relating fisher information to order parameters we study phase transitions and relevant order parameters via statistical estimation theory using the fisher information matrix the assumptions that we make limit our analysis to order parameters representable as a negative derivative of thermodynamic potential over some thermodynamic variable nevertheless the resulting representation is sufficiently general and explicitly relates elements of the fisher information matrix to the rate of change in the corresponding order parameters the obtained relationships allow us to identify in particular second order phase transitions via divergences of individual elements of the fisher information matrix a computational study of random boolean networks supports the derived relationships illustrating that fisher information of the magnetization bias that is activity level is peaked in finite size networks at the critical points and the maxima increase with the network size the framework presented here reveals the basic thermodynamic reasons behind similar empirical observations reported previously the study highlights the generality of fisher information as a measure that can be applied to a broad range of systems particularly those where the determination of order parameters is cumbersome

optimal control of evolution inclusions abstract we develop the method of discrete approximations for studying optimal control problems of the bolza type for evolution systems governed by generally nonconvex differential inclusions in infinite dimensional spaces the results obtained involve well posedness stability of discrete approximations necessary optimality conditions for general discrete time systems and necessary optimality conditions for continuous time inclusions established in the extended euler lagrange form our approach is based on the generalized differential constructions of variational analysis introduced earlier by the author

regularity of the hamiltonian along optimal trajectories this paper concerns state constrained optimal control problems in which the dynamic constraint takes the form of a differential inclusion if the differential inclusion does not depend on time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is independent of time if the differential inclusion is lipschitz continuous then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory is lipschitz continuous these two well known results are examples of the following principle the hamiltonian evaluated along the optimal state trajectory and the co state trajectory inherits the regularity properties of the differential inclusion regarding its time dependence we show that this principle also applies to another kind of regularity if the differential inclusion has bounded variation w r t time then the hamiltonian evaluated along the optimal state trajectory and the co state trajectory has bounded variation two application

nonconvex duality in optimal control we derive a representation of the minimum cost of an optimal control problem in terms of the upper envelope of generalized semisolutions of the hamilton jacobi equation

controllability and strong controllability of differential inclusions abstract in this paper we prove sufficient conditions for controllability and strong controllability in terms of the mordukhovich subdifferential for two classes of differential inclusions the first one is the class of sub lipschitz multivalued functions introduced by loewen rockafellar 1994 10 the second one introduced recently by clarke 2005 18 is the class of multivalued functions which are pseudo lipschitz and satisfy the so called tempered growth condition to do this we establish an error bound result in terms of the mordukhovich subdifferential outside asplund spaces

insights from a simple expression for linear fisher information in a recurrently connected population of spiking neurons a simple expression for a lower bound of fisher information is derived for a network of recurrently connected spiking neurons that have been driven to a noise perturbed steady state we call this lower bound linear fisher information as it corresponds to the fisher information that can be recovered by a locally optimal linear estimator unlike recent similar calculations the approach used here includes the effects of nonlinear gain functions and correlated input noise and yields a surprisingly simple and intuitive expression that offers substantial insight into the sources of information degradation across successive layers of a neural network here this expression is used to 1 compute the optimal i e information maximizing firing rate of a neuron 2 demonstrate why sharpening tuning curves by either thresholding or the action of recurrent connectivity is generally a bad idea 3 show how a single cortical expansion is sufficient to instantiate a redundant population code that can propagate across multiple cortical layers with minimal information loss and 4 show that optimal recurrent connectivity strongly depends on the covariance structure of the inputs to the network

necessary conditions for optimal control problems with state constraints necessary conditions of optimality are derived for optimal control problems with pathwise state constraints in which the dynamic constraint is modelled as a differential inclusion the novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid an euler lagrange type condition is obtained for problems where the multifunction associated with the dynamic constraint has values possibly unbounded nonconvex sets and satisfies a mild one sided lipschitz continuity hypothesis we recover as a special case the sharpest available necessary conditions for state constraint free problems proved in a recent paper by ioffe for problems where the multifunction is convex valued it is shown that the necessary conditions are still valid when the one sided lipschitz hypothesis is replaced by a milder local hypothesis a recent dualization theorem permits us to infer a strengthened form of the hamiltonian inclusion from the euler lagrange condition the necessary conditions for state constrained problems with convex valued multifunctions are derived under hypotheses on the dynamics which are significantly weaker than those invoked by loewen and rockafellar to achieve related necessary conditions for state constrained problems and improve on available results in certain respects even when specialized to the state constraint free case proofs make use of recent decoupling ideas of the authors which reduce the optimization problem to one to which pontryagin s maximum principle is applicable and a refined penalization technique to deal with the dynamic constraint

the hamiltonian inclusion for nonconvex velocity sets since clarke s 1973 proof of the hamiltonian inclusion for optimal control problems with convex velocity sets there has been speculation and more recently speculation relating to a stronger partially convexified version of the hamiltonian inclusion as to whether these necessary conditions are valid in the absence of the convexity hypothesis the issue was in part resolved by clarke himself when in 2005 he showed that l infty local minimizers satisfy the hamiltonian inclusion in this paper it is shown by counterexample that the hamiltonian inclusion and so also the stronger partially convexified hamiltonian inclusion are not in general valid for nonconvex velocity sets when the local minimizer in question is merely a w 1 1 local minimizer not an l infty local minimizer the counterexample demonstrates that the need to consider l infty local minimizers not w 1 1 local minimizers in the proof of the hamiltonian inclusion for nonconvex velocity sets is fundamental not j

the synthesis of universal feedback pursuit strategies in differential games we show how any generalized supersolution of the hamilton jacobi equation can be used to construct a feedback pursuit strategy which guarantees to any given tolerance a capture time not exceeding the solution s value if the supersolution is the value function then a near optimal pursuit strategy is obtained in this way an important feature of the construction is its universal nature i e the fact that the feedback law is uniformly effective on compact sets of initial conditions this implies in particular that the feedback construction is one that exploits nonoptimal behavior on the part of the evader

necessary and sufficient conditions for discrete and differential inclusions of elliptic type abstract this paper deals for the first time with the dirichlet problem for discrete p d discrete approximation problem on a uniform grid and differential p c inclusions of elliptic type in the form of euler lagrange inclusion necessary and sufficient conditions for optimality are derived for the problems under consideration on the basis of new concepts of locally adjoint mappings the results obtained are generalized to the multidimensional case with a second order elliptic operator

real time computation at the edge of chaos in recurrent neural networks depending on the connectivity recurrent networks of simple computational units can show very different types of dynamics ranging from totally ordered to chaotic we analyze how the type of dynamics ordered or chaotic exhibited by randomly connected networks of threshold gates driven by a time varying input signal depends on the parameters describing the distribution of the connectivity matrix in particular we calculate the critical boundary in parameter space where the transition from ordered to chaotic dynamics takes place employing a recently developed framework for analyzing real time computations we show that only near the critical boundary can such networks perform complex computations on time series hence this result strongly supports conjectures that dynamical systems that are capable of doing complex computational tasks should operate near the edge of chaos that is the transition from ordered to chaotic dynamics

second order necessary conditions for differential difference inclusion problems abstract we study an optimal control problem given by differential difference inclusions with end point constraints an approach concerning second order optimality conditions is proposed

various lipschitz like properties for functions and sets i directional derivative and tangential characterizations in this work we introduce for extended real valued functions defined on a banach space x the concept of k directionally lipschitzian behavior where k is a bounded subset of x for different types of sets k e g zero singleton or compact the k directionally lipschitzian behavior recovers well known concepts in variational analysis locally lipschitzian directionally lipschitzian or compactly epi lipschitzian properties respectively characterizations of this notion are provided in terms of the lower dini subderivatives we also adapt the concept for sets and establish characterizations of the mentioned behavior in terms of the bouligand tangent cones the special case of convex functions and sets is also studied

convergence of discrete approximations to optimization problems of neutral functional differential inclusions abstract this paper deals with the convergence of discrete approximations to the optimization problem p for a neutral functional differential inclusion subject to general endpoint constraints in the first part of the paper discrete approximations to the neutral functional differential inclusion are constructed using euler finite difference methods and the convergence of discrete approximations is proved in the second part of the paper a family of discrete optimization problems p n to p is provided and the strong convergence of optimal solutions for p n to the optimal solution of p is discussed

input anticipating critical reservoirs show power law forgetting of unexpected input events usually reservoir computing shows an exponential memory decay this letter investigates under which circumstances echo state networks can show a power law forgetting that means traces of earlier events can be found in the reservoir for very long time spans such a setting requires critical connectivity exactly at the limit of what is permissible according to the echo state condition however for general matrices the limit cannot be determined exactly from theory in addition the behavior of the network is strongly influenced by the input flow results are presented that use certain types of restricted recurrent connectivity and anticipation learning with regard to the input where power law forgetting can indeed be achieved

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

necessary conditions and non existence results for autonomous nonconvex variational problems abstract classical one dimensional autonomous lagrange problems are considered in absence of any smoothness convexity or coercivity condition on the energy density we prove a dubois reymond type necessary condition expressed as a differential inclusion involving the subdifferential of convex analysis as a consequence a non existence result is obtained

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

mean value inequalities in hilbert space we establish a new mean value theorem applicable to lower semicontinuous functions on hilbert space a novel feature of the result is its multidirectionality it compares the value of a function at a point to its values on a set we then discuss some refinements and consequences of the theorem including applications to calculus flow invariance and generalized solutions to partial differential equations rtsumt on etablit un nouveau theoreme de la valeur moyenne qui s applique aux fonctions semicontinues inferieurement sur un espace de hilbert on deduit plusieurs consequences du resultat portant par exemple sur les fonctions monotones et sur les solutions generalisees des equations aux derivees partielles

analysis and design of echo state networks the design of echo state network esn parameters relies on the selection of the maximum eigenvalue of the linearized system around zero spectral radius however this procedure does not quantify in a systematic manner the performance of the esn in terms of approximation error this article presents a functional space approximation framework to better understand the operation of esns and proposes an information theoretic metric the average entropy of echo states to assess the richness of the esn dynamics furthermore it provides an interpretation of the esn dynamics rooted in system theory as families of coupled linearized systems whose poles move according to the input signal dynamics with this interpretation a design methodology for functional approximation is put forward where esns are designed with uniform pole distributions covering the frequency spectrum to abide by the richness metric irrespective of the spectral radius a single bias parameter at the esn input adapted with the modeling error configures the esn spectral radius to the input output joint space function approximation examples compare the proposed design methodology versus the conventional design

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

discrete approximations and fixed set iterations in banach spaces we study autonomous differential inclusions with right hand sides satisfying a one sided lipschitz osl condition in banach spaces with uniformly convex duals we first show that the solution set is closed and obtain estimates for euler type discrete approximations we then use these results to derive an analogue of the exponential formula for the reachable set as well as results regarding the existence and approximation of a strongly invariant attractor in the case of a negative osl constant as a by product conditions for controllability of the reverse time system are obtained

optimal control of delay differential inclusions with functional endpoint constraints in infinite dimensions abstract this paper is devoted to the study of a general class of optimal control problems described by delay differential inclusions with equality and inequality endpoint constraints and multivalued initial conditions we use the method of discrete approximations and advanced tools of variational analysis and generalized differentiation in infinite dimensions to derive necessary optimality conditions in the extended euler lagrange form this method is fully realized for the delay differential systems under consideration

quantization dimension for condensation systems let be the attracting measure of a condensation system associated with a self similar measure we determine the upper and lower quantization dimension of under the strong separation condition

a variational principle for fractal dimensions abstract a new definition of the dimension of probability measures is introduced it is related with the fractal dimension of sets by a variational principle this principle is applied in the theory of iterated function systems

determination of the edge of criticality in echo state networks through fisher information maximization it is a widely accepted fact that the computational capability of recurrent neural networks rnns is maximized on the so called edge of criticality once the network operates in this configuration it performs efficiently on a specific application both in terms of 1 low prediction error and 2 high short term memory capacity since the behavior of recurrent networks is strongly influenced by the particular input signal driving the dynamics a universal application independent method for determining the edge of criticality is still missing in this paper we aim at addressing this issue by proposing a theoretically motivated unsupervised method based on fisher information for determining the edge of criticality in rnns it is proved that fisher information is maximized for finite size systems operating in such critical regions however fisher information is notoriously difficult to compute and requires the analytic form of the probability density function ruling the system behavior this paper takes advantage of a recently developed nonparametric estimator of the fisher information matrix and provides a method to determine the critical region of echo state networks esns a particular class of recurrent networks the considered control parameters which indirectly affect the esn performance are explored to identify those configurations lying on the edge of criticality and as such maximizing fisher information and computational performance experimental results on benchmarks and real world data demonstrate the effectiveness of the proposed method

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

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

optimal control of nonautonomous functional differential inclusions of neutral type abstract this paper deals with optimal control problems of the bolza type for functional differential systems governed by nonautonomous constrained neutral inclusions given in the hale form we develop the method of discrete approximations to derive new necessary optimality conditions for such problems in extended euler lagrange and hamiltonian forms the results obtained are expressed in terms of robust generalized differential constructions of variational analysis introduced earlier by the first author

quantization dimension for condensation systems ii the geometric mean error abstract let f i 1 n be a family of similitudes on r 1 satisfying the strong separation condition and the self similar measure associated with f i 1 n and a probability vector t 1 t n let be the attracting measure of a condensation system associated with f i 1 n and a probability vector p 0 p 1 p n we establish a relationship between the quantization dimension of and its mass distribution on cylinder sets

fisher information and criticality in the kuramoto model of nonidentical oscillators we use the fisher information to provide a lens on the transition to synchronization of the kuramoto model of nonidentical frequencies on a variety of undirected graphs we numerically solve the equations of motion for a n 400 complete graph and n 1000 small world scale free uniform random and random regular graphs for large but finite graphs of small average diameter the fisher information f as a function of coupling shows a peak closely coinciding with the critical point as determined by kuramoto s order parameter or synchronization measure r however for graphs of larger average diameter the position of the peak in f differs from the critical point determined by estimates of r on the one hand this is a finite size effect even at n 1000 however we show across a range of topologies that the fisher information peak points to a transition for smaller graphs that indicates structural changes in the numbers of locally phase synchronized clusters often directly from metastable to stable frequency synchronization solving explicitly for a two cluster ansatz subject to gaussian noise shows that the fisher infomation peaks at such a transition we discuss the implications for fisher information as an indicator for edge of chaos phenomena in finite coupled oscillator systems

quantization dimension of probability measures supported on cantor like sets let be an arbitrary probability measure supported on a cantor like set e with bounded distortion we establish a relationship between the quantization dimension of and its mass distribution on cylinder sets under a hereditary condition as an application we determine the quantization dimensions of probability measures supported on e which have explicit mass distributions on cylinder sets provided that the hereditary condition is satisfied

necessary conditions for multiobjective optimal control problems with free end time necessary conditions of optimality are derived for multiobjective optimal control problems with free end time in which the dynamics constraint is modeled as a nonconvex differential inclusion the obtained results cover some previous results on necessary conditions for multiojective and single objective optimal control problems

dualization of the euler and hamiltonian inclusions optimal control emerged as a distinct eld of research in the late 1950s with the derivation of the pontryagin maximum principle a fundamental set of necessary conditions on minimizing arcs which takes account of di erential constraints the pontryagin maximum principle directly generates the euler equation and other classical necessary conditions from the calculus of variations when specialised to problems without dynamic constraints however deeper connections between necessary conditions in optimal control and the calculus of variations were only subsequently revealed when clarke developed new analytic machinery now referred to as nonsmooth analysis a term introduced by clarke himself for extending classical necessary conditions to apply to the generalized

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

support vector echo state machine for chaotic time series prediction a novel chaotic time series prediction method based on support vector machines svms and echo state mechanisms is proposed the basic idea is replacing kernel trick with reservoir trick in dealing with nonlinearity that is performing linear support vector regression svr in the high dimension reservoir state space and the solution benefits from the advantages from structural risk minimization principle and we call it support vector echo state machines svesms svesms belong to a special kind of recurrent neural networks rnns with convex objective function and their solution is global optimal and unique svesms are especially efficient in dealing with real life nonlinear time series and its generalization ability and robustness are obtained by regularization operator and robust loss function the method is tested on the benchmark prediction problem of mackey glass time series and applied to some real life time series such as monthly sunspots time series and runoff time series of the yellow river and the prediction results are promising

statistical mechanics of letters in words we consider words as a network of interacting letters and approximate the probability distribution of states taken on by this network despite the intuition that the rules of english spelling are highly combinatorial and arbitrary we find that maximum entropy models consistent with pairwise correlations among letters provide a surprisingly good approximation to the full statistics of words capturing ensuremath sim 92 mathrm of the multi information in four letter words and even discovering words that were not represented in the data these maximum entropy models incorporate letter interactions through a set of pairwise potentials and thus define an energy landscape on the space of possible words guided by the large letter redundancy we seek a lower dimensional encoding of the letter distribution and show that distinctions between local minima in the landscape account for ensuremath sim 68 mathrm of the four letter entropy we suggest that these states provide an effective vocabulary which is matched to the frequency of word use and much smaller than the full lexicon

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

mean field theory of echo state networks dynamical systems driven by strong external signals are ubiquitous in nature and engineering here we study echo state networks networks of a large number of randomly connected nodes which represent a simple model of a neural network and have important applications in machine learning we develop a mean field theory of echo state networks the dynamics of the network is captured by the evolution law similar to a logistic map for a single collective variable when the network is driven by many independent external signals this collective variable reaches a steady state but when the network is driven by a single external signal the collective variable is non stationary but can be characterized by its time averaged distribution the predictions of the mean field theory including the value of the largest lyapunov exponent are compared with the numerical integration of the equations of motion

proximal analysis and the minimal time function under general hypotheses on the target set s and the dynamics of the system we show that the minimal time function ts cdot is a proximal solution to the hamilton jacobi equation uniqueness results are obtained with two different kinds of boundary conditions a new propagation result is proven and as an application we give necessary and sufficient conditions for ts cdot to be lipschitz continuous near s a petrov type modulus condition is also shown to be sufficient for continuity of ts cdot near s

locally adjoint mappings and optimization of the first boundary value problem for hyperbolic type discrete and differential inclusions abstract the present paper deals with discrete approximation on a uniform grid of the first boundary value problem p c for differential inclusions of hyperbolic type in the form of euler lagrange inclusions necessary and sufficient conditions for optimality are derived for the discrete p d and continuous p c problems on the basis of new concepts of locally adjoint mappings the results obtained are generalized to the multidimensional case with a second order elliptic operator

new approximation method in the proof of the maximum principle for nonsmooth optimal control problems with state constraints abstract traditional proofs of the pontryagin maximum principle pmp require the continuous differentiability of the dynamics with respect to the state variable on a neighborhood of the minimizing state trajectory when arbitrary values of control variable are inserted into the dynamic equations sussmann has drawn attention to the fact that the pmp remains valid when the dynamics are differentiable with respect to the state variable merely when the minimizing control is inserted into the dynamic equations this weakening of earlier hypotheses has been referred to as the lojasiewicz refinement arutyunov and vinter showed that these extensions of early versions of the pmp can be simply proved by finite dimensional approximations application of a lagrange multiplier rule in finite dimensions and passage to the limit this paper generalizes the finite dimensional approximation technique to a problem with state constraints where the use of needle variations of the optimal control had not been successful moreover the cost function and endpoint constraints are not assumed to be differentiable but merely locally lipschitz continuous the maximum principle is expressed in terms of michel penot subdifferential

stimulus dependent suppression of chaos in recurrent neural networks neuronal activity arises from an interaction between ongoing firing generated spontaneously by neural circuits and responses driven by external stimuli using mean field analysis we ask how a neural network that intrinsically generates chaotic patterns of activity can remain sensitive to extrinsic input we find that inputs not only drive network responses but they also actively suppress ongoing activity ultimately leading to a phase transition in which chaos is completely eliminated the critical input intensity at the phase transition is a nonmonotonic function of stimulus frequency revealing a resonant frequency at which the input is most effective at suppressing chaos even though the power spectrum of the spontaneous activity peaks at zero and falls exponentially a prediction of our analysis is that the variance of neural responses should be most strongly suppressed at frequencies matching the range over which many sensory systems operate

necessary conditions in optimal control a new approach we present necessary conditions of optimality for a general control problem formulated in terms of a differential inclusion these conditions unify and substantially extend previous results in the literature and also incorporate a stratified feature of a novel nature when specialized to the calculus of variations the results yield necessary conditions and regularity theorems that go significantly beyond the previous standard they also give rise to new and stronger maximum principles of pontryagin type the principal focus here is the development of structural criteria on the problem which guarantee a priori that the necessary conditions apply to any local minimizer

thermodynamics and signatures of criticality in a network of neurons the activity of a neural network is defined by patterns of spiking and silence from the individual neurons because spikes are relatively sparse patterns of activity with increasing numbers of spikes are less probable but with more spikes the number of possible patterns increases this tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics we construct this relationship for populations of up to n 160 neurons in a small patch of the vertebrate retina using a combination of direct and model based analyses of experiments on the response of this network to naturalistic movies we see signs of a thermodynamic limit where the entropy per neuron approaches a smooth function of the energy per neuron as n increases the form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point we suggest further tests of criticality and give a brief discussion of its functional significance

on the occurrence of intermediate local minimizers that are not strong local minimizers abstract recently it has been shown by mordukhovich that certain necessary conditions of optimality in control theory are satisfied at intermediate local minimizers in addition to strong local minimizers it is of interest to know then whether the class of intermediate local minimizers differs from that of strong local minimizers we give examples illustrating that the two concepts of minimizers are indeed distinct and also distinguish intermediate minimizers of different rank we also extend the class of necessary conditions which are known to be satisfied at an intermediate local minimizer

optimal control of semilinear unbounded differential inclusions abstract this paper concerns a general optimal control problem of the mayer type for semilinear unbounded differential inclusions with endpoint constraints in reflexive and separable banach spaces based on the method of discrete approximations and advanced generalized differential tools of variational analysis we establish necessary optimality conditions for discrete time control problems and then by passing to the limit derive intrinsic optimality conditions of the extended euler lagrange type for semilinear evolution inclusions

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

value function and necessary conditions in optimal control problems for differential difference inclusions this paper deals with the optimal control problems for differential difference inclusions subject to endpoint constraints we follow a twofold goal first we develop a method for estimating the generalized gradients of value function in the problems above second we use these estimates for obtaining necessary optimality conditions the results obtained are expressed in terms of clarke constructions for nonsmooth mappings and sets

discrete approximations to optimization of neutral functional differential inclusions the paper deals with the convergence of discrete approximations to optimization problems governed by neutral functional differential inclusions the discrete approximations through euler finite difference are constructed and the convergence of discrete approximations is proved

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

nonsmooth constrained optimization and multidirectional mean value inequalities we establish a general fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a banach space our basic tool is a constrained variational principle derived from the smooth variational principle of borwein and preiss specializing the fermat rule to the case when the convex set is a drop we obtain a multidirectional rolle type inequality from which in turn we deduce a multidirectional mean value inequality in the line of clarke and ledyaev we follow the abstract approach of our previous paper trans amer math soc 347 1995 pp 4147 4161 thus covering all standard situations met in applications while stressing the links between the results and the few key properties that are needed

a sampling method and approximation results for impulsive systems this paper studies an impulsive dynamical system that is in the form of a measure driven differential inclusion the employed solution concept depends upon a graph completion of the measure the first main result shows that a subsequence of discrete time trajectories graph converges to a solution and under lipschitz hypotheses that every solution can be obtained in this manner the second main result pursues a similar development with classical systems that approximate the given one

subdifferential analysis of differential inclusions via discretization abstract the framework of differential inclusions encompasses modern optimal control and the calculus of variations necessary optimality conditions in the literature identify potentially optimal paths but do not show how to perturb paths to optimality we first look at the corresponding discretized inclusions estimating the subdifferential dependence of the optimal value in terms of the endpoints of the feasible paths our approach is to first estimate the coderivative of the reachable map the discretized nonsmooth euler lagrange and transversality conditions follow as a corollary we obtain corresponding results for differential inclusions by passing discretized inclusions to the limit

links between directional derivatives through multidirectional mean value inequalities we prove in the general setting of lower semicontinuous functions on banach spaces the relation between the rockafellar directional derivative and the mixed lower limit of the lower dini derivatives as a byproduct we derive the famous inclusions of tangent cones of closed sets in banach spaces the results are established using as principal tool multidirectional mean value inequalities aussel et al siam j optim 9 3 690 706 1999

a proximal characterization of the reachable set we show that the graph of the reachable set of a control system given by a differential inclusion is uniquely characterized by a hamilton jacobi equation involving proximal normals

variational problems with nonconvex noncoercive highly discontinuous integrands characterization and existence of minimizers we consider the functional f v int a b f t v prime t dt in cal h p v in w 1 p v a 0 v b d p in 1 infty under only the assumption that the integrand is cal l otimes cal b n measurable we prove characterizations of strong and weak minimizers both in terms of the minimizers of the relaxed functional and by means of the euler lagrange inclusion as an application we provide necessary and sufficient conditions for the existence of the minimum expressed in terms of a limitation on the width of the slope d

on the validity of the euler lagrange equation the purpose of the present paper is to establish the validity of the euler lagrange equation for the solution x to the classical problem of the calculus of variations

existence of solutions to a global eikonal equation abstract we derive a geometric necessary and sufficient condition for the existence of solutions to a global eikonal equation we also study the existence of a minimal solution to this equation and its relation with the well known minimal time function

path dependent hamilton jacobi equations in infinite dimensions abstract we propose notions of minimax and viscosity solutions for a class of fully nonlinear path dependent pdes with nonlinear monotone and coercive operators on hilbert space our main result is well posedness existence uniqueness and stability for minimax solutions a particular novelty is a suitable combination of minimax and viscosity solution techniques in the proof of the comparison principle one of the main difficulties the lack of compactness in infinite dimensional hilbert spaces is circumvented by working with suitable compact subsets of our path space as an application our theory makes it possible to employ the dynamic programming approach to study optimal control problems for a fairly general class of delay evolution equations in the variational framework furthermore differential games associated to such evolution equations can be investigated following the krasovski subbotin approach similarly as in finite dimensions

minimax dirichlet boundary control problem with state constraints in this paper we study a minimax boundary control problem with pointwise state constraints for parabolic equations under unknown disturbances in the initial condition we establish necessary optimality conditions

an intermediate value theorem for the upper quantization dimension abstract let be a borel probability measure on r d with compact support and d r the upper quantization dimension of of order r we prove that for every t dim p dim b there exists a borel probability measure with such that d r dim b t in addition we give an example to show that the above intermediate value property may fail in the open interval dim p dim p thus we get a complete description of the dimension set d r r d 1