data assimilation for the heat equation using stabilized finite element methods we consider data assimilation for the heat equation using a finite element space semi discretization the approach is optimization based but the design of regularization operators and parameters rely on techniques from the theory of stabilized finite elements the space semi discretized system is shown to admit a unique solution combining sharp estimates of the numerical stability of the discrete scheme and conditional stability estimates of the ill posed continuous pde model we then derive error estimates that reflect the approximation order of the finite element space and the stability of the continuous model two different data assimilation situations with different stability properties are considered to illustrate the framework full detail on how to adapt known stability estimates for the continuous model to work with the numerical analysis framework is given in appendix

simultaneous exact controllability and some applications we study the exact controllability of two systems by means of a common finite dimensional input function a property called simultaneous exact controllability most of the time we consider one system to be infinite dimensional and the other finite dimensional in this case we show that if both systems are exactly controllable in time t 0 and the generators have no common eigenvalues then they are simultaneously exactly controllable in any time t t0 moreover we show that similar results hold for approximate controllability for exactly controllable systems we characterize the reachable subspaces corresponding to input functions of class h1 and h2 we apply our results to prove the exact controllability of a coupled system composed of a string with a mass at one end finally we consider an example of two infinite dimensional systems we characterize the simultaneously reachable subspace for two strings controlled from a common end the result is obtained using a recent generalization of a classical inequality of ingham

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 hierarchical neural model of matching abstract the issue of matching two fuzzy sets emerges as an essential design aspect in many algorithms of processing fuzzy information including fuzzy controllers pattern classifiers knowledge based systems etc this paper introduces a new model of matching the primordial features of the proposed model involve 1 i matching being carried out with respect to the grades of membership of the discussed fuzzy sets as well as some of their functionals like energy entropy transom 2 ii the concepts of hierarchies of matching in the model leading directly to a straightforward distinction between local and global levels of matching and 3 iii a distributed character of the model that is realized as a logic based neural network with enhanced learning capabilities

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

laplacian graph eigenvectors abstract if g is a graph its laplacian is the difference of the diagonal matrix of its vertex degrees and its adjacency matrix the main thrust of the present article is to prove several laplacian eigenvector principles which in certain cases can be used to deduce the effect on the spectrum of contracting adding or deleting edges and or of coalescing vertices one application is the construction of two isospectral graphs on 11 vertices having different degree sequences only one of which is bipartite and only one of which is decomposable

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

computation of minimal norm control asymptotics relative to the null controllability of non standard parabolic like dynamics abstract semidiscrete finite element method fem approximation schemes are presented for the null controllability of two non standard parabolic pde control systems the key feature here is that the null controllers being approximated exhibit the asymptotics of the associated minimal energy function the main driver is the appropriate use of a result in r triggiani optimal estimates of norms of fast controls in exact null controllability of two non classical abstract parabolic systems advances in differential equations 8 2 2003 189 229 which was originally devised to derive the sharp minimal norm control asymptotics for the fully continuous problem

regularization of the backward in time kuramoto sivashinsky equation we are interested in solution techniques for backward in time evolutionary pde problems arising in fluid mechanics in addition to their intrinsic interest such techniques have applications in the recently proposed retrograde data assimilation as our model system we consider the terminal value problem for the kuramoto sivashinsky equation in a 1d periodic domain such backward problems are typical examples of ill posed problems where any disturbances are amplified exponentially during the backward march hence regularization is required in order to solve such a problem efficiently in practice we consider regularization approaches in which the original ill posed problem is approximated with a less ill posed problem obtained by adding a regularization term to the original equation while such techniques are relatively well understood for simple linear problems in this work we investigate them carefully in the nonlinear setting and report on some interesting universal behavior in addition to considering regularization terms with fixed magnitudes we also mention briefly a novel approach in which these magnitudes are adapted dynamically using simple concepts from the control theory

on the observability of time discrete conservative linear systems abstract we consider various time discretization schemes of abstract conservative evolution equations of the form z a z where a is a skew adjoint operator we analyze the problem of observability through an operator b more precisely we assume that the pair a b is exactly observable for the continuous model and we derive uniform observability inequalities for suitable time discretization schemes within the class of conveniently filtered initial data the method we use is mainly based on the resolvent estimate given by burq and zworski in n burq m zworski geometric control in the presence of a black box j amer math soc 17 2 2004 443 471 electronic we present some applications of our results to time discrete schemes for wave schrodinger and kdv equations and fully discrete approximation schemes for wave equations

nonlinear damped partial differential equations and their uniform discretizations abstract we establish sharp energy decay rates for a large class of nonlinearly first order damped systems and we design discretization schemes that inherit of the same energy decay rates uniformly with respect to the space and or time discretization parameters by adding appropriate numerical viscosity terms our main arguments use the optimal weight convexity method and uniform observability inequalities with respect to the discretization parameters we establish our results first in the continuous setting then for space semi discrete models and then for time semi discrete models the full discretization is then inferred from the previous results by adapting the ideas to deal with linear systems our results cover for instance the schrodinger equation with nonlinear damping the nonlinear wave equation the nonlinear plate equation the nonlinear transport equation as well as certain classes of equations with nonlocal terms

the ordering of trees and connected graphs by algebraic connectivity abstract in this paper we first determine that the first four trees of order n 9 with the smallest algebraic connectivity are p n q n w n and z n with p n q n w n z n t where t is any tree of order n other than p n q n w n and z n then we consider the effect on the laplacian eigenvalues of connected graphs by suitably adding edges by using these results and methods we finally determine that the first six connected graphs of order n 9 with the smallest algebraic connectivity are p n q n q n w n w n and w n with p n q n q n w n w n w n g where g is any connected graph of order n other than p n q n q n w n w n and w n

uniform controllability properties for space time discretized parabolic equations this article is concerned with the analysis of semi discrete in space and fully discrete approximations of the null controllability and controllability to the trajectories for parabolic equations we propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters for time discretization we use schemes with theta in frac 1 2 1 for the proofs of controllability we rely on the strategy introduced by lebeau and robbiano comm partial differ equ 20 335 356 1995 for the null controllability of the heat equation which is based on a spectral inequality we obtain relaxed uniform observability estimates in both the semi discrete and fully discrete frameworks and associated uniform controllability properties for the practical computation of the control functions we follow j l lions hilbert uniqueness method strategy exploiting the relaxed uniform observability estimate algorithms for the computation of the controls are proposed and analysed in the semi discrete and fully discrete cases additionally we prove an error bound between the fully discrete and the semi discrete control functions this bound is however not uniform with respect to the space discretization the theoretical results are illustrated through numerical experimentations

connectives and fuzziness for classical effects abstract this article investigates various constructs on a set of classical effects measurable fuzzy sets or fuzzy events we begin by studying the properties of a connective and its dual connective which seem to have been neglected in the literature the importance of these connectives for fuzzy probability theory is pointed out we then introduce and investigate the properties of a commutator which can be employed to define a degree of relative fuzziness a special case of this commutator defines a degree of fuzziness the expectation of the commutator provides a relative entropy whose properties are delineated and the expectation of the special case gives an entropy function that has been previously studied as a measure of fuzziness

discrete carleman estimates for elliptic operators in arbitrary dimension and applications in arbitrary dimension we consider the semidiscrete elliptic operator partial t 2 mathcal a scriptscriptstyle mathfrak m where mathcal a scriptscriptstyle mathfrak m is a finite difference approximation of the operator nabla x gamma x nabla x for this operator we derive a global carleman estimate in which the usual large parameter is connected to the discretization step size we address discretizations on some families of smoothly varying meshes we present consequences of this estimate such as a partial spectral inequality of the form of that proven by g lebeau and l robbiano for mathcal a scriptscriptstyle mathfrak m and a null controllability result for the parabolic operator partial t mathcal a scriptscriptstyle mathfrak m for the lower part of the spectrum of mathcal a scriptscriptstyle mathfrak m with the control function that we construct whose norm is uniformly bounded we prove that the l 2 norm of the final state converges to zero exponentially as the step size of the discretization goes to zero a relaxed observability estimate is then deduced

a forward approach to numerical data assimilation variational data assimilation problems are concerned with computing unknown initial values for the simulation and prediction of natural phenomena most notably in weather prediction and are usually solved via an ill posed optimal control problem for the initial state at the time of the first available measurements an alternative forward approach focuses on computation of the final state after this interval which is just as suitable for prediction purposes and is well posed without additional regularization specifically it is possible to compute projections of the unknown final state on all elements of an orthonormal basis which theoretically allows for the complete reconstruction of the final state in this paper an efficient numerical method for linear evolution equations of diffusive type is presented and convergence of the numerical approximation based on a discontinuous galerkin discretization is proved the key of this method is the computation of an adaptively ordered orthonormal basis using proper orthogonal decomposition numerical examples for a scalar convection diffusion equation in two and three dimensions show the effectiveness of the method

reconstructing initial data using observers error analysis of the semi discrete and fully discrete approximations a new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in ramdani et al automatica 46 10 1616 1625 2010 based on the concept of observers also called luenberger observers this algorithm covers a large class of abstract evolution pde s in this paper we are concerned with the convergence analysis of this algorithm more precisely we provide a complete numerical analysis for semi discrete in space and fully discrete approximations derived using finite elements in space and an implicit euler method in time the analysis is carried out for abstract schrodinger and wave conservative systems with bounded observation locally distributed

an approximation method for exact controls of vibrating systems we propose a new method for the approximation of exact controls of a second order infinite dimensional system with bounded input operator the algorithm combines russell s stabilizability implies controllability principle with the galerkin method the main new feature of this work consists of giving precise error estimates in order to test the efficiency of the method we consider two illustrative examples with the finite element approximations of the wave and the beam equations and describe the corresponding simulations

on the degree of fuzziness of a fuzzy set abstract the purpose of this paper is to study the notion of transom fuzzy indefinite integral on the algebra m x of fuzzy measurable subsets of a given space x this integral is defined with respect to a usual measure the probability of a fuzzy event a defined by a lebesgue stieltjes integral introduced by zadeh 10 is obtained as a particular case this measure is used to distinguish fuzzy sets which have the same fuzzy degree thus we introduce two parameters for characterizing the fuzziness of a fuzzy set

algebraic connectivity and degree sequences of trees we investigate the structure of trees that have minimal algebraic connectivity among all trees with a given degree sequence we show that such trees are caterpillars and that the vertex degrees are non decreasing on every path on non pendant vertices starting at the characteristic set of the fiedler vector author s abstract

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

propagation observation and control of waves approximated by finite difference methods this paper surveys several topics related to the observation and control of wave propagation phenomena modeled by finite difference methods the main focus is on the property of observability corresponding to the question of whether the total energy of solutions can be estimated from partial measurements on a subregion of the domain or boundary the mathematically equivalent property of controllability corresponds to the question of whether wave propagation behavior can be controlled using forcing terms on that subregion as is often desired in engineering applications observability controllability of the continuous wave equation is well understood for the scalar linear constant coefficient case that is the focus of this paper however when the wave equation is discretized by finite difference methods the control for the discretized model does not necessarily yield a good approximation to the control for the original continuous problem in other words the classical convergence consistency stability property of a numerical scheme does not suffice to guarantee its suitability for providing good approximations to the controls that might be needed in applications observability controllability may be lost under numerical discretization as the mesh size tends to zero due to the existence of high frequency spurious solutions for which the group velocity vanishes this phenomenon is analyzed and several remedies are suggested including filtering tychonoff regularization multigrid methods and mixed finite element methods we also briefly discuss these issues for the heat beam and schrodinger equations to illustrate that diffusive and dispersive effects may help to retain the observability controllability properties at the discrete level we conclude with a list of open problems and future subjects for research

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

impulse and sampled data optimal control of heat equations and error estimates we consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multidimensional heat equation with a periodic terminal state constraint this problem has a unique optimal solution which can be characterized by an optimality system derived from the pontryagin maximum principle we define two approximations of this optimal control problem the first one is an impulse approximation and consists of considering a system of linear heat equations with impulse control the second one is obtained by the sample and hold procedure applied to the control resulting in a sampled data approximation of the controlled heat equation we prove that both problems have a unique optimal solution and we establish precise error estimates for the optimal controls and optimal states of the initial problem with respect to its impulse and sampled data approximations

exact controllability stabilization and perturbations for distributed systems exact controllability is studied for distributed systems of hyperbolic type or for petrowsky systems like plate equations the control is a boundary control or a local distributed control exact controllability consists in trying to drive the system to rest in a given finite time the solution of the problems depends on the function spaces where the initial data are taken and also depends on the function space where the control can be chosen a systematic method named hum for hilbert uniqueness method is introduced as the terminology indicates it is based on uniqueness results classical or new and on hilbert spaces constructed in infinitely many ways by using uniqueness a number of applications are indicated having a general method for exact controllability implies having a general method for stabilization this leads to new and complicated nonlinear riccati s type pdes to be compared with direct methods when available in the last part of the paper we consider how all this behaves fo

necessary conditions for exact controllability with a finite dimensional input space we derive necessary conditions for exact controllability of infinite dimensional systems described by x ax bu where the generator a has a riesz basis of eigenvectors and the input space is finite dimensional these conditions are in terms of the eigenvalues of a and the degree of unboundedness of b but no further information about b is needed our results easily imply lack of exact controllability for a wide range of distributed parameter systems it is known from partial differential equation pde examples that the addition of damping into an exactly controllable second order system can destroy exact controllability we discuss the effect of damping on such systems using our general conditions we also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension

on the spectral moments of unicyclic graphs with fixed diameter abstract unicyclic graphs are connected graphs in which the number of edges equals the number of vertices let u n d be the class of unicyclic graphs of order n and diameter d for unicyclic graphs lexicographic ordering by spectral moments s order is discussed in this paper the last d d 2 2 graphs in the s order among all unicyclic graphs in u n d 3 d n 5 are characterized in addition for the cases d n 4 and d n 3 the last 2 n 11 and last n n 2 7 unicyclic graphs in the set u n d are also given respectively

on the second largest laplacian eigenvalues of graphs the second largest laplacian eigenvalue of a graph is the second largest eigenvalue of the associated laplacian matrix in this paper we study extremal graphs for the extremal values of the second largest laplacian eigenvalue and the laplacian separator of a connected graph respectively all simple connected graphs with second largest laplacian eigenvalue at most 3 are characterized it is also shown that graphs with second largest laplacian eigenvalue at most 3 are determined by their laplacian spectrum moreover the graphs with maximum and the second maximum laplacian separators among all connected graphs are determined

an approach to the optimal time for a time optimal control problem of an internally controlled heat equation in this paper we study the optimal time for a time optimal control problem mathcal p where the state system is an internally controlled heat equation by projecting the original problem via the finite element method we obtain another time optimal control problem mathcal p hl governed by a linear system of ordinary differential equations here h and l are the mesh sizes of the finite element spaces of the state space and the control space respectively the purpose of this study is to approach the optimal time for the problem mathcal p through the optimal time for the problem mathcal p hl we prove that the optimal time for the problem mathcal p hl converges to the optimal time for the problem mathcal p when h and l are approaching zero more significantly we obtain error estimates between the optimal times in terms of h and l for certain cases

uniform exponential long time decay for the space semi discretization of a locally damped wave equation via an artificial numerical viscosity we consider the finite difference space semi discretization of a locally damped wave equation the damping being supported in a suitable subset of the domain under consideration so that the energy of solutions of the damped wave equation decays exponentially to zero as time goes to infinity the decay rate of the semi discrete systems turns out to depend on the mesh size h of the discretization and tends to zero as h goes to zero we prove that adding a suitable vanishing numerical viscosity term leads to a uniform with respect to the mesh size exponential decay of the energy of solutions this numerical viscosity term damps out the high frequency numerical spurious oscillations while the convergence of the scheme towards the original damped wave equation is kept we discuss this problem in 1d and 2d in the interval and the square respectively our method of proof relies on discrete multiplier techniques

ordering trees and graphs with few cycles by algebraic connectivity abstract several approaches for ordering graphs by spectral parameters are presented in the literature we can find graph orderings either by the greatest eigenvalue spectral radius or index or by the sum of the absolute values of the eigenvalues the energy of a graph or by the second smallest eigenvalue of the laplacian matrix the algebraic connectivity among others by considering the fact that the algebraic connectivity is related to the connectivity and shape of the graphs several structural properties of graphs relative to this parameter have been studied hence a large number of papers about ordering graphs by algebraic connectivity mainly about trees and graphs with few cycles have been published this paper surveys the significant results concerning these topics trying to focus on possible points to be investigated in order to understand the difficulties to obtain partial orderings via algebraic connectivity

edge perturbation on graphs with clusters adjacency laplacian and signless laplacian eigenvalues abstract let g be a simple undirected graph of order n a cluster in g of order c and degree s is a pair of vertex subsets c s where c is a set of cardinality c c 2 of pairwise co neighbor vertices sharing the same set s of s neighbors assuming that the graph g has k 1 clusters c 1 s 1 c k s k consider a family of k graphs h 1 h k and the graph g h 1 h k which is obtained from g after adding the edges of the graphs h 1 h k whose vertex set of each h j is identified with c j for j 1 k the laplacian eigenvalues of g h 1 h k remain the same independently of the graphs h 1 h k with the exception of c 1 c k k of them these new laplacian eigenvalues are determined using a unified approach which can also be applied to the determination of a same number of adjacency and signless laplacian eigenvalues when the graphs h 1 h k are regular the faria s lower bound on the multiplicity of the laplacian eigenvalue 1 of a graph with pendant vertices is generalized furthermore the algebraic connectivity and the laplacian index of g h 1 h k remain the same independently of the graphs h 1 h k

spectral analysis of discrete elliptic operators and applications in control theory in this paper we propose an analysis of discrete spectral properties for a finite difference discretization of quite general 1d second order self adjoint elliptic operators we particularly investigate some uniform in the discretization parameter qualitative behavior of eigenfunctions and eigenvalues with those estimates we manage to obtain new results for the construction of bounded families of controls for semi discrete parabolic pdes in particular for boundary controls of a coupled parabolic system with fewer controls than equations

uniform boundary controllability of a discrete 1 d wave equation a numerical scheme for the controlled discrete 1 d wave equation is considered we prove the convergence of the boundary controls of the discrete equations to a control of the continuous wave equation when the mesh size tends to zero when time and space steps coincide this positive result is in contrast with previous negative ones for space semi discretizations

an approximation theorem for the algebraic riccati equation for an infinite dimensional linear quadratic control problem in hilbert space approximation of the solution of the algebraic riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems as an application strong onvergence of the approximating riccati operators in case of a previously developed spline approximation scheme for delay systems is established finally convergence of the transfer functions of the approximating systems is investigated

output feedback based boundary control of uncertain coupled semilinear parabolic pde using neurodynamic programming in this paper neurodynamic programming based output feedback boundary control of distributed parameter systems governed by uncertain coupled semilinear parabolic partial differential equations pdes under neumann or dirichlet boundary control conditions is introduced first hamilton jacobi bellman hjb equation is formulated in the original pde domain and the optimal control policy is derived using the value functional as the solution of the hjb equation subsequently a novel observer is developed to estimate the system states given the uncertain nonlinearity in pde dynamics and measured outputs consequently the suboptimal boundary control policy is obtained by forward in time estimation of the value functional using a neural network nn based online approximator and estimated state vector obtained from the nn observer novel adaptive tuning laws in continuous time are proposed for learning the value functional online to satisfy the hjb equation along system trajectories while ensuring the closed loop stability local uniformly ultimate boundedness of the closed loop system is verified by using lyapunov theory the performance of the proposed controller is verified via simulation on an unstable coupled diffusion reaction process

permanental roots and the star degree of a graph abstract for undirected graphs without loops or multiple edges we define the star degree of a graph and prove that it is equal to the multiplicity of the root 1 of per xi b where b d a considering bipartite graphs we prove that per xi b per xi l where l d a and consequently that the star degree of a bipartite graph can also be characterized by the multiplicity of the root 1 of per xi l

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

on the algebraic connectivity of some caterpillars a sharp upper bound and a total ordering a caterpillar is a tree in which the removal of all pendant vertices makes it a path let d 3 and n 2 d 1 be given let p p1 p2 pd 1 with p1 1 p2 1 pd 1 1 let c p be the caterpillar obtained from the stars sp1 sp2 spd 1 and the path pd 1 by identifying the root of spi with the i vertex of pd 1 let c c p p1 p2 pd 1 n d 1 we prove that the algebraic connectivity of c p c is bounded above by 124 2 4 8 2cos d 2 d 1 moreover we prove that if d is even then c p p 1 1 p d2 1 1 p d2 n 2d 3 is the unique caterpillar in c attaining the upper bound and that if d is odd then the upper bound cannot be achieved finally for 1 k d 12 we give a total ordering by algebraic connectivity on ck c1 1 pk 1 1 pd k 1 1 c pk pd k

null controllability of the heat equation using flatness we derive in a direct and rather straightforward way the null controllability of the n dimensional heat equation in a bounded cylinder with boundary control at one end of the cylinder we use the so called flatness approach which consists in parameterizing the solution and the control by the derivatives of a flat output this yields an explicit control law achieving the exact steering to zero replacing the involved series by partial sums we obtain a simple numerical scheme for which we give explicit error bounds numerical experiments demonstrate the relevance of the approach

conjectures on index and algebraic connectivity of graphs abstract let g v e be a simple graph with vertex set v v 1 v 2 v n and edge set e g the adjacency matrix of a graph g is denoted by a g and defined as the n n matrix a ij where a ij 1 for v i v j e g and 0 otherwise the largest eigenvalue 1 of a g is called the spectral radius or the index of g the laplacian matrix of g is l g d g a g where d g is the diagonal matrix of its vertex degrees and a g is the adjacency matrix among all eigenvalues of the laplacian matrix of a graph the most studied is the second smallest called the algebraic connectivity a of a graph 12 in 1 2 aouchiche et al have given a series of conjectures on index 1 and algebraic connectivity a of g see also 3 here we prove two conjectures and disprove one by a counter example

the algebraic connectivity of graphs with given circumference abstract the algebraic connectivity of a graph is the second smallest eigenvalue of its laplacian matrix algebraic connectivity is closely related to the traditional vertex edge connectivity and it plays an important role in the design of various networks in this paper we determine the graph which has the minimum algebraic connectivity among all connected graphs with given circumference

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

convergence estimates for semidiscrete approximations of nonselfadjoint parabolic equations semidiscrete galerkin approximations of the second order nonselfadjoint parabolic equations are considered l 2 error estimates for l 2 initial conditions are derived by use of semigroup methods and of analyticity of the underlined semigroup the scheme here presented can be also freely applied to nonconforming elements

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

boundary controllability of a linear semi discrete 1 d wave equation derived from a mixed finite element method in this article one discusses the controllability of a semi discrete system obtained by discretizing in space the linear 1 d wave equation with a boundary control at one extremity it is known that the semi discrete models obtained with finite difference or the classical finite element method are not uniformly controllable as the discretization parameter h goes to zero see 8 here we introduce a new semi discrete model based on a mixed finite element method with two different basis functions for the position and velocity we show that the controls obtained with these semi discrete systems can be chosen uniformly bounded in l 2 0 t and in such a way that they converge to the hum control of the continuous wave equation i e the minimal l 2 norm control we illustrate the mathematical results with several numerical experiments