admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

a condition for finite blow up time for a volterra integral equation abstract the paper is devoted to the study of the equation u x x 0 x s 1 g u s ds 0 x 0 where g is continuous and nondecreasing some estimates of u and the interval of the existence of a nontrivial solution u are given

on optimal location of diffusion and related optimal control for null controllable heat equation abstract we consider optimal locations of heat diffusion and related optimal control to achieve null controllability for multi dimensional heat equations both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to zero in minimal time interval is actually the minimal energy of driving the system to zero in this minimal time interval and visa versa we formulate the optimal locations for time optimal control and norm optimal control into two types of shape optimization problems one is seeking the optimal domain of heat diffusion with a fixed interior actuator domain this can be considered as a domain perturbation problem in shape optimization another is to seek both the optimal locations of the optimal heat diffusion domain and the related optimal actuator domain the existences of these two types of shape optimization problems over some class of open sets in general r n space have been proved separately

actuator design for parabolic distributed parameter systems with the moment method in this paper we model and solve the problem of designing in an optimal way actuators for parabolic partial differential equations settled on a bounded open connected subset omega of r n we optimize not only the location but also the shape of actuators by finding what is the optimal distribution of actuators in omega over all possible such distributions of a given measure using the moment method we formulate a spectral optimal design problem which consists of maximizing a criterion corresponding to an average over random initial data of the largest l 2 energy of controllers since we choose the moment method to control the pde our study mainly covers one dimensional parabolic operators but we also provide several examples in higher dimensions we consider two types of controllers either internal controls modeled by characteristic functions or lumped controls that are tensorized functions in time and space under appropriate spectral assumptions we prove existence and uniqueness of an optimal actuator distribution and we provide a simple computation procedure numerical simulations illustrate our results

uniform controllability of semidiscrete approximations of parabolic control systems controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model in the present paper under the main assumptions that the discretized semigroup is uniformly analytic and that the control operator is mildly unbounded we prove that the semidiscrete approximation models are uniformly controllable moreover we provide a computationally efficient way to compute the approximation controls an example of application is implemented for the one and two dimensional heat equation with neumann boundary control

a generalized gronwall inequality and its application to a fractional differential equation this paper presents a generalized gronwall inequality with singularity using the inequality we study the dependence of the solution on the order and the initial condition of a fractional differential equation

long time behavior of a two phase optimal design for the heat equation we consider a two phase isotropic optimal design problem within the context of the transient heat equation the objective is to minimize the average of the dissipated thermal energy during a fixed time interval 0 t the time independent material properties are taken as design variables a full relaxation for this problem was established in a munch p pedregal and f periago j math pures appl 9 89 2008 pp 225 247 by using the homogenization method in this paper we study the asymptotic behavior as t goes to infinity of the solutions of the relaxed problem and prove that they converge to an optimal relaxed design of the corresponding two phase optimization problem for the stationary heat equation next we study necessary optimality conditions for the relaxed optimization problem under the transient heat equation and use those to characterize the microstructure of the optimal designs which appears in the form of a sequential laminate of rank at most n the spatial dimension an asymptotic analysis of the optimality conditions lets us prove that for t large enough the order of lamination is in fact of at most n 1 several numerical experiments in two dimensions complete our study

minimal time for the null controllability of parabolic systems the effect of the condensation index of complex sequences let a d a be a diagonalizable generator of a c0 semigroup of contractions on a complex hilbert space x b l c y y being some suitable extrapolation space of x and u l2 0 t c under some assumptions on the sequence of eigenvalues k k 1 c of a d a we prove the existence of a minimal time t0 0 depending on bernstein s condensation index of and on b such that y ay bu is null controllable at any time t t0 and not null controllable for t t0 as a consequence we solve controllability problems of various systems of coupled parabolic equations in particular new results on the boundary controllability of one dimensional parabolic systems are derived these seem to be difficult to achieve using other classical tools

sensor choice for minimum error variance estimation a kalman filter is optimal in that the variance of the error is minimized by the estimator it is shown here in an infinite dimensional context that the solution to an operator riccati equation minimizes the steady state error variance this extends a result previously known for lumped parameter systems to distributed parameter systems it is shown then that minimizing the trace of the riccati operator is a reasonable criterion for choosing sensor locations it is then shown that multiple inaccurate sensors that is those with large noise variance can provide as good an estimate as a single highly accurate but probably more expensive sensor optimal sensor location is then combined with estimator design a framework for calculation of the best sensor locations using approximations is established and sensor location as well as choice is investigated with three examples simulations indicate that the sensor locations do affect the quality of the estimation and that multiple low quality sensors can lead to better estimation than a single high quality sensor

newtonian potential and positive solutions of poisson equations abstract we study the properties of the well known newtonian potential operator including its domain codomain compactness relation with the poisson s equation and the spectral radii of the weight newtonian potential operators these new properties are then applied to study the existence of positive classical solutions of nonlinear poisson s equations in bounded open sets which are required neither to be connected nor smooth on their boundaries the connectedness and smoothness are common requirements for studying the existence of classical or weak solutions of linear or nonlinear elliptic boundary value problems in the literature some applications to population models are provided

convergence of controllers designed using state space techniques in this paper coprime factorization theory is used to examine convergence of finite dimensional controllers for infinite dimensional systems designed using approximations under assumptions of uniform stabilizability detectability the controllers stabilize the original system for large enough model order and the corresponding sequence of closed loops converge

admissible input elements for systems in hilbert space and a carleson measure criterion we study the control system dot x ax bu quad x in x quad u text scalar where a generates a semigroup on the hilbert space x but in general the control input element b notin x many boundary value control systems point control force situations etc can be studied in this context we define and analyze admissible input elements b and develop sufficient conditions for b to be admissible in terms of the carleson measure theorem of h p theory

on the nonlearnability of a single spiking neuron we study the computational complexity of training a single spiking neuron n with binary coded inputs and output that in addition to adaptive weights and a threshold has adjustable synaptic delays a synchronization technique is introduced so that the results concerning the nonlearnability of spiking neurons with binary delays are generalized to arbitrary real valued delays in particular the consistency problem for n with programmable weights a threshold and delays and its approximation version are proven to be np complete it follows that the spiking neurons with arbitrary synaptic delays are not properly pac learnable and do not allow robust learning unless rp np in addition the representation problem for n a question whether an n variable boolean function given in dnf or as a disjunction of o n threshold gates can be computed by a spiking neuron is shown to be conp hard

weakly singular gronwall inequalities and applications to fractional differential equations abstract we obtain some new gronwall type inequalities which are applicable to some weakly singular volterra integral equations similar to the ones first studied by d henry the main interest is that we consider cases with a double singularity and we obtain explicit l bounds rather than l 1 bounds furthermore our bounds involve the exponential function and not the mittag leffler function as in some previous works we give applications to some volterra integral equations with a doubly singular kernel that arise from caputo fractional differential equations where as opposed to previous papers we have a singularity in the nonlinearity

mesh independence of kleinman newton iterations for riccati equations in hilbert space in this paper we consider the convergence of the infinite dimensional version of the kleinman newton algorithm for solving the algebraic riccati operator equation associated with the linear quadratic regulator problem in a hilbert space we establish mesh independence for this algorithm and apply the result to systems governed by delay equations numerical examples are presented to illustrate the results

continuous dependence of optimal control to controlled domain of actuator for heat equation abstract in this paper we consider continuous dependence of the optimal control with respect to the actuator domain which is varying as open subset in the spatial domain for a multi dimensional heat equation both time optimal control and norm optimal control problems are considered the reason behind combining these two problems together is that these two problems are actually equivalent the energy to be used to drive the system to target set in minimal time interval is actually the minimal energy of driving the system to target set in this minimal time interval and visa versa it is shown that both optimal control and optimal cost are continuous with respect to open controlled actuator domain under the lebesgue measure

the linear regulator problem for parabolic systems we present an approximation framework for computation in finite dimensional spaces of riccati operators that can be guaranteed to converge to the riccati operator in feedback controls for abstract evolution systems in a hilbert space it is shown how these results may be used in the linear optimal regulator problem for a large class of parabolic systems

a spillover phenomenon in the optimal location of actuators in this paper we are interested in finding the optimal location and shape of the actuators in a stabilization problem namely we consider the one dimensional wave equation damped by an internal feedback supported on a subdomain omega of given length the criterion we want to optimize represents the rate of decay of the total energy of the system it theoretically involves all the eigenmodes of the operator from an engineering point of view it seems more realistic to consider only a finite number of modes say the n first ones in that context we are able to prove existence and uniqueness of an optimal domain omega n it is the better possible location for the actuators we characterize this optimal domain and we point out the following strange phenomenon at least for small lengths the optimal domain omega n which is the better one for the n first modes is actually the worse one for the n 1th mode this looks like the well known spillover phenomenon in control theory at last we will give some possible extension and open problems in higher dimension

numerical approximation of bang bang controls for the heat equation an optimal design approach this work is concerned with the numerical computation of null controls of minimal l infty norm for the linear heat equation with a bounded potential both the cases of internal and boundary dirichlet and neumann controls are considered dual arguments allow to reduce the search of controls to the unconstrained minimization of a conjugate function with respect to the initial condition of a backward heat equation however as a consequence of the regularizing property of the heat operator this initial final condition lives in a huge space that can not be approximated with robustness for this reason very specific to the parabolic situation the minimization is severally ill posed on the other hand the optimality conditions for this problem show that in general the unique control v of minimal l infty norm has a bang bang structure as he takes only two values this allows to reformulate the problem as an optimal design problem where the new unknowns are the amplitude of the bang bang control and the space time regions where the control takes its two possible values this second optimization variable is modeled through a characteristic function since the admissibility set for this new control problem is not convex we obtain a relaxed formulation of it which leads to a well posed relaxed problem and lets use a gradient descent method for the numerical resolution of the problem numerical experiments for the inner and boundary controllability cases are described within this new approach

time optimal boundary controls for the heat equation the fact that the time optimal controls for parabolic equations have the bang bang property has been recently proved for controls distributed inside the considered domain interior control the main result in this article asserts that the boundary controls for the heat equation have the same property at least in rectangular domains this result is proved by combining methods from traditionally distinct fields the lebeau robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems on one side and a remez type inequality for muntz spaces and a generalization of turan s inequality on the other side

linear quadratic optimal control of hereditary differential systems infinite dimensional riccati equations and numerical approximations recent theory of infinite dimensional riccati equations is applied to the linear quadratic optimal control problem for hereditary differential systems and it is shown that for most such problems the operator solutions of the riccati equations are of trace class i e nuclear with special attention to trace norm convergence an abstract approximation theory is developed and applied to a particular approximation scheme numerical examples are given problems on both finite and infinite time intervals are studied for both the hereditary system and the approximating systems in the infinite time problem characteristic equations are derived for the closed loop eigenvalues and formulas for the corresponding eigenvectors are given

bounds on functions biorthogonal to sets of complex exponentials control of damped elastic systems abstract in this article we obtain explicit bounds on the norms of biorthogonal functions to sets of complex exponentials exp k t where the k belong to a sector of the positive real axis the bounds obtained are uniform within a class of sequences which exhibit similar growth and separation properties the dependence of the bounds upon the growth and separation parameters is carefully examined the main theorem then leads to some results regarding the solvability of moment problems of the form 0 t f t exp k t dt c k which arise in control problems associated with damped or dissipative systems as an application we establish an exact null controllability result for boundary control of a rectangular structurally damped plate

optimal distribution of the internal null control for the one dimensional heat equation abstract this paper addresses the problem of optimizing the distribution of the support of the internal null control of minimal l 2 norm for the 1 d heat equation a measure constraint is imposed on the support but no topological assumption such as the number of connected components therefore the problem typically lacks of solution in the class of characteristic functions and needs of relaxation we show that the relaxed formulation is obtained by replacing the set of characteristic functions by its convex envelope the proof requires that the observability constant related to the control problem be uniform with respect to the support property which is obtained by the control transmutation method the optimality conditions of the relaxed problem as well as the case where the number of connected components is fixed a priori are also discussed several numerical experiments complete the study and suggest the ill posedness of the problem in contrast to the wave situation

a new approach to an analysis of henry type integral inequalities and their bihari type versions abstract in this paper we propose a new method to solve integral inequalities of henry gronwall type and their bihari nonlinear version nonlinear integral inequalities with weakly singular kernels and with multiple integrals as well as a modification of the ou iang pachpatte inequality are also treated

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

null controllability of the linearized compressible navier stokes system in one dimension in this paper we consider the one dimensional compressible navier stokes equations linearized around a constant steady state q0 v0 q0 0 v0 0 with periodic boundary conditions in the interval i2 0 2 we explore the controllability of this linearized system using a control only for the velocity equation we prove that the linearized system with homogeneous periodic boundary conditions is null controllable in h 1 per i2 l 2 i2 by a localized interior control when time is sufficiently large where h 1 per i2 denotes the sobolev space of periodic functions with mean value zero we show null controllability of the system by proving an observability inequality with the help of two types of ingham inequality

h optimal actuator location in many control applications there is freedom to place the actuators the actuator location should be chosen to optimize certain performance objectives in this paper h performance with state feedback is considered that is both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output a framework for calculating h optimal actuator locations is developed conditions for well posedness of the h optimal actuator location problem are presented many optimal actuator problems involve systems modelled by partial differential equations and conditions under which approximations yield reliable results are given a derivative free optimization algorithm to calculate h optimal actuator locations is described the results are illustrated using several examples

optimal shape and location of sensors for parabolic equations with random initial data in this article we consider parabolic equations on a bounded open connected subset omega of mathbb r n we model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure this problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation for instance what is the optimal location and shape of a thermometer we show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data where the unknown ranges over the set of all possible measurable subsets of omega of fixed measure we prove that under appropriate sufficient spectral assumptions this optimal design problem has a unique solution depending only on a finite number of modes and that the optimal domain is semi analytic and thus has a finite number of connected components this result is in strong contrast with hyperbolic conservative equations wave and schrodinger studied in privat et al j eur math soc 2015 for which relaxation does occur we also provide examples of applications to anomalous diffusion or to the stokes equations in the case where the underlying operator is any positive possible fractional power of the negative of the dirichlet laplacian we show that surprisingly enough the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power the results are illustrated with several numerical simulations

sensor placement for optimal estimation of vector valued diffusion processes abstract diffusion processes are commonplace in many scientific disciplines as they describe a broad range of physical phenomenon consider a diffusion process observed through linear sensors with additive white noise we derive the optimal placement of these sensors for estimating this process where optimality is defined in terms of the mean squared estimation error mse of the state given past observations we consider two cases first we assume the sensors to be orthogonal we show in this case that the minimum mse is related to the nuclear norm of the system matrix of the process second we remove the orthogonality constraint and show that the mse is related to the schatten p norm of the system matrix of the process and the optimal sensors are proportional its matrix cube root we present simulation results illustrating the fact that the gain afforded by optimizing the choice of sensors depends on the ratio p n where n is the dimension of the system and p the dimension of the wiener processes driving it and this gain is in general large especially when p n is small

approximation of the algebraic riccati equation in the hilbert space of hilbert schmidt operators this paper deals with the problem of approximating the infinite dimensional algebraic riccati equation considered as an abstract equation in the hilbert space of hilbert schmidt operators two kin

linear quadratic optimal actuator location in control of vibrations diffusion and many other problems governed by partial differential equations there is freedom in the choice of actuator location the actuator location should be chosen to optimize performance objectives in this paper we consider linear quadratic performance two types of cost are considered the choice depends on whether the response to the worst initial condition is to be minimized or whether the initial condition is regarded as random in practice approximations are used in controller design and thus in selection of the actuator locations the optimal cost and location of the approximating sequence should converge to the exact optimal cost and location in this work conditions for this convergence are given in the case of linear quadratic control examples are provided to illustrate that convergence may fail when these conditions are not satisfied

space time topology optimization for one dimensional wave propagation abstract a space time extension of the topology optimization method is presented the formulation with design variables in both the spatial and temporal domains is used to create structures with an optimized distribution of material properties that can vary in time the method is outlined for one dimensional transient wave propagation in an elastic rod with time dependent young s modulus by two simulation examples it is demonstrated how dynamic structures can display rich dynamic behavior such as wavenumber frequency shifts and lack of energy conservation the optimization method s potential for creating structures with novel dynamic behavior is illustrated by a simple example it is shown that an elastic rod in which the optimized stiffness distribution is allowed to vary in time can be much more efficient in prohibiting wave propagation compared to a static bandgap structure optimized designs in form of spatio temporal laminates and checkerboards are generated and discussed the example lays the foundation for creating designs with more advanced functionalities in future work

the turnpike property in finite dimensional nonlinear optimal control turnpike properties have been established long time ago in finite dimensional optimal control problems arising in econometry they refer to the fact that under quite general assumptions the optimal solutions of a given optimal control problem settled in large time consist approximately of three pieces the first and the last of which being transient short time arcs and the middle piece being a long time arc staying exponentially close to the optimal steady state solution of an associated static optimal control problem we provide in this paper a general version of a turnpike theorem valuable for nonlinear dynamics without any specific assumption and for very general terminal conditions not only the optimal trajectory is shown to remain exponentially close to a steady state but also the corresponding adjoint vector of the pontryagin maximum principle the exponential closedness is quantified with the use of appropriate normal forms of riccati equations we show then how the property on the adjoint vector can be adequately used in order to initialize successfully a numerical direct method or a shooting method in particular we provide an appropriate variant of the usual shooting method in which we initialize the adjoint vector not at the initial time but at the middle of the trajectory

boundary control of a linear thermoelastic beam abstract we show in this paper that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam the optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted as in the euler bernoulli beam theory

on the complexity of learning for spiking neurons with temporal coding abstract spiking neurons are models for the computational units in biological neural systems where information is considered to be encoded mainly in the temporal patterns of their activity in a network of spiking neurons a new set of parameters becomes relevant which has no counterpart in traditional neural network models the time that a pulse needs to travel through a connection between two neurons also known as delay of a connection it is known that these delays are tuned in biological neural systems through a variety of mechanisms in this article we consider the arguably most simple model for a spiking neuron which can also easily be implemented in pulsed vlsi we investigate the vapnik chervonenkis vc dimension of networks of spiking neurons where the delays are viewed as programmable parameters and we prove tight bounds for this vc dimension thus we get quantitative estimates for the diversity of functions that a network with fixed architecture can compute with different settings of its delays in particular it turns out that a network of spiking neurons with k adjustable delays is able to compute a much richer class of functions than a threshold circuit with k adjustable weights the results also yield bounds for the number of training examples that an algorithm needs for tuning the delays of a network of spiking neurons results about the computational complexity of such algorithms are also given

on the controllability of a fractional order parabolic equation the null controllability property of a 1 d parabolic equation involving a fractional power of the laplace operator delta alpha is studied the control is a scalar time dependent function g g t acting on the system through a given space profile f f x on the interior of the domain thus the control g determines the intensity of the space control f applied to the system the latter being given a priori we show that if alpha leq 1 2 and the shape function f is say in l 2 no initial datum belonging to any sobolev space of negative order may be driven to zero in any time this is in contrast with the existing positive results for the case alpha 1 2 and in particular for the heat equation that corresponds to alpha 1 this negative result exhibits a new phenomenon that does not arise either for finite dimensional systems or in the context of the heat equation on the contrary if more regularity of the shape function f is assumed then we show that there are initial data in any sobolev space h m that may be controlled once again this is precisely the opposite behavior with respect to the control properties of the heat equation in which when increasing the regularity of the control profile the space of controllable data decreases these results show that in order for the control properties of the heat equation to be true the dynamical system under consideration has to have a sufficiently strong smoothing effect that is critical when alpha 1 2 for the fractional powers of the dirichlet laplacian in 1 d the results we present here are in nature and with respect to techniques of proof similar to those on the control of the heat equation in unbounded domains in s micu and e zuazua trans amer math soc 353 2000 pp 1635 1659 and s micu and e zuazua portugal math 58 2001 pp 1 24 we also discuss the hyperbolic counterpart of this problem considering a fractional order wave equation and some other models

the hilbert schmidt property of feedback operators in this article we prove new sufficient conditions under which the feedback operator associated with the linear quadratic regulator control design for distributed parameter systems is nuclear or hilbert schmidt examples illustrating the main results are also given

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

a spatio temporal design problem for a damped wave equation we analyze in this work a spatio temporal optimal design problem governed by a linear damped one dimensional wave equation the problem consists of simultaneously seeking the spatio temporal layout of two isotropic materials and the static position of the damping set in order to minimize a functional depending quadratically on the gradient of the state the lack of classical solutions for this kind of nonlinear problem is well known we examine a well posed relaxation by using the representation of a two dimensional divergence free vector as a rotated gradient we transform the original optimal design problem into a nonconvex vector variational problem by means of gradient young measures we compute an explicit form of the constrained quasi convexification of the cost density moreover this quasi convexification is recovered by first order laminates which give the optimal distribution of materials and damping set at every point finally we analyze the relaxed problem and some numerical experiments ar

a post treatment of the homogenization method for shape optimization we propose an alternative to the classical post treatment of the homogenization method for shape optimization rather than penalize the material density once the optimal composite shape is obtained by the homogenization method in order to produce a workable shape close to the optimal one we macroscopically project the microstructure of the former through an appropriate procedure that roughly consists in laying the material along the directions of lamination of the composite we have tested our approach in the framework of compliance minimization in two dimensional elasticity numerical results are provided

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

optimal design in two dimensional conductivity for a general cost depending on the field we explore an optimal design problem in two dimensional conductivity for a rather general cost depending on the underlying field through a typical variational reformulation that has been explored recently we provide a simplified relaxed version which is amenable to numerical simulation and prove that it is a true relaxation under a main structural hypothesis several important cases are covered including a linear cost in the gradient and a convex isotropic functional in particular the pth power of the field for any p geqq1 for the isotropic non quadratic case our computations do not require an explicit form of the constrained quasiconvexification of the equivalent vector variational problem that structural assumption ties together the underlying state equation and the integral cost

shape turnpike for linear parabolic pde models abstract we introduce and study the turnpike property for time varying shapes within the viewpoint of optimal control we focus here on second order linear parabolic equations where the shape acts as a source term and we seek the optimal time varying shape that minimizes a quadratic criterion we first establish existence of optimal solutions under some appropriate sufficient conditions we then provide necessary conditions for optimality in terms of adjoint equations and using the concept of strict dissipativity we prove that state and adjoint satisfy the measure turnpike property meaning that the extremal time varying solution remains essentially close to the optimal solution of an associated static problem we show that the optimal shape enjoys the exponential turnpike property in terms of hausdorff distance for a mayer quadratic cost we illustrate the turnpike phenomenon in optimal shape design with several numerical simulations

classification of linearly nonseparable patterns by linear threshold elements learning and convergence properties of linear threshold elements or perceptrons are well understood for the case where the input vectors or the training sets to the perceptron are linearly separable little is known however about the behavior of the perceptron learning algorithm when the training sets are linearly nonseparable we present the first known results on the structure of linearly nonseparable training sets and on the behavior of perceptrons when the set of input vectors is linearly nonseparable more precisely we show that using the well known perceptron learning algorithm a linear threshold element can learn the input vectors that are provably learnable and identify those vectors that cannot be learned without committing errors we also show how a linear threshold element can be used to learn large linearly separable subsets of any given nonseparable training set in order to develop our results we first establish formal characterizations of linearly nonseparable training sets and define learnable structures for such patterns we also prove computational complexity results for the related learning problems next based on such characterizations we show that a perceptron does the best one can expect for linearly nonseparable sets of input vectors and learns as much as is theoretically possible

training a single sigmoidal neuron is hard we first present a brief survey of hardness results for training feedforward neural networks these results are then completed by the proof that the simplest architecture containing only a single neuron that applies a sigmoidal activation function r satisfying certain natural axioms e g the standard logistic sigmoid or saturated linear function to the weighted sum of n inputs is hard to train in particular the problem of finding the weights of such a unit that minimize the quadratic training error within 2 or its average over a training set within 5 2 12n of its infimum proves to be np hard hence the well known backpropagation learning algorithm appears not to be efficient even for one neuron which has negative consequences in constructive learning

multi order fractional differential equations and their numerical solution we consider the numerical solution of possibly nonlinear fractional differential equations of the form y a t f t y t y b 1 t y b 2 t y b n t with a b n b n 1 b 1 and a b n 1 b j b j 1 1 0 b 1 1 combined with suitable initial conditions the derivatives are understood in the caputo sense we begin by discussing the analytical questions of existence and uniqueness of solutions and we investigate how the solutions depend on the given data moreover we propose convergent and stable numerical methods for such initial value problems

on nonconvergence of adjoint semigroups for control systems with delays it is shown that the adjoints of a spline based approximation scheme for delay equations do not converge strongly

ingham type inequalities for complex frequencies and applications to control theory abstract for complex valued sequences n n 1 of the form n a n i b n with a n r and b n 0 we prove inequalities of the form 0 t n p x n e i t n 2 d t n p x n 2 1 b n for all sequences x n with n 1 x n 2 1 b n we apply these to prove exact null controllability for a class of hinged beam equations with mild internal damping with either boundary control or internal control

integral inequalities and global solutions of semilinear evolution equations a criterion for the nonexplosion of solutions to semilinear evolution equations on banach spaces is proved the result is obtained by applying a modification of the bihari type inequality to the case of a weakly singular nonlinear integral inequality

on the approximation of the boundary layers for the controllability problem of nonlinear singularly perturbed systems abstract a new systematic approach to the construction of approximate solutions to a class of nonlinear singularly perturbed feedback control systems using the boundary layer functions especially with regard to the possible occurrence of the boundary layers is proposed for example problems with feedback control such as the steady states of the thermostats where the controllers add or remove heat depending upon the temperature registered in another place of the heated bar can be interpreted with a second order ordinary differential equation subject to a nonlocal three point boundary condition the o accurate approximation of behavior of these nonlinear systems in terms of the exponentially small boundary layer functions is given at the end of this paper we formulate the unsolved controllability problem for nonlinear systems

closed form h infinity optimal control for a class of infinite dimensional systems h infinity optimal control and estimation are addressed for a class of systems governed by partial differential equations with bounded input and output operators diffusion equations are an important example in this class explicit formulas for the optimal state feedback controller as well as the optimal state estimator are given unlike traditional methods for h infinity synthesis no iteration is needed to obtain the optimal solution moreover the optimal performance for both the state feedback and state estimation problems are explicitly calculated this is shown to be useful for problems of h infinity optimal actuator and sensor location furthermore the results can be used in testing and bench marking of general purpose algorithms for h infinity synthesis the results also apply to finite dimensional systems

non structural controllability of linear elastic systems with structural damping this paper proves that any initial condition in the energy space for the plate equation with square root damping 2 u on a smooth bounded domain with hinged boundary conditions 0 can be steered to zero by a square integrable input function u supported in arbitrarily small time interval 0 t and subdomain as t tends to zero for initial states with unit energy norm the norm of this u grows at most like exp cp tp for any real p 1 and some cp 0 indeed this fast controllability cost estimate is proved for more general linear elastic systems with structural damping and non structural controls satisfying a spectral observability condition moreover under some geometric optics condition on the subdomain allowing to apply the control transmutation method this estimate is improved into p 1 and the dependence of cp on the subdomain is made explicit these results are analogous to the optimal ones known for the heat flow

optimal actuator sensor location for active noise regulator and tracking control problems in this paper we investigate the problem of finding the optimal location of sensors and actuators to achieve reduction of the noise field in an acoustic cavity we offer two control strategies the first is based on linear quadratic tracking where the offending noise is tracked and the second considers the formulation of the harmonic control strategy as a periodic static output feedback control problem the first method which is based on full state information is suitable for optimal location of actuators while the second strategy can extend the results to finding optimal location of sensors as well as actuators for both methods we consider the optimization of an appropriate quadratic performance criterion with respect to the location of the actuators and or the sensors numerical examples are presented to compare the effectiveness of each control strategy and also the effect of optimal placement of actuators and sensors

optimal actuator location of minimum norm controls for heat equation with general controlled domain abstract in this paper we study optimal actuator location of the minimum norm controls for a multi dimensional heat equation with control defined in the space l 2 0 t the actuator domain is time varying in the sense that it is only required to have a prescribed lebesgue measure for any moment we select an optimal actuator location so that the optimal control takes its minimal norm over all possible actuator domains we build a framework of finding the nash equilibrium so that we can develop a sufficient and necessary condition to characterize the optimal relaxed solutions for both actuator location and corresponding optimal control of the open loop system the existence and uniqueness of the optimal classical solutions are therefore concluded as a result we synthesize both optimal actuator location and corresponding optimal control into a time varying feedbacks

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

regularity issues for the null controllability of the linear 1 d heat equation the fact that the heat equation is controllable to zero in any bounded domain of the euclidean space any timet 0 and from any open subset of the boundary is well known on the other hand numerical experiments show the ill posedness of the problem in this paper we develop a rigorous analysis of the 1 d problem which provides a sharp description of this ill posedness to be more precise to each initial data y 0 2 l 2 0 1 of the 1 d linear heat equation it corresponds a boundary control of minimal l 2 0 t norm which drives the state to zero in time t 0 this control is given by a solution of the homogeneous adjoint equation with some initial data b 0 minimizing a suitable quadratic cost our aim is to study the relation between the regularity of y 0 and that of b 0 we show that there are regular data y 0 for which the corresponding b 0 are highly irregular not belonging to any negative exponent sobolev space moreover the class of such initial data y 0 is dense in l 2 0 1 this explains the severe ill posedness of the numerical algorithms developed for the approximation of the minimal l 2 0 t norm control of y 0 based on the computation of b 0 the lack of polynomial convergence rates for tychono regularization processes is a consequence of this phenomenon too

a nonstandard approach to a data assimilation problem and tychonov regularization revisited we consider evolution problems such as diffusion convection equations or the linearized navier stokes system or a weak coupling of them which we would like to predict on a time interval t 0 t 0 t but for which on one hand the initial value of the state variable is unknown on the other hand measurements of the solutions are known on a time interval 0 t 0 and for example on a subdomain in the space variable the classical approach in variational data assimilation is to look for the initial value at time 0 and this is known to be an ill posed problem which has to be regularized here we propose to look for the value of the state variable at time t 0 the end time of the measurements and we prove on some basic examples that this is a well posed problem we give a result of exact reconstruction of the value at t 0 which is based on global carleman inequalities and we give an approximation algorithm which uses classical optimal control auxiliary problems using the same mathematical arguments we also show why tychonov regularization for variational data assimilation works in practical situations corresponding to realistic applications

compactness and nuclearity of the hankel operator and internal stability of infinite dimensional state linear systems we examine the relationships between the exponential or strong stability of certain classes of regular linear systems and the compactness and nuclearity properties of the hankel operator new sufficient conditions for nuclearity are given for exponentially stable regular linear systems with an analytic semigroup

computational limitations on learning from examples the computational complexity of learning boolean concepts from examples is investigated it is shown for various classes of concept representations that these cannot be learned feasibly in a distribution free sense unless r np these classes include a disjunctions of two monomials b boolean threshold functions and c boolean formulas in which each variable occurs at most once relationships between learning of heuristics and finding approximate solutions to np hard optimization problems are given

sharp observability estimates for heat equations the goal of this article is to derive new estimates for the cost of observability of heat equations we have developed a new method allowing one to show that when the corresponding wave equation is observable the heat equation is also observable this method allows one to describe the explicit dependence of the observability constant on the geometry of the problem the domain in which the heat process evolves and the observation subdomain we show that our estimate is sharp in some cases particularly in one space dimension and in the multi dimensional radially symmetric case our result extends those in fattorini and russell arch rational mech anal 43 272 292 1971 to the multi dimensional setting and improves those available in the literature namely those by miller j differ equ 204 1 202 226 2004 siam j control optim 45 2 762 772 2006 atti accad naz lincei cl sci fis mat natur rend lincei 9 mat appl 17 4 351 366 2006 and tenenbaum and tucsnak j differ equ 243 1 70 100 2007 our approach is based on an explicit representation formula of some solutions of the wave equation in terms of those of the heat equation in contrast to the standard application of transmutation methods which uses a reverse representation of the heat solution in terms of the wave one we shall also explain how our approach applies and yields some new estimates on the cost of observability in the particular case of the unit square observed from one side we will also comment on the applications of our techniques to controllability properties of heat type equations