averaged controllability of parameter dependent conservative semigroups abstract we consider the problem of averaged controllability for parameter depending either in a discrete or continuous fashion control systems the aim being to find a control independent of the unknown parameters so that the average of the states is controlled we do it in the context of conservative models both in an abstract setting and also analysing the specific examples of the wave and schrodinger equations our first result is of perturbative nature assuming the averaging probability measure to be a small parameter dependent perturbation in a sense that we make precise of an atomic measure given by a dirac mass corresponding to a specific realisation of the system we show that the averaged controllability property is achieved whenever the system corresponding to the support of the dirac is controllable similar tools can be employed to obtain averaged versions of the so called ingham inequalities particular attention is devoted to the 1d wave equation in which the time periodicity of solutions can be exploited to obtain more precise results provided the parameters involved satisfy diophantine conditions ensuring the lack of resonances

the instability of spiky steady states for a competing species model with cross diffusion this paper is concerned with the stability instability of a class of positive spiky steady states for a quasi linear cross diffusion system describing two species competition by detailed spectral analysis it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross diffusion system are non linearly unstable

stability analysis on a type of steady state for the skt competition model with large cross diffusion abstract this paper is concerned with the existence and the stability of steady state solutions for the skt biological competition model with cross diffusion by applying the higher order expansion and some detailed spectral analysis to the limiting system as the cross diffusion rate tends to infinite it is proved that the nontrivial positive steady states with some special bifurcating structure are unstable further the existence and the instability of the corresponding nontrivial positive steady states for the original cross diffusion system are proved by applying perturbation argument finally we show the global existence of solutions for the limiting system and present some numerical simulation on the large time behavior of the solution with more general initial data

global superconvergence of finite element methods for biharmonic equations and blending surfaces abstract in this paper new numerical algorithms of finite element methods fem are reported for both biharmonic equations and 3d blending surfaces to achieve the global superconvergence o h3 o h4 in h2 norms this is significant compared with the optimal convergence o h2 the algorithms are simple because only an a posteriori interpolant solution is needed such a global convergence method was originated by lin and his colleagues in 1 3 for only the clamped boundary conditions recently we extended the global superconvergence to other boundary conditions such as the simple support condition the periodic boundary and the natural boundary condition moreover we apply in 4 6 this global superconvergence to the fem using the penalty techniques for biharmonic equations and blending problems also to reach o h3 and o h4 for quasiuniform and uniform ij respectively currently we develop in 7 8 and in this paper the fem using the penalty plus hybrid techniques to reduce the condition number down to o h 4 o h 5 of the associated matrix while retaining superconvergence o h3 o h4 since instability is severe for biharmonic equations any reduction of the condition number is crucial by the new algorithms in this paper not only can a great deal of cpu time be saved but also the complicated biharmonic equations and blending surfaces may be solved in double precision numerical experiments are carried out to support the theoretical conclusions

distributed control for identical dynamically coupled systems a decomposition approach we consider the problem of designing distributed controllers for a class of systems which can be obtained from the interconnection of a number of identical subsystems if the state space matrices of these systems satisfy a certain structural property then it is possible to derive a procedure for designing a distributed controller which has the same interconnection pattern as the plant this procedure is basically a multiobjective optimization under linear matrix inequality constraints with system norms as performance indices the explicit expressions for computing these controllers are given for both h infin or h 2 performance and both for static state feedback and dynamic output feedback in discrete time at the end of the paper two application examples illustrate the effectiveness of the approach

ensemble control of bloch equations in this article we study a class of control problems which involves controlling a continuum of dynamical systems with different values of parameters characterizing the system dynamics by using the same control signal we call such problems control of ensembles the motivation for looking into these problems comes from the manipulation of an ensemble of nuclear spins in nuclear magnetic resonance nmr spectroscopy and imaging mri with dispersions in natural frequencies and the strengths of the applied radio frequency rf field from the standpoint of mathematical control theory the challenge is to simultaneously steer a continuum of systems between points of interest with the same control input this raises some new and unexplored questions about controllability of such systems we show that controllability of an ensemble can be understood by the study of the algebra of polynomials defined by the noncommuting vector fields that govern the system dynamics a systematic study of these systems has immediate applications to broad areas of control of ensembles of quantum systems as arising in coherent spectroscopy and quantum information processing the new mathematical structures appearing in such problems are excellent motivation for new developments in control theory

free endpoint optimal control of inhomogeneous bilinear ensemble systems optimal control of bilinear systems has been a well studied subject in the areas of mathematical and computational optimal control however effective methods for solving emerging optimal control problems involving an ensemble of deterministic or stochastic bilinear systems are underdeveloped these burgeoning problems arise in diverse applications from quantum control and molecular imaging to neuroscience in this work we develop an iterative method to find optimal controls for an inhomogeneous bilinear ensemble system with free endpoint conditions the central idea is to represent the bilinear ensemble system at each iteration as a time varying linear ensemble system and then solve it in an iterative manner we analyze convergence of the iterative procedure and discuss optimality of the convergent solutions the method is directly applicable to solve the same class of optimal control problems involving a stochastic bilinear ensemble system driven by independent additive noise processes we demonstrate the robustness and applicability of the developed iterative method through practical control designs in neuroscience and quantum control

distributed control of spatially invariant systems we consider distributed parameter systems where the underlying dynamics are spatially invariant and where the controls and measurements are spatially distributed these systems arise in many applications such as the control of vehicular platoons flow control microelectromechanical systems mems smart structures and systems described by partial differential equations with constant coefficients and distributed controls and measurements for fully actuated distributed control problems involving quadratic criteria such as linear quadratic regulator lqr h sub 2 and h sub spl infin optimal controllers can be obtained by solving a parameterized family of standard finite dimensional problems we show that optimal controllers have an inherent degree of decentralization and this provides a practical distributed controller architecture we also prove a general result that applies to partially distributed control and a variety of performance criteria stating that optimal controllers inherit the spatial invariance structure of the plant connections of this work to that on systems over rings and systems with dynamical symmetries are discussed

ensemble observability of linear systems we address the observability problem for ensembles that are described by probability distributions the problem is to reconstruct a probability distribution of the initial state from the time evolution of the probability distribution of the output under a classical finite dimensional linear system we present two solutions to this problem one based on formulating the problem as an inverse problem and the other one based on reconstructing all the moments of the distribution the first approach leads us to a connection between the reconstruction problem and mathematical tomography problems in the second approach we use the framework of tensor systems to describe the dynamics of the moments which leads to a more systems theoretic treatment of the reconstruction problem furthermore we show that both frameworks are inherently related the appeal of having two dual viewpoints the first being more geometric and the second one being more systems theoretic is illuminated in several examples of theoretical or practical importance

control and synchronization of neuron ensembles synchronization of oscillations is a phenomenon prevalent in natural social and engineering systems controlling synchronization of oscillating systems is motivated by a wide range of applications from surgical treatment of neurological diseases to the design of neurocomputers in this paper we study the control of an ensemble of uncoupled neuron oscillators described by phase models we examine controllability of such a neuron ensemble for various phase models and furthermore study the related optimal control problems in particular by employing pontryagin s maximum principle we analytically derive optimal controls for spiking single and two neuron systems and analyze the applicability of the latter to an ensemble system finally we present a robust computational method for optimal control of spiking neurons based on pseudospectral approximations the methodology developed here is universal to the control of general nonlinear phase oscillators

superconvergence analysis of approximate boundary flux calculations certain projection post processing techniques have been proposed for computing the boundary flux for two dimensional problems e g see carey et al 5 in a series of numerical experiments on elliptic problems they observed that these post processing formulas for approximate fluxes were almost o h 2 accurate for linear triangular elements in this paper we prove that the computed boundary flux iso h 2 ln 1 h accurate in the maximum norm for the partial method of 5 if the solutionu h 3 then the boundary flux error iso h 3 2 in thel 2 norm

on discontinuous continuity equation and impulsive ensemble control abstract the note is devoted to a class of singular ensemble control problems we consider the continuity equation driven by a control affine vector field subject to the constraint on the l 1 norm of the input function since no geometric constraints on control are imposed solutions to the continuity equation can potentially tend somehow to discontinuous measure valued functions we elaborate a representation of such discontinuous generalized solutions through an ordinary control equation being a time reparameterization of the characteristic ordinary differential equation ode of the original continuity equation by virtue of standard results from impulsive control theory of odes we perform a constructive formula for generalized solutions in terms of a measure differential equation finally we state an optimal impulsive control problem for the singular continuity equation and prove the existence of its solution the obtained results can be useful for modeling and analysis of impulsive control problems with uncertain initial data

a diffusive competition model with a protection zone this paper is concerned with a two species diffusive competition model with a protection zone for the weak competitor our mathematical results imply that when the protection zone is above a certain critical patch size determined by the birth rate of the weak competitor the weak species almost always survives but it cannot survive when the protection zone is below the critical size and its competitor is strong enough while this is the main feature of the model the actual dynamical behavior of the reaction diffusion system is more complicated the key to reveal the main feature of the system lies in a detailed analysis of the attracting regions of its steady state solutions our mathematical analysis shows that compared with the predator prey model discussed in yihong du junping shi a diffusive predator prey model with a protect zone j differential equations 226 2006 63 91 the protection zone has some essentially different effects on the fine dynamics of the competition model

averaged control we analyze the problem of controlling parameter dependent systems we introduce the notion of averaged control according to which the quantity of interest is the average of the states with respect to the parameter first we consider the problem of controllability for linear finite dimensional systems and show that a necessary and sufficient condition for averaged controllability is an averaged rank condition in the spirit of the classical rank condition for linear control systems but involving averaged momenta of any order of the matrices generating the dynamics and representing the control action we also describe some open problems and directions of possible research in particular on the average controllability of evolution partial differential equations in this context we analyze also the averaged version of a classical optimal control problem for a parameter dependent elliptic equation and derive the corresponding optimality system

robust controllability assessment via semi definite programming abstract this paper presents a new framework for systematic assessment of the controllability of uncertain linear time invariant lti systems the objective is to evaluate controllability of an uncertain system with norm bounded perturbation over the entire uncertain region the method is based on a singular value minimization problem over the entire complex plane to solve the problem first a necessary and sufficient condition is proposed to avert the difficulties of griding over the complex plane and to verify the controllability of directed and undirected networks in a single step secondly the results are utilized to formulate the problem as two lyapunov based linear matrix inequalities lmis for undirected networks and four lmis for directed ones the proposed approach is subsequently extended to evaluate the maximum guaranteed distance to uncontrollability of a system through a quasi convex optimization problem whose solution is guaranteed to be globally optimal by duality analogous results are established for robust observability of directed and undirected networks the proposed framework is implemented efficiently using lmi tools and provides a fast and reliable tool for the assessment of robust controllability and by duality robust observability it is also extendable to robust control system design problems such as control node selection for uncertain systems

on the phase reduction and response dynamics of neural oscillator populations weundertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli recalling and extending results from the literature we compute phase response curves prcs valid near bifurcations to periodic firing for hindmarsh rose hodgkin huxley fitzhugh nagumo and morris lecar models encompassing the four generic codimension one bifurcations phase density equations are then used to analyze the role of the bifurcation and the resulting prc in responses to stimuli in particular we explore the interplay among stimulus duration baseline firing frequency and population level response patterns we interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions

a moment based approach to ensemble controllability of linear systems abstract in this paper we introduce a new perspective on the l 2 ensemble controllability problem of linear time invariant systems using an l 2 framework the reformulation results from focussing on the controllability of the ensemble s moments which evolve under a linear system defined on the space of square summable sequences for a specific class of ensembles a necessary and sufficient condition can be stated in terms of truncations of infinite kalman matrices we illustrate the analysis in this framework on several examples which highlight the possibility of using elementary structural arguments in proving or disproving the controllability of an ensemble s moments as well as essential differences to the uniform ensemble controllability property which is mostly considered in the literature

ensemble control of finite dimensional time varying linear systems in this article we investigate the problem of simultaneously steering an uncountable family of finite dimensional time varying linear systems with the same control signal this class of control problems motivates further research in the new subject of control theory called ensemble control a notion coming from the study of complex spin dynamics in nuclear magnetic resonance spectroscopy and imaging we derive the necessary and sufficient controllability conditions and an accompanying analytical optimal control law for an ensemble of finite dimensional time varying linear systems we show that ensemble controllability is in connection with singular values of the operator characterizing the system dynamics in addition the problem of optimal ensemble control of harmonic oscillators is studied to demonstrate the controllability results we show that the optimal solutions are pertinent to the study of time frequency limited signals and prolate spheroidal wave functions a systematic study of ensemble control systems has immediate applications to dynamical systems with parameter uncertainty as well as to wide ranging areas such as neuroscience and quantum control the work in ensemble control will foster further developments in mathematical control and systems theory

controllability issues for continuous spectrum systems and ensemble controllability of bloch equations we study the controllability of the bloch equation for an ensemble of non interactinghalf spins inastaticmagneticfield withdispersioninthelarmorfrequency this system may be seen as a prototype for infinite dimensional bilinear systems with continuous spectrum whose controllability is not well understood we provide several mathematical answers with discrimination between approximate and exact controlla bility and between finite time or infinite time controllability this system is not exactly controllable in finite time t with bounded controls in l 2 0 t but it is approximately controllable in l in finite time with unbounded controls in l 0 moreover we propose explicit controls realizing the asymptotic exact controllability to a uniform state of spin 1 2o r 1 2

ensemble control of linear systems with parameter uncertainties abstractin this paper we study the optimal control problem for a class of four dimensional linear systems based on quaternionic and fourier analysis when the control is unconstrained the optimal ensemble controller for this linear ensemble control systems is given in terms of prolate spheroidal wave functions for the constrained convex optimisation problem of such systems the quadratic programming is presented to obtain the optimal control laws simulations are given to verity the effectiveness of the proposed theory

sampled observability and state estimation of linear discrete ensembles we consider the problem of reconstructing the initial states of a finite group of structurally identical linear systems in the situation that output measurements of the individual systems are received at discrete time steps and in an anonymized manner while we do know all output measurements of the individual systems in the group we do not know which output measurement corresponds to which system this state estimation problem addresses the essence of state estimation problems for populations in which the output measurements of the individual systems are given only as statistics we adopt a measure theoretical approach in which the group is modelled by an lti system describing the structure of the individual systems and an initial state which is expressed by a discrete measure in this framework we derive a geometric characterization for the state estimation to admit a unique solution which combined with a result on the observability of linear systems under irregular sampling yields a sufficient condition for the sampled observability of discrete ensembles as a supplement to our theoretical findings we provide illustrations by means of simulation examples furthermore we consider the practical state estimation problem under noisy output measurements

ensemble observability of bloch equations with unknown population density abstract we introduce in the paper a novel observability problem for a large population in the limit a continuum ensemble of nonholonomic control systems with unknown population density we address the problem by focussing on a prototype namely the ensemble of bloch equations which is known for its use of describing the evolution of the bulk magnetization of a collective of non interacting nuclear spins in a static field modulated by a radio frequency rf field the dynamics of the equations are structurally identical but show variations in larmor dispersion and rf inhomogeneity we assume that the initial state of any individual system i e individual bloch equation is unknown and moreover the population density of these individual systems is also unknown furthermore we assume that at any time there is only one scalar measurement output at our disposal the measurement output integrates a certain observation function common to all individual systems over the continuum ensemble the observability problem we pose in the paper is thus the following whether one is able to use the common control input i e the rf field and the single measurement output to estimate both the initial states of the individual systems and the population density amongst other things we establish a sufficient condition for the ensemble system to be observable we show that if the common observation function is any harmonic homogeneous polynomial of positive degree then the ensemble system is observable the proof of the result utilizes tools from representation theory of lie algebras although the results established in the paper are for the specific ensemble of bloch equations the approach developed along the analysis can be generalized to investigate observability of other ensemble systems with single integrated measurement outputs

instability of spiky steady states for s k t biological competing model with cross diffusion abstract this paper is concerned with the stability instability of a type of positive spiky steady states for a quasi linear cross diffusion system describing two species competition by detailed spectral analysis it is proved that the spiky steady states for the related shadow system are linearly unstable when the diffusion rate is small enough and the positive spiky steady states for the original cross diffusion system are unstable when the cross diffusion rate for one species is large enough and the diffusion rate of another species is small

penalty combinations of the ritz galerkin and finite difference methods for singularity problems abstract penalty combination of the ritz galerkin and finite difference methods is presented for solving elliptic boundary value problems with singularities the superconvergence rate o h2 of solution derivatives by the combination can be achieved while using quasiuniform rectangular difference grids where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number it is due to its simplicity that the penalty combination of the ritz galerkin and finite difference methods is highly recommended for solving the complicated problems with multiple singularities and multiple interfaces

diffusion self diffusion and cross diffusion u1 x 0 u 1 0 x u2 x 0 u 2 0 x i n 0 where 2 n 1 2 x 2 is the laplace operator 0 is a bounded smooth domain of r n with n1 0 and 0 are the boundary and the closure of 0 respectively 0t 0 0 t and 0t 0 0 t for some t 0 is the outward unit normal vector on 0 di ai bi ci i 1 2 are all positive constants while ij i j 1 2 denote non negative constants the initial values u1 0 and u2 0 are non negative smooth func tions which are not identically zero the mathematical model 1 1 was proposed by shigesada et al 27 in their study of spatial segregation of interacting species where u1 and u2 represent the densities of two competing species d1 and d2 are their diffu sion rates a1 and a2 denote the intrinsic growth rates b1 and c2 account for intra specific competitions b2 and c1 are the coefficients of inter specific competitions 11 and 22 are usually referred as self diffusion pressures and 12 and 21 are cross diffusion pressures an interesting feature of 1 1 is that the movements of these two competing species are affected by the population pressures due to the mutual interference between the individuals and it was hoped that the nonlinear dispersive force due to the

a characterization of von neumann rings in terms of linear systems abstract a new characterization of commutative von neumann regular rings is given in terms of linear control systems having locally a brunovsky canonical form the problem of enumerating reachable systems over certain von neumann regular ring is solved extending classical results over fields

spatial chaotic structure of attractors of reaction diffusion systems the dynamics described by a system of reaction diffusion equations with a nonlinear potential exhibits complicated spatial patterns these patterns emerge from preservation of homotopy classes of solutions with bounded energies chaotically arranged stable patterns exist because of realizability of all elements of a fundamental homotopy group of a fixed degree this group corresponds to level sets of the potential the estimates of homotopy complexity of attractors are obtained in terms of geometric characteristics of the potential and other data of the problem

notes on sliding mode control of two level quantum systems in dong d and petersen i r 2012 sliding mode control of two level quantum systems automatica 48 725 735 a sliding mode control approach has been proposed for two level quantum systems to deal with bounded uncertainties in the system hamiltonian this paper further extends these results in two directions one extension is to consider the effect of uncertainties during the process of driving the system s state back to the sliding mode domain from outside and we propose two approaches to accomplish this control task the other extension generalizes the previous design approach to consider uncertainties described as perturbations in the free hamiltonian

asymptotic ensemble stabilizability of the bloch equation in this paper we are concerned with the stabilizability to an equilibrium point of an ensemble of non interacting half spins we assume that the spins are immersed in a static magnetic field with dispersion in the larmor frequency and are controlled by a time varying transverse field our goal is to steer the whole ensemble to the uniform down position two cases are addressed for a finite ensemble of spins we provide a control function in feedback form that asymptotically stabilizes the ensemble in the down position generically with respect to the initial condition for an ensemble containing a countable number of spins we construct a sequence of control functions such that the sequence of the corresponding solutions pointwise converges asymptotically in time to the target state generically with respect to the initial conditions the control functions proposed are uniformly bounded and continuous

optimal ensemble control of stochastic time varying linear systems abstract we consider the optimal guidance of an ensemble of independent structurally identical finite dimensional stochastic linear systems with variation in system parameters between initial and target states of interest by applying a common control function without the use of feedback our exploration of such ensemble control systems is motivated by practical control design problems in which variation in system parameters and stochastic effects must be compensated for when state feedback is unavailable such as in pulse design for nuclear magnetic resonance spectroscopy and imaging in this paper we extend the notion of ensemble control to stochastic linear systems with additive noise and jumps which we model using white gaussian noise and poisson counters respectively and investigate the optimal steering problem in our main result we prove that the minimum norm solution to a fredholm integral equation of the first kind provides the optimal control that simultaneously minimizes the mean square error mse and the error in the mean of the terminal state the optimal controls are generated numerically for several example ensemble control problems and monte carlo simulations are used to illustrate their performance this work has immediate applications to the control of dynamical systems with parameter dispersion or uncertainty that are subject to additive noise which are of interest in quantum control neuroscience and sensorless robotic manipulation

on a global superconvergence of the gradient of linear triangular elements abstract we study a simple superconvergent scheme which recovers the gradient when solving a second order elliptic problem in the plane by the usual linear elements the recovered gradient globally approximates the true gradient even by one order of accuracy higher in the l 2 norm than the piecewise constant gradient of the ritz galerkin solution a superconvergent approximation to the boundary flux is presented as well

on controllability of diagonal systems with one dimensional input space abstract this paper deals with diagonal systems on a hilbert state space with a one dimensional input space and a possibly unbounded control operator a priori it is not assumed that the input operator is admissible necessary and sufficient conditions for different notions of controllability such as null controllability exact controllability and approximate controllability are presented these conditions which are given in terms of the eigenvalues of the diagonal operator and in terms of the control operator are linked with the theory of interpolation in hardy spaces

global superconvergence in combinations of ritz galerkin fem for singularity problems abstract this paper combines the piecewise bilinear elements with the singular functions to seek the corner singular solution of elliptic boundary value problems the global superconvergence rates o h 2 can be achieved by means of the techniques of lin and yan the construction and analysis of high efficient fem hobei university publishing hobei 1996 for different coupling strategies such as the nonconforming constraints the penalty integrals and the penalty plus hybrid integrals where 0 is an arbitrarily small number and h is the maximal boundary length of quasiuniform rectangles q ij used a little effort in computation is paid to conduct a posteriori interpolation of the numerical solutions u h only on the subregion used in finite element methods this paper also explores an equivalence of superconvergence between this paper and z c li internat j numer methods eng 39 1996 1839 1857 and j comput appl math 81 1997 1 17

superconvergence of coupling techniques in combined methods for elliptic equations with singularities abstract several coupling techniques such as the nonconforming constraints penalty and hybrid integrals of the ritz galerkin and finite difference methods are presented for solving elliptic boundary value problems with singularities based on suitable norms involving discrete solutions at specific points superconvergence rates on solution derivatives are exploited by using five combinations e g the nonconforming combination the penalty combination combinations i and ii and symmetric combination for quasi uniform rectangular grids the superconvergence rates o h 2 of solution derivatives by all five combinations can be achieved where h is the maximal mesh length of difference grids used in the finite difference method and 0 is an arbitrarily small number superconvergence analysis in this paper lies in estimates on error bounds caused by the coupling techniques and their incorporation with finite difference methods therefore a similar analysis and conclusions may be extended to linear finite element methods using triangulation by referring to existing references moreover the five combinations having o h 2 of solution derivatives are well suited to solving engineering problems with multiple singularities and multiple interfaces

ensemble control of time invariant linear systems with linear parameter variation in this paper we study the control of a class of time invariant linear ensemble systems whose natural dynamics are linear in the system parameter this class of ensemble control systems arises from practical engineering and physical applications such as transport of quantum particles and control of uncertain harmonic systems we establish explicit algebraic criteria to examine controllability of such ensemble systems our derivation is based on the notion of polynomial approximation where the elements of the reachable set of the ensemble system are represented in polynomials of the system parameter and used to approximate the desired state of interest in addition we highlight the role of the spectra of the system matrices play in the determination of ensemble controllability finally illustrative examples and numerical simulations for optimal control of this class of linear ensemble systems are presented to demonstrate the theoretical results

a nonconforming combined method for solving laplace s boundary value problems with singularities for solving laplace s boundary value problems with singularities a nonconforming combined approach of the ritz galerkin method and the finite element method is presented in this approach singular functions are chosen to be admissible functions in the part of a solution domain where there exist singularities and piecewise linear functions are chosen to be admissible functions in the rest of the solution domain in addition the admissible functions used here are constrained to be continuous only at the element nodes on the common boundary of both methods this method is nonconforming however the nonconforming effect does not result in larger errors of numerical solutions as long as a suitable coupling strategy is used

superconvergence phenomenon in the finite element method arising from averaging gradients we study a superconvergence phenomenon which can be obtained when solving a 2nd order elliptic problem by the usual linear elements the averaged gradient is a piecewise linear continuous vector field the value of which at any nodal point is an average of gradients of linear elements on triangles incident with this nodal point the convergence rate of the averaged gradient to an exact gradient in thel 2 norm can locally be higher even by one than that of the original piecewise constant discrete gradient

uniform ensemble controllability for one parameter families of time invariant linear systems abstract in this paper we derive necessary as well as sufficient conditions for approximate controllability of parameter dependent linear systems in the supremum norm using tools from complex approximation theory we prove the existence of parameter independent open loop controls that steer the zero initial state of an ensemble of linear systems uniformly to a prescribed family of terminal states new necessary conditions for uniform ensemble controllability of single input systems are derived our results extend earlier ones of li for ensemble controllability of linear systems