolds mem amer math soc 104 497 1993 j roe index theory coarse geometry and the topology of manifolds cbms reg conf ser math vol 90 amer math soc providence ri 1996 if x is a simply connected complete riemannian manifold with nonpositive sectional curvature and is a torsion free discrete group acting on x properly and isometrically we prove that the equivariant higher index map is injective

decomposition of homogeneous polynomials with low rank let f be a homogeneous polynomial of degree d in m 1 variables defined over an algebraically closed field of characteristic 0 and suppose that f belongs to the s th secant variety of the d uple veronese embedding of mathbb p m into mathbb p tiny left begin array c rm m d rm d end array right 1 but that its minimal decomposition as a sum of d th powers of linear forms m1 mr is f m 1 d cdots m r d with r s we show that if s r 2d 1 then such a decomposition of f can be split in two parts one of them is made by linear forms that can be written using only two variables the other part is uniquely determined once one has fixed the first part we also obtain a uniqueness theorem for the minimal decomposition of f if r is at most d and a mild condition is satisfied

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

data driven zero sum neuro optimal control for a class of continuous time unknown nonlinear systems with disturbance using adp this paper is concerned with a new data driven zero sum neuro optimal control problem for continuous time unknown nonlinear systems with disturbance according to the input output data of the nonlinear system an effective recurrent neural network is introduced to reconstruct the dynamics of the nonlinear system considering the system disturbance as a control input a two player zero sum optimal control problem is established adaptive dynamic programming adp is developed to obtain the optimal control under the worst case of the disturbance three single layer neural networks including one critic and two action networks are employed to approximate the performance index function the optimal control law and the disturbance respectively for facilitating the implementation of the adp method convergence properties of the adp method are developed to show that the system state will converge to a finite neighborhood of the equilibrium the weight matrices of the critic and the two action networks are also convergent to finite neighborhoods of their optimal ones finally the simulation results will show the effectiveness of the developed data driven adp methods

a relative higher index theorem diffeomorphisms and positive scalar curvature abstract we prove a general relative higher index theorem for complete manifolds with positive scalar curvature towards infinity we apply this theorem to study riemannian metrics of positive scalar curvature on manifolds for every two metrics of positive scalar curvature on a closed manifold and a galois cover of the manifold we define a secondary higher index class non vanishing of this higher index class is an obstruction for the two metrics to be in the same connected component of the space of metrics of positive scalar curvature in the special case where one metric is induced from the other by a diffeomorphism of the manifold we obtain a formula for computing this higher index class in particular it follows that the higher index class lies in the image of the baum connes assembly map

existence stability and optimality for optimal control problems governed by maximal monotone operators abstract we study optimal control problems governed by maximal monotone differential inclusions with mixed control state constraints in infinite dimensional spaces we obtain some existence results for this kind of dynamics and construct the discrete approximations that allows us to strongly approximate optimal solutions of the continuous type optimal control problems by their discrete counterparts our approach allows us to apply our results for a wide class of mappings that are applicable in mechanics and material sciences

discrete approximations relaxation and optimization of one sided lipschitzian differential inclusions in hilbert spaces we study discrete approximations of nonconvex differential inclusions in hilbert spaces and dynamic optimization optimal control problems involving such differential inclusions and their discrete approximations the underlying feature of the problems under consideration is a modified one sided lipschitz condition imposed on the right hand side i e on the velocity sets of the differential inclusion which is a significant improvement of the conventional lipschitz continuity our main attention is paid to establishing efficient conditions that ensure the strong approximation in the w1 p norm as p 1 of feasible trajectories for the one sided lipschitzian differential inclusions under consideration by those for their discrete approximations and also the strong convergence of optimal solutions to the corresponding dynamic optimization problems under discrete approximations to proceed with the latter issue we derive a new extension of the bogolyubov type relaxation density theorem to the case of differential inclusions satisfying the modified one sided lipschitzian condition all the results obtained are new not only in the infinite dimensional hilbert space framework but also in finite dimensional spaces

neural network based finite horizon optimal control of uncertain affine nonlinear discrete time systems in this paper the finite horizon optimal control design for nonlinear discrete time systems in affine form is presented in contrast with the traditional approximate dynamic programming methodology which requires at least partial knowledge of the system dynamics in this paper the complete system dynamics are relaxed utilizing a neural network nn based identifier to learn the control coefficient matrix the identifier is then used together with the actor critic based scheme to learn the time varying solution referred to as the value function of the hamilton jacobi bellman hjb equation in an online and forward in time manner since the solution of hjb is time varying nns with constant weights and time varying activation functions are considered to properly satisfy the terminal constraint an additional error term is incorporated in the novel update law such that the terminal constraint error is also minimized over time policy and or value iterations are not needed and the nn weights are updated once a sampling instant the uniform ultimate boundedness of the closed loop system is verified by standard lyapunov stability theory under nonautonomous analysis numerical examples are provided to illustrate the effectiveness of the proposed method

the coarse geometric novikov conjecture for subspaces of non positively curved manifolds abstract in this paper we prove the coarse geometric novikov conjecture for metric spaces with bounded geometry which admit a coarse embedding into a simply connected complete riemannian manifold of non positive sectional curvature

baire s category and the bang bang property for evolution differential inclusions of contractive type abstract the baire category method is employed in order to establish that the bang bang property holds for a class of evolution differential inclusions of contractive type in reflexive and separable real banach spaces

neural network based online simultaneous policy update algorithm for solving the hji equation in nonlinear h_ infty control it is well known that the nonlinear h state feedback control problem relies on the solution of the hamilton jacobi isaacs hji equation which is a nonlinear partial differential equation that has proven to be impossible to solve analytically in this paper a neural network nn based online simultaneous policy update algorithm spua is developed to solve the hji equation in which knowledge of internal system dynamics is not required first we propose an online spua which can be viewed as a reinforcement learning technique for two players to learn their optimal actions in an unknown environment the proposed online spua updates control and disturbance policies simultaneously thus only one iterative loop is needed second the convergence of the online spua is established by proving that it is mathematically equivalent to newton s method for finding a fixed point in a banach space third we develop an actor critic structure for the implementation of the online spua in which only one critic nn is needed for approximating the cost function and a least square method is given for estimating the nn weight parameters finally simulation studies are provided to demonstrate the effectiveness of the proposed algorithm

data driven h_ infty control for nonlinear distributed parameter systems the data driven h infty control problem of nonlinear distributed parameter systems is considered in this paper an off policy learning method is developed to learn the h infty control policy from real system data rather than the mathematical model first karhunen loeve decomposition is used to compute the empirical eigenfunctions which are then employed to derive a reduced order model rom of slow subsystem based on the singular perturbation theory the h infty control problem is reformulated based on the rom which can be transformed to solve the hamilton jacobi isaacs hji equation theoretically to learn the solution of the hji equation from real system data a data driven off policy learning approach is proposed based on the simultaneous policy update algorithm and its convergence is proved for implementation purpose a neural network nn based action critic structure is developed where a critic nn and two action nns are employed to approximate the value function control and disturbance policies respectively subsequently a least square nn weight tuning rule is derived with the method of weighted residuals finally the developed data driven off policy learning approach is applied to a nonlinear diffusion reaction process and the obtained results demonstrate its effectiveness

brief paper a neural network solution for fixed final time optimal control of nonlinear systems in this paper fixed final time optimal control laws using neural networks and hjb equations for general affine in the input nonlinear systems are proposed the method utilizes kronecker matrix methods along with neural network approximation over a compact set to solve a time varying hjb equation the result is a neural network feedback controller that has time varying coefficients found by a priori offline tuning convergence results are shown the results of this paper are demonstrated on an example

model free optimal tracking control via critic only q learning model free control is an important and promising topic in control fields which has attracted extensive attention in the past few years in this paper we aim to solve the model free optimal tracking control problem of nonaffine nonlinear discrete time systems a critic only q learning coql method is developed which learns the optimal tracking control from real system data and thus avoids solving the tracking hamilton jacobi bellman equation first the q learning algorithm is proposed based on the augmented system and its convergence is established using only one neural network for approximating the q function the coql method is developed to implement the q learning algorithm furthermore the convergence of the coql method is proved with the consideration of neural network approximation error with the convergent q function obtained from the coql method the adaptive optimal tracking control is designed based on the gradient descent scheme finally the effectiveness of the developed coql method is demonstrated through simulation studies the developed coql method learns with off policy data and implements with a critic only structure thus it is easy to realize and overcome the inadequate exploration problem

optimal control of the sweeping process over polyhedral controlled sets abstract the paper addresses a new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping moreau process while using controls to determine the best shape of moving convex polyhedra in order to optimize the given bolza type functional which depends on control and state variables as well as their velocities besides the highly non lipschitzian nature of the unbounded differential inclusion of the controlled sweeping process the optimal control problems under consideration contain intrinsic state constraints of the inequality and equality types all of this creates serious challenges for deriving necessary optimality conditions we develop here the method of discrete approximations and combine it with advanced tools of first order and second order variational analysis and generalized differentiation this approach allows us to establish constructive necessary optimality conditions for local minimizers of the controlled sweeping process expressed entirely in terms of the problem data under fairly unrestrictive assumptions as a by product of the developed approach we prove the strong w 1 2 convergence of optimal solutions of discrete approximations to a given local minimizer of the continuous time system and derive necessary optimality conditions for the discrete counterparts the established necessary optimality conditions for the sweeping process are illustrated by several examples

the coarse geometric novikov conjecture and uniform convexity abstract the coarse geometric novikov conjecture provides an algorithm to determine when the higher index of an elliptic operator on a noncompact space is nonzero the purpose of this paper is to prove the coarse geometric novikov conjecture for spaces which admit a coarse uniform embedding into a uniformly convex banach space

a bott periodicity theorem for infinite dimensional euclidean space abstract we formulate and prove an equivariant bott periodicity theorem for infinite dimensional euclidean vector spaces the main features of our argument are i the construction of a non commutative c algebra to play the role of the algebra of functions on infinite dimensional euclidean space and ii the construction of a certain index one elliptic partial differential operator which provides the basis for an inverse to the bott periodicity map these tools have applications to index theory and the novikov conjecture notably a proof of the novikov conjecture for amenable groups the applications will be considered elsewhere

topological theory of non autonomous parabolic evolution inclusions on a noncompact interval and applications in this paper we consider the topological structure of the solution set of non autonomous parabolic evolution inclusions with time delay defined on non compact intervals the result restricted to compact intervals is then extended to non autonomous parabolic control problems with time delay moreover as the applications of the information about the structure we establish the existence result of global integral solutions for non autonomous cauchy problems subject to nonlocal condition and prove the invariance of a reachability set for non autonomous control problems under single valued nonlinear perturbations finally some illustrating examples are supplied

online actor critic algorithm to solve the continuous time infinite horizon optimal control problem in this paper we discuss an online algorithm based on policy iteration for learning the continuous time ct optimal control solution with infinite horizon cost for nonlinear systems with known dynamics that is the algorithm learns online in real time the solution to the optimal control design hj equation this method finds in real time suitable approximations of both the optimal cost and the optimal control policy while also guaranteeing closed loop stability we present an online adaptive algorithm implemented as an actor critic structure which involves simultaneous continuous time adaptation of both actor and critic neural networks we call this synchronous policy iteration a persistence of excitation condition is shown to guarantee convergence of the critic to the actual optimal value function novel tuning algorithms are given for both critic and actor networks with extra nonstandard terms in the actor tuning law being required to guarantee closed loop dynamical stability the convergence to the optimal controller is proven and the stability of the system is also guaranteed simulation examples show the effectiveness of the new algorithm

optimal control of a nonconvex perturbed sweeping process abstract the paper concerns optimal control of discontinuous differential inclusions of the normal cone type governed by a generalized version of the moreau sweeping process with control functions acting in both nonconvex moving sets and additive perturbations this is a new class of optimal control problems in comparison with previously considered counterparts where the controlled sweeping sets are described by convex polyhedra besides a theoretical interest a major motivation for our study of such challenging optimal control problems with intrinsic state constraints comes from the application to the crowd motion model in a practically adequate planar setting with nonconvex but prox regular sweeping sets based on a constructive discrete approximation approach and advanced tools of first order and second order variational analysis and generalized differentiation we establish the strong convergence of discrete optimal solutions and derive a complete set of necessary optimality conditions for discrete time and continuous time sweeping control systems that are expressed entirely via the problem data

generic properties of multifunctions application to differential inclusions abstract we prove that almost all in baire sense caratheodory multifunctions in finite dimensional space are kamke continuous further the main properties of differential inclusions with kamke and one sided kamke right hand sides are studied as a corollary we prove that for almost all optimal control problems the relaxation and relaxation stability properties hold

finite horizon control constrained nonlinear optimal control using single network adaptive critics to synthesize fixed final time control constrained optimal controllers for discrete time nonlinear control affine systems a single neural network nn based controller called the finite horizon single network adaptive critic is developed in this paper inputs to the nn are the current system states and the time to go and the network outputs are the costates that are used to compute optimal feedback control control constraints are handled through a nonquadratic cost function convergence proofs of 1 the reinforcement learning based training method to the optimal solution 2 the training error and 3 the network weights are provided the resulting controller is shown to solve the associated time varying hamilton jacobi bellman equation and provide the fixed final time optimal solution performance of the new synthesis technique is demonstrated through different examples including an attitude control problem wherein a rigid spacecraft performs a finite time attitude maneuver subject to control bounds the new formulation has great potential for implementation since it consists of only one nn with single set of weights and it provides comprehensive feedback solutions online though it is trained offline

successive galerkin approximation algorithms for nonlinear optimal and robust control nonlinear optimal control and nonlinear h infinity control are two of the most significant paradigms in nonlinear systems theory unfortunately these problems require the solution of hamilton jacobi equations which are extremely difficult to solve in practice to make matters worse approximation techniques for these equations are inherently prone to the so called curse of dimensionality while there have been many attempts to approximate these equations solutions resulting in closed loop control with well defined stability and robustness have remained elusive this paper describes a recent breakthrough in approximating the hamilton jacobi bellman and hamilton jacobi isaacs equations successive approximation and galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing closed loop control laws with well defined stability regions in addition we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to po

infinite horizon self learning optimal control of nonaffine discrete time nonlinear systems in this paper a novel iterative adaptive dynamic programming adp based infinite horizon self learning optimal control algorithm called generalized policy iteration algorithm is developed for nonaffine discrete time dt nonlinear systems generalized policy iteration algorithm is a general idea of interacting policy and value iteration algorithms of adp the developed generalized policy iteration algorithm permits an arbitrary positive semidefinite function to initialize the algorithm where two iteration indices are used for policy improvement and policy evaluation respectively it is the first time that the convergence admissibility and optimality properties of the generalized policy iteration algorithm for dt nonlinear systems are analyzed neural networks are used to implement the developed algorithm finally numerical examples are presented to illustrate the performance of the developed algorithm

groups acting properly on bolic spaces and the novikov conjecture we introduce a class of metric spaces which we call bolic they include hyperbolic spaces simply connected complete manifolds of nonpositive curvature euclidean buildings etc we prove the novikov conjecture on higher signatures for any discrete group which admits a proper isometric action on a bolic weakly geodesic metric space of bounded geometry

adiabatic groupoid and secondary invariants in k theory abstract in this paper we define new k theoretic secondary invariants attached to a lie groupoid g the receptacle for these invariants is the k theory of c r g a d where g a d is the adiabatic deformation g restricted to the interval 0 1 our construction directly generalises the cases treated in 29 30 in the setting of the coarse geometry to more involved geometrical situations such as foliations moreover we tackle the problem of producing a wrong way functoriality between adiabatic deformation groupoid k groups associated to transverse maps this extends the construction of the lower shriek map in 6 furthermore we prove a lie groupoid version of the delocalized aps index theorem of piazza and schick finally we give a product formula for secondary invariants

geometrization of the strong novikov conjecture for residually finite groups in this paper we prove that the strong novikov conjecture for a residually finite group is essentially equivalent to the coarse geometric novikov conjecture for a certain metric space associated to the group as an application we obtain the coarse geometric novikov conjecture for a large class of sequences of expanders

variational analysis of evolution inclusions the paper is devoted to optimization problems of the bolza and mayer types for evolution systems governed by nonconvex lipschitzian differential inclusions in banach spaces under endpoint constraints described by finitely many equalities and inequalities with generally nonsmooth functions we develop a variational analysis of such problems mainly based on their discrete approximations and the usage of advanced tools of generalized differentiation satisfying comprehensive calculus rules in the framework of asplund and hence any reflexive banach spaces in this way we establish extended results on stability of discrete approximations with the strong w 1 2 convergence of optimal solutions under consistent perturbations of endpoint constraints and derive necessary optimality conditions for nonconvex discrete time and continuous time systems in the refined euler lagrange and weierstrass pontryagin forms accompanied by the appropriate transversality inclusions in contrast to the case of geometric endpoint constraints in infinite dimensions the necessary optimality conditions obtained in this paper do not impose any nonempty interiority finite codimension normal compactness assumptions the approach and results developed in the paper make a bridge between optimal control dynamic optimization and constrained mathematical programming problems in infinite dimensional spaces

finite horizon h_ infty tracking control for unknown nonlinear systems with saturating actuators in this paper a neural network nn based online model free integral reinforcement learning algorithm is developed to solve the finite horizon h infty optimal tracking control problem for completely unknown nonlinear continuous time systems with disturbance and saturating actuators constrained control input an augmented system is constructed with the tracking error system and the command generator system a time varying hamilton jacobi isaacs hji equation is formulated for the augmented problem which is extremely difficult or impossible to solve due to its time dependent property and nonlinearity then an actor critic disturbance nn structure based scheme is proposed to learn the time varying solution to the hji equation in real time without using the knowledge of system dynamics since the solution to the hji equation is time dependent the form of nns representation with constant weights and time dependent activation functions is considered furthermore an extra error is incorporated in order to satisfy the terminal constraints in the weight update law convergence and stability proofs are given based on the lyapunov theory for nonautonomous systems two simulation examples are provided to demonstrate the effectiveness of the designed algorithm

on the homotopy invariance of higher signatures for manifolds with boundary if m is a compact oriented manifold with boundary whose fundamental group is virtually nilpotent or gromov hyperbolic we show that the higher signatures of m are oriented homotopy invariants we give applications to the question of when higher signatures of closed manifolds are cut and paste invariant

difference methods for differential inclusions a survey the main objective of this survey is to study convergence properties of difference methods applied to differential inclusions it presents in a unified way a number of results scattered in the li

approximate optimal trajectory tracking for continuous time nonlinear systems adaptive dynamic programming has been investigated and used as a method to approximately solve optimal regulation problems however the extension of this technique to optimal tracking problems for continuous time nonlinear systems has remained a non trivial open problem the control development in this paper guarantees ultimately bounded tracking of a desired trajectory while also ensuring that the enacted controller approximates the optimal controller

e theory and kk theory for groups which act properly and isometrically on hilbert space a good deal of research in c algebra k theory in recent years has been devoted to the baum connes conjecture 3 which proposes a formula for the k theory of group c algebras that blends group homology with the representation theory of compact subgroups the conjecture has brought c algebra theory into close contact with manifold theory through its obvious similarity to the borel conjecture of surgery theory 9 31 and its links with the theory of positive scalar curvature 27 in addition there are points of contact with harmonic analysis particularly the tempered representation theory of semisimple groups although the proper relation between the baum connes conjecture and representation theory is not well understood the conjecture is most easily formulated for groups which are discrete and torsion free for such a group g there is a natural homomorphism

compression functions of uniform embeddings of groups into hilbert and banach spaces we construct finitely generated groups with arbitrary prescribed hilbert space compression alpha from the interval 0 1 for a large class of banach spaces e including all uniformly convex banach spaces the e compression of these groups coincides with their hilbert space compression moreover the groups that we construct have asymptotic dimension at most 3 hence they are exact in particular the first examples of groups that are uniformly embeddable into a hilbert space respectively exact of finite asymptotic dimension with hilbert space compression 0 are given these groups are also the first examples of groups with uniformly convex banach space compression 0

an iterative adaptive dynamic programming method for solving a class of nonlinear zero sum differential games in this paper a new iterative adaptive dynamic programming adp method is proposed to solve a class of continuous time nonlinear two person zero sum differential games the idea is to use the adp technique to obtain the optimal control pair iteratively which makes the performance index function reach the saddle point of the zero sum differential games if the saddle point does not exist the mixed optimal control pair is obtained to make the performance index function reach the mixed optimum stability analysis of the nonlinear systems is presented and the convergence property of the performance index function is also proved two simulation examples are given to illustrate the performance of the proposed method

policy iteration adaptive dynamic programming algorithm for discrete time nonlinear systems this paper is concerned with a new discrete time policy iteration adaptive dynamic programming adp method for solving the infinite horizon optimal control problem of nonlinear systems the idea is to use an iterative adp technique to obtain the iterative control law which optimizes the iterative performance index function the main contribution of this paper is to analyze the convergence and stability properties of policy iteration method for discrete time nonlinear systems for the first time it shows that the iterative performance index function is nonincreasingly convergent to the optimal solution of the hamilton jacobi bellman equation it is also proven that any of the iterative control laws can stabilize the nonlinear systems neural networks are used to approximate the performance index function and compute the optimal control law respectively for facilitating the implementation of the iterative adp algorithm where the convergence of the weight matrices is analyzed finally the numerical results and analysis are presented to illustrate the performance of the developed method

the novikov conjecture for groups with finite asymptotic dimension recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology page 28 14 more precisely the asymptotic dimension for a metric space is the smallest integer n such that for any r 0 there exists a uniformly bounded cover c ui iei of the metric space for which the r multiplicity of c is at most n 1 i e no ball of radius r in the metric space intersects more than n 1 members of c 14 the class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word length metric then its finitely generated subgroups also have finite asymptotic dimension as metric spaces with word length metrics cf section 6 this together with a result of gromov in 14 implies that finitely generated subgroups of gromov s hyperbolic groups have finite asymptotic dimension currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known it should also be noted that two different definitions of asymptotic dimension