the functional analytic foundation of colombeau algebras abstract colombeau algebras constitute a convenient framework for performing nonlinear operations like multiplication on schwartz distributions many variants and modifications of these algebras exist for various applications we present a functional analytic approach placing these algebras in a unifying hierarchy which clarifies their structural properties as well as their relation to each other

nonsmooth differential geometry and algebras of generalized functions algebras of generalized functions offer possibilities beyond the purely distributional approach in modelling singular quantities in nonsmooth differential geometry this article presents an introductory survey of recent developments in this field and highlights some applications in mathematical physics

moving horizon state estimation for nonlinear systems using neural networks moving horizon mh state estimation is addressed for nonlinear discrete time systems affected by bounded noises acting on system and measurement equations by minimizing a sliding window least squares cost function such a problem is solved by searching for suboptimal solutions for which a certain error is allowed in the minimization of the cost function nonlinear parameterized approximating functions such as feedforward neural networks are employed for the purpose of design thanks to the offline optimization of the parameters the resulting mh estimation scheme requires a reduced online computational effort simulation results are presented to show the effectiveness of the proposed approach in comparison with other estimation techniques

boundary value problems for second order nonlinear differential equations on infinite intervals in this paper we consider boundary value problems for nonlinear differential equations on the semi axis 0 and also on the whole axis under the assumption that the left hand side being a second order linear differential expression belongs to the weyl limit circle case the boundary value problems are considered in the hilbert spaces l2 0 and l2 and include boundary conditions at infinity the existence and uniqueness results for solutions of the considered boundary value problems are established

receding horizon estimation for discrete time linear systems the problem of estimating the state of a discrete time linear system can be addressed by minimizing an estimation cost function dependent on a batch of recent measure and input vectors this problem has been solved by introducing a receding horizon objective function that includes also a weighted penalty term related to the prediction of the state for such an estimator convergence results and unbiasedness properties have been proved the issues concerning the design of this filter are discussed in terms of the choice of the free parameters in the cost function the performance of the proposed receding horizon filter is evaluated and compared with other techniques by means of a numerical example

spline smoothing the equivalent variable kernel method on considere une approche spline pour la regression non parametrique et l estimation de courbe on montre que dans un certain sens le lissage spline correspond approximativement au lissage par une methode du noyau avec une largeur de bande dependant de la densite locale des points du plan de regression

regularization theory and neural networks architectures we had previously shown that regularization principles lead to approximation schemes that are equivalent to networks with one layer of hidden units called regularization networks in particular s

observer design for nonlinear systems with discrete time measurements this paper focuses on the development of asymptotic observers for nonlinear discrete time systems it is argued that instead of trying to imitate the linear observer theory the problem of constructing a nonlinear observer can be more fruitfully studied in the context of solving simultaneous nonlinear equations in particular it is shown that the discrete newton method properly interpreted yields an asymptotic observer for a large class of discrete time systems while the continuous newton method may be employed to obtain a global observer furthermore it is analyzed how the use of broyden s method in the observer structure affects the observer s performance and its computational complexity an example illustrates some aspects of the proposed methods moreover it serves to show that these methods apply equally well to discrete time systems and to continuous time systems with sampled outputs

nonlinear generalized functions on manifolds in this work we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators this produces a generaliz

very weak solutions to hypoelliptic wave equations abstract in this paper we study the cauchy problem for the wave equations for hypoelliptic homogeneous left invariant operators on graded lie groups when the time dependent non negative propagation speed is regular holder and distributional for holder coefficients we derive the well posedness in the spaces of ultradistributions associated to rockland operators on graded groups in the case when the propagation speed is a distribution we employ the notion of very weak solutions to the cauchy problem that was already successfully used in similar contexts in 12 and 20 we show that the cauchy problem for the wave equation with the distributional coefficient has a unique very weak solution in an appropriate sense which coincides with classical or distributional solutions when the latter exist examples include the time dependent wave equation for the sub laplacian on the heisenberg group or on general stratified lie groups or p evolution equations for higher order operators on r n or on groups the results already being new in all these cases

fast homoclinic solutions for some second order non autonomous systems abstract this paper is concerned with the existence of homoclinic solutions for the following second order non autonomous system fhs q c q u q t q 0 where q r n c 0 is a constant and u t q c 1 r r n r under some reasonable assumptions for the first time we introduce the concept of fast homoclinic solutions of fhs and establish some criteria to guarantee the existence of fast homoclinic solutions of fhs besides that in some particular case the uniqueness of homoclinic solutions of fhs is also obtained recent results in the literature are generalized and significantly improved

further results on nonlinear receding horizon observers further results are proposed that concern the design and the convergence of receding horizon nonlinear observers the key feature is the definition of observability radius in relation with a prespecified compact set of initial configurations this enables a semiglobal convergence result to be derived that turns out to be a global convergence result when appropriate regularity assumptions are made a simple example is proposed to illustrate the basic features of the note

neural approximations for infinite horizon optimal control of nonlinear stochastic systems a feedback control law is proposed that drives the controlled vector v sub t of a discrete time dynamic system to track a reference v sub t over an infinite time horizon while minimizing a given cost function the behavior of v sub t over time is completely unpredictable random noises act on the dynamic system and the state observation channel which may be nonlinear it is assumed that all such random vectors are mutually independent and that their probability density functions are known so general a non lqg optimal control problem is very difficult to solve the proposed solution is based on three main approximating assumptions 1 the problem is stated in a receding horizon framework where v sub t is assumed to remain constant within a shifting time window 2 the control law is assigned a given structure that of a multilayer feedforward neural net in which a finite number of parameters have to be determined so as to minimize the cost function and 3 the control law is given a limited memory which prevents the amount of data to be stored from increasing over time errors resulting from the second and third assumptions are discussed due to the very general assumptions under which the control law is derived we are not able to report stability results however simulation results show that the proposed method may constitute an effective tool for solving to a sufficient degree of accuracy a wide class of control problems traditionally regarded as difficult ones an example of freeway traffic optimal control is given

optimum nonlinear filtering this paper is composed of two parts the first part surveys the literature regarding optimum nonlinear filtering from the continuous time stochastic analysis point of view and the other part explores the impact of recent applications of neural networks in a discrete time context to nonlinear filtering in particular the results obtained by using a regularized form of radial basis function rbf networks are presented in fair detail

on estimation error bounds for receding horizon filters using quadratic boundedness quadratic boundedness is used to deal with stability and design of receding horizon estimators upper bounds on the norm of the estimation error have been found by means of invariant sets that can be constructed by using quadratic boundedness moreover these bounds are expressed in terms of linear matrix inequalities and are well suited to being minimized for the purpose of design

error estimates for approximate optimization by the extended ritz method an alternative to the classical ritz method for approximate optimization is investigated in the extended ritz method sets of admissible solutions are approximated by their intersections with sets of linear combinations of all n tuples of functions from a given basis this alternative scheme called variable basis approximation includes functions computable by trigonometric polynomials with free frequencies free node splines neural networks and other nonlinear approximating families estimates of rates of approximate optimization by the extended ritz method are derived upper bounds on rates of convergence of suboptimal solutions to the optimal one are expressed in terms of the degree n of variable basis functions the modulus of continuity of the functional to be minimized the modulus of tikhonov well posedness of the problem and certain norms tailored to the type of basis the results are applied to convex best approximation and to kernel methods in machine learning

generalized pseudo riemannian geometry generalized tensor analysis in the sense of colombeau s construction is employed to introduce a nonlinear distributional pseudo riemannian geometry in particular after deriving several characterizations of invertibility in the algebra of generalized functions we define the notions of generalized pseudo riemannian metric generalized connection and generalized curvature tensor we prove a fundamental lemma of pseudo riemannian geometry in this setting and define the notion of geodesics of a generalized metric finally we present applications of the resulting theory to general relativity

scalar products of elementary distributions the schrodinger equation with point interaction in one dimension is revisited in a simple framework where the singular potential is defined as a symmetric operator in a natural way the main tool is a scalar product of the elementary distributions constructed after a commutative and well ordered field extension of r followed by complexification a contact with the hyper real numbers that arise in nonstandard analysis is possible but not essential our extensions of r and c being obtained by a quite elementary method

symmetry group analysis of weak solutions methods of lie group analysis of differential equations are extended to weak solutions of linear and non linear partial differential equations where the term weak solution comprises the following settings distributional solutions solutions in generalized function algebras solutions in the sense of association corresponding to a number of weak or integral solution concepts in classical analysis factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed thereby unifying and extending previous work in this area 2000 mathematical subject classification 46f30 22e70 35dxx 35a30

intrinsic characterization of manifold valued generalized functions the concept of generalized functions taking values in a differentiable manifold is extended to a functorial theory we establish several characterization results which allow a global intrinsic formulation both of the theory of manifold valued generalized functions and of generalized vector bundle homomorphisms as a consequence a characterization of equivalence that does not resort to derivatives analogous to scalar valued cases of colombeau s construction is achieved these results are employed to derive a point value description of all types of generalized functions valued in manifolds and to show that composition can be carried out unrestrictedly finally a new concept of association adapted to the present setting is introduced

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

nonlinear black box models in system identification mathematical foundations we discuss several aspects of the mathematical foundations of the nonlinear black box identification problem we shall see that the quality of the identification procedure is always a result of a certain trade off between the expressive power of the model we try to identify the larger the number of parameters used to describe the model the more flexible is the approximation and the stochastic error which is proportional to the number of parameters a consequence of this trade off is the simple fact that a good approximation technique can be the basis of a good identification algorithm from this point of view we consider different approximation methods and pay special attention to spatially adaptive approximants we introduce wavelet and neuron approximations and show that they are spatially adaptive then we apply the acquired approximation experience to estimation problems finally we consider some implications of these theoretical developments for the practically implemented versions of the spatially adaptive algorithms

sheaves of nonlinear generalized functions and manifold valued distributions this paper is part of an ongoing program to develop a theory of generalized differential geometry we consider the space g x y of colombeau generalized functions defined on a manifold x and taking values in a manifold y this space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics we introduce an embedding of the space of continuous mappings c x y into g x y and study the sheaf properties of g x y similar results are obtained for spaces of generalized vector bundle homomorphisms based on these constructions we propose the definition of a space d 0 x y of distributions on x taking values in y d 0 x y is realized as a quotient of a certain subspace of g x y

generalized group actions in a global setting we study generalized group actions on differentiable manifolds in the colombeau framework extending previous work on flows of generalized vector fields and symmetry group analysis of generalized solutions as an application we analyze group invariant generalized functions in this setting

on transitional solutions of second order nonlinear differential equations this work was supported by the research grant of the italian ministry of foreign affairs p a 31524 02 and carried out while the third author was visiting the university of modena and reggio emilia and the polytechnic university of marche corresponding author tel 39 0522 522616 fax 39 0522 522609 e mail addresses malaguti luisa unimore it l malaguti marcelli dipmat univpm it c marcelli ninopa rmi acnet ge n partsvania

symmetric hyperbolic systems in algebras of generalized functions and distributional limits we study existence uniqueness and distributional aspects of generalized solutions to the cauchy problem for first order symmetric or hermitian hyperbolic systems of partial differential equations with colombeau generalized functions as coefficients and data the proofs of solvability are based on refined energy estimates on lens shaped regions with spacelike boundaries we obtain several variants and also partial extensions of previous results in oberguggenberger 1989 lafon and oberguggenberger 1991 and hormann 2004 26 23 16 and provide aspects accompanying related recent work in oberguggenberger 2009 garetto and oberguggenberger 2011 28 10 9

a nonlinear theory of distributional geometry this paper builds on the theory of nonlinear generalized functions begun in nigsch vickers nigsch vickers 2021 proc r soc a 20200640 doi 10 1098 rspa 2020 0640 and extends this to a diffe

neural approximators for nonlinear finite memory state estimation no general analytical tools are available to estimate the state of a nonlinear stochastic system observed through a nonlinear noisy channel this problem is addressed in this paper under the assumption that the statistics of the random variables are unknown hence a statistical approach is followed instead of a probabilistic one the following approximations are enforced i the state estimator is a finite memory one ii the estimation functions are given fixed structures in which a certain number of parameters have to be optimized multilayer feedforward neural networks are chosen from among various possible nonlinear approximators iii the algorithms for optimizing the parameters i e the network weights rely on a stochastic approximation simulation results are reported to compare the behaviour of the proposed estimator with the extended kalman filter and the estimators based on the on line minimization of the estimation error

feedback stabilization using two hidden layer nets the representational capabilities of one hidden layer and two hidden layer nets consisting of feedforward interconnections of linear threshold units are compared it is remarked that for certain problems two hidden layers are required contrary to what might be in principle expected from the known approximation theorems the differences are not based on numerical accuracy or number of units needed nor on capabilities for feature extraction but rather on a much more basic classification into direct and inverse problems the former correspond to the approximation of continuous functions while the latter are concerned with approximating one sided inverses of continuous functions and are often encountered in the context of inverse kinematics determination or in control questions a general result is given showing that nonlinear control systems can be stabilized using two hidden layers but not in general using just one

numerical solutions to the witsenhausen counterexample by approximating networks approximate solutions to the witsenhausen counterexample 1968 are derived by constraining the unknown control functions to take on fixed structures containing free parameters to be optimized such structures are given by nonlinear approximating networks i e linear combinations of parametrized basis functions that benefit by density properties in normed linear spaces this reduces the original functional problem to a nonlinear programming one which is solved via stochastic approximation the method yields lower values of the costs than the ones achieved so far in the literature and most of all provides rather a complete overview of the shapes of the optimal control functions when the two parameters that characterize the witsenhausen counterexample vary one hidden layer neural networks are chosen as approximating networks

hilbert ℂ modules structural properties and applications to variational problems abstract we develop a theory of hilbert c over tilde modules which forms the core of a new functional analytic approach to algebras of generalized functions particular attention is given to nitely generated submodules projection operators representation theorems for c over tilde linear functionals and c over tilde sesquilinear forms we establish a generalized lax milgram theorem and use it to prove existence and uniqueness theorems for variational problems involving a generalized bilinear or sesquilinear form

observability of systems under uncertainty the evolution of the state x cdot of a system under uncertainty governed by a differential inclusion text for almost all t in 0 t qquad x t in f t x t is observed through an observation map h forall t in 0 t quad y t in h x t the set valued character due to the uncertainty leads to the introduction of the following sharp input output map which is the usual product forall x 0 in x quad i x 0 h circ mathcal s x 0 mathop cup limits x cdot in mathcal s x 0 h x cdot hazy input output map which is the square product forall x 0 in x quad i x 0 h square mathcal s x 0 mathop cap limits x cdot in mathcal s x 0 h x cdot where mathcal s denotes the solution map recovering the input x 0 from the outputs i x 0 or i x 0 means that these input output maps are injective in the sense that locally x 1 ne x 2 rightarrow i x 1 cap i x 2 emptyset criteria for bo

suppleness of the sheaf of algebras of generalized functions on manifolds we show that the sheaves of algebras of generalized functions g and g are open sets in a manifold x are supple contrary to the non suppleness of the sheaf of distributions

neural approximations for multistage optimal control of nonlinear stochastic systems two main approximations are used to solve a nonlinear quadratic gaussian lqg optimal control problem the control law is assigned a given structure in which a finite number of parameters have to be determined to minimize the cost function the chosen structure is that of a multilayer feedforward neural network and the control law is given a limited memory the errors resulting front both assumptions are discussed simulation results show that the proposed method may constitute a simple and effective tool for solving to a sufficient degree of accuracy optimal control problems traditionally regarded as difficult ones

a note on a nonautonomous o d e related to the fisher equation abstract we give elementary proofs of the existence of a heteroclinic for the nonautonomous analogue of an o d e arising in connection with the fisher equation in particular we give a simple technique to compute lower bounds of the values of the admissible speeds

a rigorous solution concept for geodesic and geodesic deviation equations in impulsive gravitational waves the geodesic as well as the geodesic deviation equation for impulsive gravitational waves involve highly singular products of distributions 2 2 a solution concept for these equations based on embedding the distributional metric into the colombeau algebra of generalized functions is presented using a universal regularization procedure we prove existence and uniqueness results and calculate the distributional limits of these solutions explicitly the obtained limits are regularization independent and display the physically expected behavior

state observation by on line minimization a new approach is presented to observing the state of a nonlinear control system the class considered consists of nonlinear observable systems described by an ordinary differential equation in combination with a measurement equation the system functions are required to be twice differentiable based on newton s method an iterative process of state approximation is derived the algorithm computes a piecewise continuous function which is an approximation of the true state the convergence of the proposed algorithm is proved finally several examples show the performance characteristics of the method

distributed moving horizon estimation for linear constrained systems this paper presents a novel distributed estimation algorithm based on the concept of moving horizon estimation under weak observability conditions we prove convergence of the state estimates computed by any sensors to the correct state even when constraints on noise and state variables are taken into account in the estimation process simulation examples are provided in order to show the main features of the proposed method

generalized functions and multiplication of distributions on c manifolds abstract an improvement in definition permits us to make invariant under the action of c diffeomorphisms a concept of generalized functions introduced by the first author to define arbitrary products of distributions

removable singularities in yang mills fields we show that a field satisfying the yang mills equations in dimension 4 with a point singularity is gauge equivalent to a smooth field if the functional is finite we obtain the result that every yang mills field overr4 with bounded functional l2 norm may be obtained from a field ons4 r4 hodge or coulomb gauges are constructed for general small fields in arbitrary dimensions including 4

moving horizon observers and observer based control in this paper two topics are explored a new approach to the problem of obtaining an estimate of the state of a nonlinear system is proposed the moving horizon observer produces an estimate of the state of the nonlinear system at time t either by minimizing or approximately minimizing a cost function over the preceding interval horizon t t t as t advances so does the horizon convergence of the estimator is established under the assumption that the corresponding global optimization problem can be approximately solved and a uniform reconstructability condition is satisfied the latter condition is automatically satisfied for linear observable systems the utility of the estimator for receding horizon control is explored in particular stability of a composite moving horizon system comprising a moving horizon regulator and a moving horizon observer is established

the elastoplastic shock problem as an example of the resolution of ambiguities in the multiplication of distributions one dimensional collisions between two isotropic solids in which the equations of physics lead to multiplications of distributions are considered based on this example a general physicomathematical method to be adapted to each particular case is proposed to resolve the ambiguity inherent in such products this can be achieved with the aid of a new mathematical theory of generalized functions which permits dealing with mathematical phenomena of a microscopic nature that govern products of distributions having singularities at the same point this tool has recently been applied in various situations in continuum mechanics in which the equations of physics lead to heuristic products of distributions one obtains new algebraic formulas in the simplest cases and new numerical schemes in more general cases the key to the resolution of ambiguities lies in more precise statements of the laws of physics than are permitted within distribution theory and have no analog in classical analysis so that in general a resolution cannot be obtained from formal calculations

a global theory of algebras of generalized functions abstract we present a geometric approach to defining an algebra g m the colombeau algebra of generalized functions on a smooth manifold m containing the space d m of distributions on m based on differential calculus in convenient vector spaces we achieve an intrinsic construction of g m g m is a differential algebra its elements possessing lie derivatives with respect to arbitrary smooth vector fields moreover we construct a canonical linear embedding of d m into g m that renders c m a faithful subalgebra of g m finally it is shown that this embedding commutes with lie derivatives thus g m retains all the distinguishing properties of the local theory in a global context

geodesics and geodesic deviation for impulsive gravitational waves the geometry of impulsive pp waves is explored via the analysis of the geodesic and geodesic deviation equation using the distributional form of the metric the geodesic equation involves formally ill defined products of distributions due to the nonlinearity of the equations and the presence of the dirac distribution in the space time metric thus strictly speaking it cannot be treated within schwartz s linear theory of distributions to cope with this difficulty we proceed by first regularizing the singularity then solving the regularized equation within classical smooth functions and finally obtaining a distributional limit as solution to the original problem furthermore it is shown that this limit is independent of the regularization without requiring any additional condition thereby confirming earlier results in a mathematically rigorous fashion we also treat the jacobi equation which despite being linear in the deviation vector field involves even more delicate singular expressions like the square of the dirac distribution again the same regularization procedure provides us with a perfectly well behaved smooth regularization and a regularization independent distributional limit hence it is concluded that the geometry of impulsive pp waves can be described consistently using distributions as long as careful regularization procedures are used to handle the ill defined products

hinging hyperplanes for regression classification and function approximation a hinge function y h x consists of two hyperplanes continuously joined together at a hinge in regression prediction classification pattern recognition and noiseless function approximation use of sums of hinge functions gives a powerful and efficient alternative to neural networks with computation times several orders of magnitude less than is obtained by fitting neural networks with a comparable number of parameters a simple and effective method for finding good hinges is presented

heteroclinic connections for fully non linear non autonomous second order differential equations we investigate the solvability of the following strongly non linear non autonomous boundary value problem p a x t x t f t x t x t a e t r x x with given constants where a x is a generic continuous positive function and f is a caratheodory non linear function we show that the solvability of p is strictly connected to a sharp relation between the behaviors of f t x as x 0 and f x x as t such a relation is optimal for a wide class of problems for which we prove that p is not solvable when it does not hold

nonlinear generalized sections of vector bundles abstract we present an extension of j f colombeau s theory of nonlinear generalized functions to spaces of generalized sections of vector bundles our construction builds on classical functional analytic notions which is the key to having a canonical geometric embedding of vector bundle valued distributions into spaces of generalized sections this permits to have tensor products invariance under diffeomorphisms covariant derivatives and the sheaf property while retaining as much compatibility to l schwartz theory of distributions as possible our theory provides the basis for a rigorous and general treatment of singular pseudo riemannian geometry in the setting of colombeau nonlinear generalized functions

generalized solutions of nonlinear parabolic equations with distributions as initial conditions abstract we consider the nonlinear parabolic equation u t u u 3 0 in q 0 t t 0 open set in r d d 1 2 with the boundary condition u x t 0 on 0 t and the initial condition u x 0 x in where is the dirac mass at the origin of r it is known that this problem has no weak solution in any known classical sense within the distribution theory using a theory of generalized functions we obtain existence uniqueness and consistence results which describe mathematically the behaviour of the solutions ue obtained with smooth initial conditions u e x 0 e x e d and e when e 0

on the observer problem for discrete time control systems this paper studies the construction of observers for nonlinear time varying discrete time systems in a general context where a certain function of the states must be estimated appropriate notions of robust complete observability are proposed under which a constructive proof of existence of an observer is developed moreover a transitive observer property is proven according to which a state observer can be generated as the series connection of two observers the analysis and the results are developed in general normed linear spaces to cover both finite dimensional and infinite dimensional systems

algebraic theory of colombeauʼs generalized numbers abstract let k denote the commutative ring of colombeau s full generalized numbers endowed with scarpalezos sharp topology it becomes a topological ring we study the algebraic and topological properties of this topological ring in particular we prove that the group of units of k is dense in the sharp topology determine its boolean algebra show that it has minimal primes describe them completely which results in a complete classification of the maximal ideals from the description of the prime and maximal ideals it becomes clear that they should be determined by certain ultra filters

robust receding horizon state estimation for uncertain discrete time linear systems an approach to robust receding horizon state estimation for discrete time linear systems is presented estimates of the state variables can be obtained by minimizing a worst case quadratic cost function according to a sliding window strategy this leads to state the estimation problem in the form of a regularized least squares one with uncertain data the optimal solution involving on line scalar minimization together with a suitable closed form approximation are given the stability properties of the estimation error for both the optimal filter and the approximate one have been studied and conditions to select the design parameters are proposed simulation results are reported to show the effectiveness of the proposed approach

classification of singular sobolev connections by their holonomy for a connection on a principalsu 2 bundle over a base space with a codimension two singular set a limit holonomy condition is stated in dimension four finite action implies that the condition is satisfied and an a priori estimate holds which classifies the singularity in terms of holonomy if there is no holonomy then a codimension two removable singularity theorem is obtained

hyperbolic systems with discontinuous coefficients generalized solutions and a transmission problem in acoustics where v v v g g g a and f are n x n matrices a is real valued and diagonal in the case of discontinuous coeffkients more precisely we wish to assume solely that a is a bounded measurable function and f is the distributional derivative of such a function the example of hurd and sattinger 3 p 1661 shows that in general such a system will not have solutions in the sense of distributions even if it is in the form of a conservation law observe that a priori 8 v will not be a function and so multiplication of distributions is involved in 0 1 our approach will be to solve the problem in the colombeau algebra y a differential algebra which contains the distributions and in addition has the algebra of infinitely differentiable functions as a sub algebra in section 1 we shall show that if a f g are generalized functions belonging to 9 lr2 and a belongs to 9 r then system 0 1 has a unique solution ve a r provided the generalized functions a and f satisfy some assumptions corresponding to a e l lr fe w l m lr distributional solutions are recovered from the generalized solutions via a partial projection 452 0022 247x 89 3 00

system of nonlinear volterra s integral equations with polar kernel and singularities abstract we study an existence uniqueness theorem for a system of nonlinear volterra type integral equations with polar kernel and singularities to do so we use a method of regularization of fractional derivatives with a delta sequence to avoid singularities over diagonal y x y 0 the other singularities we regularize with delta sequences with different growth as a framework we use colombeau special algebra

nsm constrained approximation of lipschitz functions from data the problem of approximating a function from a set of its values in the presence of constraints is considered solving this problem is useful in system identification state estimation and control the typical approach to this problem consists in 1 deriving an approximation without considering the constraints and 2 using the constraints to clip the approximation however using this approach may give poor results and in any case no optimality properties can be guaranteed in this paper a nonlinear set membership method is developed allowing optimal constrained approximation the effectiveness of the proposed method is shown through its application to the problem of model predictive control approximation

design of asymptotic estimators an approach based on neural networks and nonlinear programming a methodology to design state estimators for a class of nonlinear continuous time dynamic systems that is based on neural networks and nonlinear programming is proposed the estimator has the structure of a luenberger observer with a linear gain and a parameterized in general nonlinear function whose argument is an innovation term representing the difference between the current measurement and its prediction the problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error toward this end if a neural network is used to take on this function the parameters i e the neural weights are chosen together with the gain by constraining the derivative of a quadratic lyapunov function for the estimation error to be negative definite on a given compact set it is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points the gain is determined by solving a lyapunov equation the neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid point constraints that are not satisfied techniques based on low discrepancy sequences are applied to deal with a small number of sampling points and hence to reduce the computational burden required to optimize the parameters numerical results are reported and comparisons with those obtained by the extended kalman filter are made

constrained state estimation for nonlinear discrete time systems stability and moving horizon approximations state estimator design for a nonlinear discrete time system is a challenging problem further complicated when additional physical insight is available in the form of inequality constraints on the state variables and disturbances one strategy for constrained state estimation is to employ online optimization using a moving horizon approximation we propose a general theory for constrained moving horizon estimation sufficient conditions for asymptotic and bounded stability are established we apply these results to develop a practical algorithm for constrained linear and nonlinear state estimation examples are used to illustrate the benefits of constrained state estimation our framework is deterministic

moving horizon observer with regularisation for detectable systems without persistence of excitation a constrained moving horizon observer is developed for nonlinear discrete time systems the algorithm is proved to converge exponentially under a detectability assumption with the data being exciting at all times however in many practical estimation problems such as combined state and parameter estimation the data may not be exciting for every period of time the algorithm therefore has regularisation mechanisms to ensure robustness and graceful degradation of performance in cases when the data are not exciting this includes the use of a priori estimates in the moving horizon cost function and the use of thresholded singular value decomposition to avoid ill conditioned or ill posed inversion of the associated nonlinear algebraic equations that define the moving horizon cost function the latter regularisation relies on monitoring of the rank of an estimate of a hessian like matrix and conditions for uniform exponential convergence are given the method is in particular useful with augmented state sp

the wave equation on singular space times we prove local unique solvability of the wave equation for a large class of weakly singular locally bounded space time metrics in a suitable space of generalised functions

comparative study of non linear state observation techniques abstract this paper contains a comparative study of four techniques for observing the slates of non linear systems the first technique examined is inspired by bestle and zeitz and icrener and respondek in this method a non linear transformation is found that brings the system into a canonical form from where observer design is facilitated the second technique is attributable to thau in this method the error between the system s true state and the output of the observer is shown to be asymptotically convergent to zero provided that an additional assumption is valid the third technique is due to baumann and rugh in this method extended or pseudolinearization of the error differential equation about a family of equilibrium points results in an observer design such that the eigenvalues of the linearized error equation are locally invariant finally techniques from variable structure systems are utilized to design an observer that yields an exponentially decaying error like thau s observer but unl

moving horizon state estimation for nonlinear discrete time systems new stability results and approximation schemes a moving horizon state estimation problem is addressed for a class of nonlinear discrete time systems with bounded noises acting on the system and measurement equations as the statistics of such disturbances and of the initial state are assumed to be unknown we use a generalized least squares approach that consists in minimizing a quadratic estimation cost function defined on a recent batch of inputs and outputs according to a sliding window strategy for the resulting estimator the existence of bounding sequences on the estimation error is proved in the absence of noises exponential convergence to zero is obtained moreover suboptimal solutions are sought for which a certain error is admitted with respect to the optimal cost value the approximate solution can be determined either on line by directly minimizing the cost function or off line by using a nonlinear parameterized function simulation results are presented to show the effectiveness of the proposed approach in comparison with the extended kalman filter

a neural state estimator with bounded errors for nonlinear systems a neural state estimator is described acting on discrete time nonlinear systems with noisy measurement channels a sliding window quadratic estimation cost function is considered and the measurement noise is assumed to be additive no probabilistic assumptions are made on the measurement noise nor on the initial state novel theoretical convergence results are developed for the error bounds of both the optimal and the neural approximate estimators to ensure the convergence properties of the neural estimator a minimax tuning technique is used the approximate estimator can be designed offline in such a way as to enable it to process on line any possible measure pattern almost instantly

neural networks for feedback feedforward nonlinear control systems this paper deals with the problem of designing feedback feedforward control strategies to drive the state of a dynamic system in general nonlinear so as to track any desired trajectory joining the points of given compact sets while minimizing a certain cost function in general nonquadratic due to the generality of the problem conventional methods are difficult to apply thus an approximate solution is sought by constraining control strategies to take on the structure of multilayer feedforward neural networks after discussing the approximation properties of neural control strategies a particular neural architecture is presented which is based on what has been called the linear structure preserving principle the original functional problem is then reduced to a nonlinear programming one and backpropagation is applied to derive the optimal values of the synaptic weights recursive equations to compute the gradient components are presented which generalize the classical adjoint system equations of n stage optimal control theory simulation results related to nonlinear nonquadratic problems show the effectiveness of the proposed method

sup norm approximation bounds for networks through probabilistic methods we consider the problem of approximating a smooth target function and its derivatives by networks involving superpositions and translations of a fixed activation function the approximation is with respect to the sup norm and the rate is shown to be of order o n sup 1 2 that is the rate is independent of the dimension d the results apply to neural and wavelet networks and extend the work of barren see proc 7th yale workshop on adaptive and learning systems may 1992 and ibid vol 39 p 930 1993 the approach involves probabilistic methods based on central limit theorems for empirical processes indexed by classes of functions

brief paper moving horizon partition based state estimation of large scale systems this paper presents three novel moving horizon estimation mhe methods for discrete time partitioned linear systems i e systems decomposed into coupled subsystems with non overlapping states the mhe approach is used due to its capability of exploiting physical constraints on states and noise in the estimation process in the proposed algorithms each subsystem solves reduced order mhe problems to estimate its own state and different estimators have different computational complexity accuracy and transmission requirements among subsystems in all cases proper tuning of the design parameters i e the penalties on the states at the beginning of the estimation horizon guarantees convergence of the estimation error to zero numerical simulations demonstrate the viability of the approach

from continuous time design to sampled data design of observers in this work a sampled data nonlinear observer is designed using a continuous time design coupled with an inter sample output predictor the proposed sampled data observer is a hybrid system it is shown that under certain conditions the robustness properties of the continuous time design are inherited by the sampled data design as long as the sampling period is not too large the approach is applied to linear systems and to triangular globally lipschitz systems

linear moving horizon estimation with pre estimating observer in this note a moving horizon estimation mhe strategy for detectable linear systems is proposed like the idea of pre stabilizing model predictive control the states are estimated by a forward simulation with a pre estimating observer in the mhe formulation compared with standard linear mhe approaches it has more degrees of freedom to optimize the noise filtering tuning parameters are chosen to minimize the effects of measurement noise and model errors which is implemented by finding tightest estimation error bounds the performance of the proposed observer is demonstrated on one linear discrete time example