on continuous dependence for an inverse initial boundary value problem for the wave equation abstract we show that the bounded coefficient q x of the wave equation u tt u qu x t 0 t u x 0 u t x 0 0 x u v x x x t 0 t can be continuously determined by its neumann to dirichlet map in certain function spaces

decentralized adaptive output feedback stabilization for large scale stochastic nonlinear systems in this paper the problem of decentralized adaptive output feedback stabilization is investigated for large scale stochastic nonlinear systems with three types of uncertainties including parametric uncertainties nonlinear uncertain interactions and stochastic inverse dynamics under the assumption that the inverse dynamics of the subsystems are stochastic input to state stable an adaptive output feedback controller is constructively designed by the backstepping method it is shown that under some general conditions the closed loop system trajectories are bounded in probability and the outputs can be regulated into a small neighborhood of the origin in probability in addition the equilibrium of interest is globally stable in probability and the outputs can be regulated to the origin almost surely when the drift and diffusion vector fields vanish at the origin the contributions of the work are characterized by the following novel features 1 even for centralized single input single output systems this paper presents a first result in stochastic nonlinear adaptive output feedback asymptotic stabilization 2 the methodology previously developed for deterministic large scale systems is generalized to stochastic ones at the same time novel small gain conditions for small signals are identified in the setting of stochastic systems design 3 both drift and diffusion vector fields are allowed to be dependent not only on the measurable outputs but some unmeasurable states 4 parameter update laws are used to counteract the parametric uncertainty existing in both drift and diffusion vector fields which may appear nonlinearly 5 the concept of stochastic input to state stability and the method of changing supply functions are adapted for the first time to deal with stochastic and nonlinear inverse dynamics in the context of decentralized control

links between functions and subdifferentials abstract a function in a class f x is said to be subdifferentially determined in f x if it is equal up to an additive constant to any function in f x with the same subdifferential a function is said to be subdifferentially representable if it can be recovered from a subdifferential we identify large classes of lower semicontinuous functions that possess these properties

maximal quasimonotonicity and dense single directional properties of quasimonotone operators the concept of maximal quasimonotonicity of set valued map is introduced and studied regularity properties of this class of operators is investigated in particular the ccvc property an adaptation to the quasimonotone case of the classical notion of cusco map in an asplund space we provide sufficient conditions for a ccvc quasimonotone operator to be single directional on a g delta dense subset

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

when is the space of minimal usco cusco maps a topological vector space abstract we will define a vector structure on the space of minimal usco resp cusco maps from a locally compact space to a stegall locally convex space we will prove that these spaces equipped with the topology of uniform convergence on compacta are locally convex spaces

null sets and essentially smooth lipschitz functions in this paper we extend the notion of a lebesgue null set to a notion which is valid in any completely metrizable abelian topological group we then use this definition to introduce and study the class of essentially smooth functions these are roughly speaking those lipschitz functions which are smooth in each direction almost everywhere

recovery of a source term or a speed with one measurement and applications we study the problem of recovery the source a t x f x in the wave equation in anisotropic medium with a known so that a 0 x 6d 0 with a single measurement we use carleman estimates combined with geometric arguments and give sharp conditions for uniqueness we also study the non linear problem of recovery the sound speed in the equation utt c x u d 0with one measurement we give sharp conditions for stability as well an application to thermoacoustic tomography is also presented

stable recovery of time dependent coefficient from arbitrary measurements for wave equation abstract this paper concerns the stability in the inverse problem of determining the time dependent coefficient in the wave equation from partial knowledge of the dirichlet to neumann map at the boundary of a bounded domain of interest we first establish a stability estimate in the determination of the time dependent coefficient from partial boundary observations then we prove that by enlarging the set of data the unknown coefficient can be stably determined in larger regions

robust nonlinear control of systems with input unmodeled dynamics we consider systems with unmodeled dynamics appearing at the input and present a redesign that preserves global asymptotic stability the admissible unmodeled dynamics are relative degree zero and minimum phase the redesign is illustrated on an analytical example

integration of subdifferentials of nonconvex functions in nonsmooth analysis and optimization subgradients come in many different flavors e g approximate dini proximal clarke generalized see 2 5 6 12 14 191 these subgradients are important and valuable tools however many questions remain unsolved concerning the exact link between the function and its subgradients for instance can two functions not differing by an additive constant have the same subgradients in this paper we study the fundamental problem of determining functions that can be recovered up to an additive constant from the knowledge of their subgradients this integration problem is not very well understood and very few functions or classes of functions are known to be recoverable from their subgradients in section 4 we show that if the basic constraint qualification holds at r then the composition of a closed i e lowersemicontinuous proper convex function with a twice continuously differentiable mapping is determined up to an additive constant by its generalized subgradients actually in this case all above mentioned subgradients are the same beside the obvious theoretical interest of this integration problem it is our hope or perhaps our long term goal that once this problem is better understood we can then tackle the question of uniqueness of solutions to generalized differential equations involving subgradients in place of partial derivatives an example of such an equation that well deserves study is the extended hamilton jacobi equation used in optimal control see clarke 2 let us also mention that a problem similar to the integration problem is the one of determining the set valued mappings that are in fact subgradient set valued mappings uniqueness is not mandatory for a contribution to this problem see janin 8 before we look at some of the known cases where the function can be recovered from its subgradients let us look at some negative examples it is clear that not every function can be recovered up to an additive constant from its subgradients we only need to look at the following two functions 0 x50 0 x10 f x 1 x o g x 2 x o

clf based designs with robustness to dynamic input uncertainties abstract the problem of robust stabilization of nonlinear systems in the presence of input uncertainties is of great importance in practical implementation stabilizing control laws may not be robust to this type of uncertainty especially if cancellation of nonlinearities is used in the design by exploiting a connection between robustness and optimality domination redesign of the control lyapunov function clf based sontag s formula has been shown to possess robustness to static and dynamic input uncertainties in this paper we provide a sufficient condition for the domination redesign to apply this condition relies on properties of local homogeneous approximations of the system and of the clf we show that an inverse optimal control law may not exist when these conditions are violated and illustrate how these conditions may guide the choice of a clf which is suitable for domination redesign

gap metric based robustness analysis of nonlinear systems with full and partial feedback linearisation abstractthis paper uses gap metric analysis to derive robustness and performance margins for feedback linearising controllers distinct from previous robustness analysis it incorporates the case of output unstructured uncertainties and is shown to yield general stability conditions which can be applied to both stable and unstable plants it then expands on existing feedback linearising control schemes by introducing a more general robust feedback linearising control design which classifies the system nonlinearity into stable and unstable components and cancels only the unstable plant nonlinearities this is done in order to preserve the stabilising action of the inherently stabilising nonlinearities robustness and performance margins are derived for this control scheme and are expressed in terms of bounds on the plant nonlinearities and the accuracy of the cancellation of the unstable plant nonlinearity by the controller case studies then confirm reduced conservatism compared with standard methods

stability in the determination of a time dependent coefficient for wave equations from partial data let be a c2 bounded domain of rn n 2 and fix q 0 t with t 0 we consider the stability in the inverse problem of determining a time dependent coefficient of order zero q appearing in a dirichlet initial boundary value problem for a wave equation t2u xu q t x u 0 in q from partial observations on q the observation is given by a boundary operator associated to the wave equation using suitable geometric optics solutions and carleman estimates we prove a stability estimate in the determination of q from the boundary operator

stability estimates for the hyperbolic dirichlet to neumann map in anisotropic media abstract in this paper we prove stability estimates for the hyperbolic dirichlet to neumann map associated to the wave equation in anisotropic media

completeness of products of solutions and some inverse problems for pde abstract in this paper we give conditions which guarantee that products of solutions of partial differential equations pu au 0 are complete in l 2 here p is a linear partial differential operator with constant coefficients and a is a function in l we check these conditions for elliptic parabolic and hyperbolic equations of second order and give applications to inverse problems for related equations where a is to be found

achieving nonvanishing stability regions with high gain cheap control using h techniques the second order case abstract this paper demonstrates how to use an asymptotically h optimal controller to stabilize a second order system subject to unknown disturbances such that the stability region does not vanish as the feedback gains increase the high gain feedback arises when one attempts to achieve the lowest achievable limit of the disturbance attenuation under the h design this type of gain increase can cause the stability region to vanish if the disturbance contains nonlinear terms the analysis using lyapunov techniques derives a sufficient condition on the design parameters to prevent the stability region from vanishing in addition to describing exact solutions for six different cases the paper provides simulations to illustrate the results

inverse optimal design of input to state stabilizing nonlinear controllers we show that input to state stabilizability as defined by sontag 1989 1995 is both necessary and sufficient for the solvability of a hamilton jacobi isaacs equation associated with a meaningful differential game problem similar to but more general than the nonlinear h sub spl infin problem the significance of the result stems from the fact that constructive solutions to the input to state stabilization problem are available presented in the paper and that as shown here inverse optimal controllers possess margins on input to state stability against a certain class of input unmodeled dynamics rather than completion of squares the main tools in our analysis are legendre fenchel transformations and the general form of young s inequality

a constructive proof of the existence of a utility in revealed preference theory within the context of the standard model of rationality within economic modelling we show the existence of a utility function that rationalises a demand correspondence hence completely characterizes the associated preference structure by taking a dense demand sample this resolves the problem of revealed preferences under some very mild assumptions on the demand correspondence which are closely related to a number of established axioms in preference theory the proof establishes the existence of a limit of a sequences of indirect utilities that rationalise finite data sets where the sample size increases to infinity this limiting utility provides a rationalisation of the demand relation up to a rescaling this limiting indirect utility is unique and continuous on a set of full measure

the differentiability of real functions on normed linear space using generalized subgradients abstract the modification of the clarke generalized subdifferential due to michel and penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space the g teaux differentiability of any real function can be deduced from the g teaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative for any distance function on a space with uniformly g teaux differentiable norm the clarke and michel penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for g teaux differentiability however on a banach space with rotund dual the frechet differentiability of a distance function implies that it is a convex function a mean value theorem for the modified generalized subdifferential has implications for g teaux differentiability

integration of multivalued operators and cyclic submonotonicity we introduce a notion of cyclic submonotonicity for multivalued operators from a banach space x to its dual we show that if the clarke subdifferential of a locally lipschitz function is strictly submonotone on an open subset u of x then it is also maximal cyclically submonotone on u and conversely that every maximal cyclically submonotone operator on u is the clarke subdifferential of a locally lipschitz function which is unique up to a constant if u is connected in finite dimensions these functions are exactly the lower c 1 functions considered by spingarn and rockafellar

brief paper robust adaptive observer for nonlinear systems with unmodeled dynamics unmodeled dynamics exist in almost all applications of observers due to the impossibility of using exact and detailed models it is highly desired that the observers can dominate the effects of unmodeled dynamics independently to prevent the state estimations from diverging and to get the precise estimations based on adaptive nonlinear damping this paper presents a robust adaptive observer for multiple input multiple output nonlinear systems with unknown parameters uncertain nonlinearities disturbances and unmodeled dynamics the observer only has one adaptive parameter no matter how high the order of the system is and how many unknown parameters there are with the proposed observer neither estimating the unknown parameters or solving linear matrix inequalities is needed it is shown that the state estimation error is uniformly bounded and can be made arbitrarily small

technical results for the study of robustness of lagrange stability abstract we study the robustness of boundedness of solutions of nonlinear dynamical systems a sufficient coordinate free technical condition on the characterization of unmodelled effects is given this characterization with some analogy with the input to state stability of 16 has the potential to encompass most classical uncertainties in this context we establish lagrange stability results which look very much like a small gain theorem we illustrate the use of these technical results in the robust stabilization problem by a class of interconnected systems

essentially smooth lipschitz functions in this paper we address some of the most fundamental questions regarding the differentiability structure of locally lipschitz functions defined on separable banach spaces for example we examine the relationship between integrability d representability and strict differentiability in addition to this we show that on any separable banach space there is a significant family of locally lipschitz functions that are integrable d representable and possess desirable differentiability properties we also present some striking applications of our results to distance functions

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

adaptive fuzzy decentralized control for a class of strong interconnected nonlinear systems with unmodeled dynamics the state feedback decentralized stabilization problem is considered for interconnected nonlinear systems in the presence of unmodeled dynamics the functional relationship in affine form between the strong interconnected functions and error signals is established which makes backstepping based fuzzy control successfully generalized to strong interconnected nonlinear systems by combining adaptive control with both backstepping design and the approximation property of fuzzy systems an adaptive decentralized control algorithm is developed it is demonstrated by both theoretical analysis and simulation study that the proposed control strategy ensures semiglobally uniformly ultimately bounded of all signals within the closed loop systems

minimal usco and minimal cusco maps and compactness abstract we prove a nice generalization of the arzela ascoli theorem from continuous functions to minimal usco cusco maps into metric spaces let x be a locally compact space y d be a metric space k y be the space of all nonempty compact subsets of y and mu x y be the space of minimal usco maps from x to y the family e mu x y is compact in k y x equipped with the topology uc of uniform convergence on compact sets if and only if e is closed in mu x y uc pointwise bounded and densely equicontinuous the same result holds also for compact subsets of mc x y uc the space of minimal cusco maps from x to a banach space y

stability estimate for an inverse wave equation and a multidimensional borg levinson theorem abstract we consider the stability in an inverse problem of determining the potential q entering the wave equation t 2 u u q x u 0 in a bounded smooth domain of r d from boundary observations the observation is given by a hyperbolic dynamic dirichlet to neumann map associated to a wave equation we prove a log type stability estimate in determining q from a partial dirichlet to neumann map provided that q is a priori known in a neighbourhood of the boundary of the spatial domain and satisfies an additional condition next we use this result to establish a stability estimate related to the multidimensional borg levinson theorem

the maximal normal operator space and integration of subdifferentials on nonconvex functions on definit des operateurs normaux maximaux et des fonctions primales sur des ensembles normaux on trouve que la normalite maximale est conservee dans plusieurs operations et que les operateurs normaux maximaux ont de bonnes proprietes de fermeture et de limitation

lipschitz functions with prescribed derivatives and subderivatives in general it is difficult to construct lipschitz functions which are not directly built up from either convex or distance functions one impediment to such constructions is that outside of the real line it is difficult to find anti derivatives the main result of this paper provides under suitable circumstances a technique for constructing such anti derivatives more precisely we show that if f 1 f 2 dots f n are continuously differentiable real valued locally lipschitz functions defined on a non empty open subset a of a separable banach space x then there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co nabla f 1 x nabla f 2 x dots nabla f n x this same construction also shows that for any finite family t 1 t 2 dots t n of maximal cyclically monotone mappings from a into non empty subsets of x there exists a real valued locally lipschitz function g defined on a such that at each point x in a the clarke subgradient of g at x equals co t 1 x t 2 x dots t n x moreover we show that g is convex if and only if t 1 t 2 cdots t n

a continuity property related to kuratowski s index of non compactness its relevance to the drop property and its implications for differentiability theory abstract we define and study the properties of upper semi continuity a new continuity property for set valued mappings from a topological space into subsets of a metric space expressed in terms of kuratowski s index of non compactness this upper semi continuity is related closely to the usual upper semi continuity but significantly upper semi continuous minimal weak cuscos from a baire space into subsets of the dual of any banach space are generically single valued kuratowski s index of non compactness has been used to study the drop property and upper semi continuity is dual to property studied there uniform upper semi continuity of the duality mapping is dual to nearly uniform rotundity properties importantly upper semi continuity has application in differentiability theory providing another characterisation for asplund spaces an examination of an associated concept denting point for sets yields still further advances concerning the differentiability of convex functions on a large class of banach spaces

trigonometric rbf neural robust controller design for a class of nonlinear system with linear input unmodeled dynamics considered both the situation with unknown control function matrices and the situation with linear unmodeled input dynamics adaptive neural robust controller was designed by using adaptive backstepping method for a class of multi input to multi output nonlinear systems which could be turned to standard block control type it was proved by constructing lyapunov function step by step that all signals of the system are bounded and exponentially converge to the neighborhood of the origin globally and by adopting the trigonometric function as basis function the input need not be force to between 1 and 1 and there is no need to choose the centre of basis function which reduced the difficulty of doing simulation and made the neural net work more practical and the variable structure control is adopt to eliminate the error of approximation also the method of differential reconstruction of neural network is used to increase the damp of neural network and it makes the system more stable finally simulation study is given to demonstrate that the proposed method is effective and the known information of system was made use of as maximally as possible by introducing the pid control

a geometrical insight on pseudoconvexity and pseudomonotonicity generalised convexity is revisited from a geometric point of view a substitute to the subdifferential is proposed then generalised monotonicity is considered a representation of generalised monotone maps allows us to obtain a symmetry between maps and their inverses finally maximality of generalised monotone maps is analysed

differentiability of lipschitz functions on banach spaces abstract the main result of this note says that if the norm of a banach space e is differentiable frechet gateaux or in some intermediate sense away from the origin then every locally lipschitz function defined on an open subset g of e is differentiable in the same sense at every point of some subset of g dense in g as the main corollary we prove that every lipschitz function on an asplund space is frechet differentiable on a dense set moreover in all these results the mean value theorem for the points of differentiability which is equivalent to the possibility of computing the clarke subdifferential from the points of differentiability is shown to hold

a chain rule for essentially smooth lipschitz functions in this paper we introduce a new class of real valued locally lipschitz functions that are similar in nature and definition to valadier s saine functions which we call arcwise essentially smooth and we show that if g rm rightarrow r is arcwise essentially smooth on rm and each function f j r n rightarrow r 1 leq j leq m is strictly differentiable almost everywhere in rn then g circ f is strictly differentiable almost everywhere in rn where f equiv f 1 f 2 ldots f m we also show that all the semismooth and all the pseudoregular functions are arcwise essentially smooth thus we provide a large and robust lattice algebra of lipschitz functions whose generalized derivatives are well behaved

robust input output feedback linearization to make input output feedback linearization a practical and systematic design methodology for single input nonlinear systems two problems need to be addressed one is to handle systematically the difficulties associated with the internal dynamics or zero dynamics when the relative degree is less than the system order the other is to account for the effect of model uncertainties in the successive differentiations of the output of interest while the first problem has recently received considerable attention the second has been largely unexplored this paper represents a preliminary study of a systematic methodology to account robustly for parametric uncertainties in the original system model the approach is based on combining sliding control ideas with the recursive construction of a closed loop lyapunov function and is illustrated with a simple example

integrability of subdifferentials of certain bivariate functions abstract we study integrability properties of bivariate functions defined on the product of banach spaces

an inverse problem of finding two time dependent coefficients in second order hyperbolic equations from dirichlet to neumann map abstract in the present paper we consider a non self adjoint hyperbolic operator with a vector field and an electric potential that depend not only on the space variable but also on the time variable more precisely we attempt to stably and simultaneously retrieve the real valued velocity field and the real valued potential from the knowledge of neumann measurements performed on the whole boundary of the domain we establish in dimension n greater than two stability estimates for the problem under consideration thereafter by enlarging the set of data we show that the unknown terms can be stably retrieved in larger regions including the whole domain the proof of the main results is mainly based on the reduction of the inverse problem under investigation to an equivalent and classic inverse problem for an electro magnetic wave equation

design of softer robust nonlinear control laws abstract we show that the lyapunov function used in backstepping feedback designs for uncertain nonlinear systems leads to unnecessarily hard control laws having undesirable high gain properties we present a new lyapunov function and use it to design softer control laws which exhibit the high gain properties to a much lesser extent we show that the soft designs eliminate the chattering exhibited by the hard designs and achieve the same or better performance with less control effort

stability estimate for the hyperbolic inverse boundary value problem by local dirichlet to neumann map abstract in this paper we consider the stability of the inverse problem of determining a function q x in a wave equation t 2 u u q x u 0 in a bounded smooth domain in r n from boundary observations this information is enclosed in the hyperbolic dynamic dirichlet to neumann map associated to the solutions to the wave equation we prove in the case of n 2 that q x is uniquely determined by the range restricted to a subboundary of the dirichlet to neumann map whose stability is a type of double logarithm

a new class of stabilizing controllers for uncertain dynamical systems this paper is concerned with the problem of designing a stabilizing controller for a class of uncertain dynamical systems the vector of uncertain parameters q cdot is time varying and its values q t lie within a prespecified bounding set q in r p furthermore no statistical description of q cdot is assumed and the controller is shown to render the closed loop system practically stable in a so called guaranteed sense that is the desired stability properties are assured no matter what admissible uncertainty q cdot is realized within the perspective of previous research in this area this paper contains one salient feature the class of stabilizing controllers which we characterize is shown to include linear controllers when the nominal system happens to be linear and time invariant in contrast in much of the previous literature see for example 1 2 7 and 9 a linear system is stabilized via nonlinear control another feature of this paper is the fact that the

robust control of nonlinear systems with input unmodeled dynamics in the rich repertoire of design methodologies for uncertain nonlinear systems the problem of unmodeled dynamics has thus far received little attention the authors initiate the investigation of nonlinear systems with input unmodeled dynamics first the authors show that even in their simplest forms they can result in dramatic shrinking of the region of attraction and in finite escape time in this paper the authors introduce a dynamic nonlinear damping design which guarantees global boundedness in the presence of input unmodeled dynamics with this design the authors achieve global asymptotic stability for strict feedback systems and output feedback systems

property c and an inverse problem for a hyperbolic equation let utt u q x t u 0 in d 0 t where d r3 is a bounded domain with a smooth boundary d t d d diam d q x t c 0 t l d suppose that for every u d f x t c1 d 0 t the value un d h s t is known where n is the outer normal to d u solves and and satisfies the initial conditions u ut 0 at t 0 then q x t is uniquely determined by the data f h f c1 d 0 t in the subset s of d 0 t consisting of the lines which make 45 with the t axis and which meet the planes t 0 and t t outside d 0 t provided that q x t is known outside s here d is the closure of d

adaptive sliding mode control of multi input nonlinear systems with perturbations to achieve asymptotical stability a design of adaptive sliding surfaces for a class of multi input mi nonlinear systems with matched and mismatched perturbations is proposed in this note to solve regulation problems based on the lyapunov stability theorem and backstepping technique by using some adaptive gains designed in the sliding surface function the virtual input and the controllers not only the mismatched perturbations are automatically overcome during the sliding mode but also the property of asymptotical stability of controlled systems is achieved at the same time furthermore the knowledge of the upper bound of partial perturbations is not required controlling a two link robot manipulator is given to demonstrate the feasibility of the proposed control scheme

a derivative for complex lipschitz maps with generalised cauchy riemann equations a r t i c l e i n f o a b s t r a c t we introduce the lipschitz derivative or the l derivative of a locally lipschitz complex map it is a scott continuous compact and convex set valued map that extends the classical derivative to the bigger class of locally lipschitz maps and allows an extension of the fundamental theorem of calculus and a new generalisation of cauchy riemann equations to these maps which form a continuous scott domain we show that a complex lipschitz map is analytic in an open set if and only if its l derivative is a singleton at all points in the open set the calculus of the l derivative for sum product and composition of maps is derived the notion of contour integration is extended to scott continuous non empty compact convex valued functions on the complex plane and by using the l derivative the fundamental theorem of contour integration is extended to these functions 2014 the author published by elsevier b v this is an open access article under the cc by license http creativecommons org licenses by 3 0

determination of coefficients for a dissipative wave equation via boundary measurements in this paper we consider the inverse problem of recovering the viscosity coefficient in a dissipative wave equation via boundary measurements we obtain stability estimates by considering all possible measurements implemented on the boundary we also prove that the viscosity coefficient is uniquely determined by a finite number of measurements on the boundary provided that it belongs to a given finite dimensional vector space

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

quantitative stability of variational systems i the epigraphical distance this paper proposes a global measure for the distance between the elements of a variational system parametrized families of optimization problems

stable determination outside a cloaking region of two time dependent coefficients in an hyperbolic equation from dirichlet to neumann map abstract in this paper we treat the inverse problem of determining two time dependent coefficients appearing in a dissipative wave equation from measured neumann boundary observations we establish in dimension n 2 stability estimates with respect to the dirichlet to neumann map of these coefficients provided that they are known outside cloaking regions moreover we prove that they can be stably recovered in larger subsets of the domain by enlarging the set of data