accurate calculation of the solutions to the thomas fermi equations abstract we obtain highly accurate solutions to the thomas fermi equations for atoms and atoms in very strong magnetic fields we apply the pade hankel method numerical integration power series with pade and hermite pade approximants and chebyshev polynomials both the slope at origin and the location of the right boundary in the magnetic field case are given with unprecedented accuracy

global superconvergence for the solution of linear boundary value problems das verfahren von i h sloan das eine konvergenzverbesserung bei der numerischen losung von integralgleichungen liefert kann auch auf lineare randwertaufgaben gewohnlicher differentialgleichungen angewendet werden die ergebnisse sind jedoch nur bedingt vergleichbar

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

a unified treatment of superconvergent recovered gradient functions for piecewise linear finite element approximations piecewise linear finite element approximations to two dimensional poisson problems are treated for simplicity consideration is restricted to problems having dirichlet boundary conditions and defined on rectangular domains which are partitioned by a uniform triangular mesh it is also required that the solutions u h3 a method is proposed for recovering the gradients of the finite element approximations to a root mean square accuracy of o h2 both at element edge mid points and element vertices using simple averaging schemes over adjacent elements piecewise linear interpolants respectively discontinuous and continuous are then fitted to these recovered gradients and are shown to be o h2 estimates for u in the l2 norm and thus superconvergent a discussion is given of the extension of the results to problems with more general region and mesh geometries boundary conditions and with solutions of lower regularity and also to other second order elliptic boundary value problems e g the problem of planar linear elasticity

on the necessity of the invariance conditions for reach control on polytopes abstract we study the reach control problem rcp to make the solutions of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope so called invariance conditions are used to prevent solutions from leaving the polytope through facets which are not designated as the exit facet these conditions are known to be necessary for solvability of the rcp on polytopes by continuous state feedback we study whether the invariance conditions are also necessary for solvability of the rcp on polytopes by open loop controls we show by way of a counterexample that surprisingly the answer is negative we identify a suitable class of polytopes for which the invariance conditions remain necessary conditions

on the divergence of gradient expansions for kinetic energy functionals in the potential functional theory we consider the density of a fermionic system as a functional of the potential in one dimensional case where it is approximated by the thomas fermi term plus semiclassical corrections through the gradient expansion we compare this asymptotic series with the exact answer for the case of the harmonic oscillator and the morse potential it is found that the leading thomas fermi term is in agreement with the exact density but the subdominant term does not agree in terms of the asymptotic behavior because of the presence of oscillations in the exact density but their absence in the gradient expansion however after regularization of the density by convolution with a gaussian the agreement can be established even in the subdominant term moreover it is found that the expansion is always divergent and its terms grow proportionally to the factorial function of the order similar to the well known divergence of perturbation series in field theory and the quantum anharmonic oscillator pade hermite approximants allow summation of the series and one of the branches of the approximants agrees with the density

a supraconvergent scheme for nonlinear hyperbolic systems abstract supraconvergent difference schemes are methods for which the local truncation error degrades when applied to an irregular mesh yet the observed error in the solution is as good as if the mesh were uniform for a version of the lax wendroff scheme requiring the computation of the jacobian we show that superconvergence holds on an arbitrary grid one dimensional for the standard version we present a computation which shows that second order accuracy is not always maintained but the convergence is better than first order

monotonic reach control on polytopes the technical note studies the problem of making the trajectories of an affine system defined on a polytopic state space reach a prescribed facet of the polytope in finite time without first leaving the polytope the focus is on solvability by continuous piecewise affine feedback and we formulate a variant of the problem in which trajectories exit in a monotonic sense this allows to obtain necessary and sufficient conditions for solvability in certain geometric situations

high order analysis of the limit cycle of the van der pol oscillator we have applied the lindstedt poincare method to study the limit cycle of the van der pol oscillator obtaining the numerical coefficients of the series for the period and for the amplitude to order 859 hermite pade approximants have been used to extract the location of the branch cut of the series with unprecedented accuracy 100 digits both series have then been resummed using an approach based on pade approximants where the exact asymptotic behaviors of the period and the amplitude are taken into account our results improve drastically all previous results obtained on this subject

language guided controller synthesis for linear systems this paper considers the problem of controlling discrete time linear systems from specifications given as formulas of syntactically co safe linear temporal logic over linear predicates in the state variables a systematic procedure is developed for the automatic computation of sets of initial states and feedback controllers such that all the resulting trajectories of the closed loop system satisfy the given specifications the procedure is based on the iterative construction and refinement of an automaton that enforces the satisfaction of the formula linear programming based approaches are proposed to compute the polytope to polytope controllers that label the transitions of the automaton extensions to discrete time piecewise affine systems and specifications given as formulas of full linear temporal logic are included the algorithms developed in this paper were implemented as a software package that is available for download their application and effectiveness are demonstrated for several case studies

controlling a class of nonlinear systems on rectangles in this paper we focus on a particular class of nonlinear affine control systems of the form xdot f x bu where the drift f is a multi affine vector field i e affine in each state component the control distribution b is constant and the control u is constrained to a convex set for such a system we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant we then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time if the control constraints are polyhedral we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle this work is motivated by the need to construct discrete abstractions for continuous and hybrid systems in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs we show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

a control problem for affine dynamical systems on a full dimensional polytope given an affine system on a full dimensional polytope the problem of reaching a particular facet of the polytope using continuous piecewise affine state feedback is studied necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the polytope special attention is paid to affine systems on full dimensional simplices in this case the necessary and sufficient conditions are equivalent and a constructive procedure yields an affine feedback control law that solves the reachability problem under consideration

superconvergence for rectangular mixed finite elements in this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems for the rectangular finite elements of raviart and thomas 19 and for those of brezzi et al 4 we prove that the distance inl 2 between the approximate solution and a projection of the exact one is of higher order than the error itself this result is exploited to obtain superconvergence at gaussian points and to construct higher order approximations by a local postprocessing

brief paper tracking and synchronisation for a class of pwa systems in this paper the tracking problem for a class of discontinuous piecewise affine pwa systems is addressed we propose an observer based output feedback control design consisting of a feedforward a piecewise affine feedback law and a model based observer solving the tracking problem these synthesis results can also be employed to tackle the master slave synchronisation problem for pwa systems it is shown that for certain classes of pwa systems the design is characterised in terms of linear matrix inequalities the results are illustrated by application to mechanical systems with discontinuous friction characteristics

global dynamics for switching systems and their extensions by linear differential equations abstract switching systems use piecewise constant nonlinearities to model gene regulatory networks this choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of morse graphs associated to nodes of a parameter graph the parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical morse graphs however there are many cellular processes that do not exhibit threshold like behavior and thus are not well described by a switching system we consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations we show that the parameter graphs associated to the switching system and any of its extensions are identical for each parameter graph node there is an order preserving map from the morse graph of the switching system to the morse graph of any of its extensions we provide counterexamples that show why possible stronger relationships between the morse graphs are not valid

an input output simulation approach to controlling multi affine systems for linear temporal logic specifications this article presents an input output simulation approach to controlling multi affine systems for linear temporal logic ltl specifications which consists of the following steps first the state space is partitioned into rectangles each of which satisfies atomic ltl propositions then we study the control of multi affine systems on rectangles including the control based on the exit sub region to drive all trajectories starting from a rectangle to exit through a facet and the control to stabilise the multi affine system towards a desired point with the proposed controllers a finitely abstracted transition system is constructed which is shown to be input output simulated by the rectangular transition system of the multi affine system since the input output simulation preserves ltl properties the controller synthesis of the multi affine system for ltl specifications is achieved by designing a nonblocking supervisor for the abstracted transition system and by implementing the resulting supervisor to

chebyshev expansion on intervals with branch points with application to the root of kepler s equation a chebyshev hermite pade method when two or more branches of a function merge the chebyshev series of u l will converge very poorly with coefficients a n of t n l falling as o 1 n a for some small positive exponent a however as shown in j p boyd chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one dimensional bratu equation appl math comput 143 2002 189 200 it is possible to obtain approximations that converge exponentially fast in n if the roots that merge are denoted as u 1 l and u 2 l then both branches can be written without approximation as the roots of u u 1 l u u 2 l u email protected l email protected l by expanding the nonsingular coefficients of the quadratic b l and c l as chebyshev series and then applying the usual roots of a quadratic formula we can approximate both branches simultaneously with error that decreases proportional to exp email protected for some constant s 0 where n is the truncation of the chebyshev series this is dubbed the chebyshev shafer or chebyshev hermite pade method because it substitutes chebyshev series for power series in the generalized pade approximants known variously as shafer or hermite pade approximants here we extend these ideas first we explore square roots with branches that are both real valued and complex valued in the domain of interest illustrated by meteorological baroclinic instability second we illustrate triply branched functions via roots of the kepler equation f u l e email protected u email protected 0 only one of the merging roots is real valued and the root depends on two parameters l e rather than one nonetheless the chebyshev hermite pade scheme is successful over the whole two dimensional parameter plane we also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain

summation of the eigenvalue perturbation series by multi valued pade approximants application to resonance problems and double wells quadratic pade approximants are used to obtain energy levels both for the anharmonic oscillator x2 2 lambda x4 and for the double well x2 2 lambda x4 in the first case the complex valued energy of the resonances is reproduced by summation of the real terms of the perturbation series the second case is treated formally as an anharmonic oscillator with a purely imaginary frequency we use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function then we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants in this way we can reconstruct the real energy by summation of the complex series such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory around the minima of the double well potential incorrect

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

error analysis of the combination technique the combination technique is a method to reduce the computational time in the numerical approximation of partial differential equations in this paper we present a new technique to analyze the convergence rate of the combination technique this technique is applied to general second order elliptic differential equations in two dimensions furthermore it is proved that the combination technique for poisson s equation convergences in arbitrary dimensions

extrapolation of the iterated collocation method for integral equations of the second kind for integral equations of the second kind with appropriate smoothness of both kernel and exact solution it is known that the iterated collocation method based on discontinuous piecewise polynomials of order r and on collocation at the gauss points can exhibit a uniform superconvergent error of order 2r this paper shows that the order of the iterated collocation approximation on an arbitrary mesh can be further improved by one step of richardson extrapolation assuming the calculation to be repeated with each subinterval halved

brief paper reachability and stabilization of discrete time affine systems with disturbances this paper studies the fundamental problems whether an affine system affected by additive disturbances is robustly transferable from a source set simplex to a target set polytope and whether it is robustly stabilizable with its state constrained in a simplex first a necessary and sufficient condition is derived for the existence of affine feedback control that solves the robust reachability problem further investigation is provided for two situations relying on whether the union of the source set and the target set is convex or non convex for the former one a necessary and sufficient condition is obtained in the form of linear inequalities while for the latter several computationally feasible sufficient conditions are found second we show that robust stabilization subject to a state constraint is equivalent to find a feasible solution to a linear equation once it is known that either of the problems has a solution by checking the derived conditions design of control laws is then straightforward

a fully automated framework for control of linear systems from temporal logic specifications we consider the following problem given a linear system and a linear temporal logic ltl formula over a set of linear predicates in its state variables find a feedback control law with polyhedral bounds and a set of initial states so that all trajectories of the closed loop system satisfy the formula our solution to this problem consists of three main steps first we partition the state space in accordance with the predicates in the formula and construct a transition system over the partition quotient which captures our capability of designing controllers second using a procedure resembling model checking we determine runs of the transition system satisfying the formula third we generate the control strategy illustrative examples are included

an algorithm to find maximal state constraint sets for discrete time linear dynamical systems with bounded controls and states an algorithm to find a polyhedral approximant to the maximal state constraint set given state and input bounds is suggested for linear discrete time dynamical systems

hyperasymptotics and the linear boundary layer problem why asymptotic series diverge the simplest problem with boundary layers epsilon 2 u xx u f x is used to illustrate i why the perturbation series in powers of epsilon is asymptotic but divergent ii why the optimally truncated expansion is superasymptotic in the sense that that error is proportional to exp mbox constant epsilon and iii how to obtain an improved hyperasymptotic approximation

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

superconvergence of a collocation type method for simple turning points of hammerstein equations approximation d un point de retournement simple de l equation d hammerstein dependant d un parametre

centroidal voronoi tessellation based finite element superconvergence in this article a finding on finite element superconvergence is reported the laplacian operator with dirichlet boundary condition is considered the linear finite element solutions have an o h2 0 5 superconvergence in l2 norm at nodes on an almost equilateral triangular mesh generated based on centroidal voronoi tessellation for an arbitrary 2d bounded domain extensive numerical examples are presented to demonstrate the superconvergence property copyright 2008 john wiley sons ltd

reach control problem for linear differential inclusion systems on simplices this technical note concerns the reach control problem for the dynamical systems represented by linear differential inclusions the goal is to derive conditions for the trajectories of a differential inclusion defined on a full dimensional simplex to reach exit facets in finite time using affine feedback to achieve this goal an invariance condition is firstly derived for lipschitz differential inclusions as an application of the obtained invariance condition the reach control problem rcp is solved for two classes of differential inclusions norm bounded linear differential inclusion and polytypic linear differential inclusion in the sense of strong and weak reachability respectively

a note on a superconvergence result for the generalized airfoil equation we obtain convergence results and error estimates for calculating linear functionals of solutions of a class of cauchy singular integral equations via galerkin s method these results generalize the superconvergence estimates obtained by miel 7 and involve extending the class of functionals relaxing the smoothness conditions imposed and determining the effects of quadrature errors in the numerical implementation of this method in particular it is shown that full superconvergence can be restored by evaluating the integrals sufficiently accurately

on the superconvergence of the satisfying babuska brezzi method a simple two point boundary value problem written in a mixed form is shown to have superconvergence characteristics using the satisfying babuska brezzi sbb method convergence and accuracy analyses reveal superior performance of the method compared to the usual galerkin method

on invariant polyhedra of continuous time linear systems this note presents some conditions of existence of positively invariant polyhedra for linear continuous time systems these conditions are first described algebraically then interpreted on the basis of the system eigenstructure then a simple state feedback placement method is described for solving some linear regulation problems under constraints

the finite element method for parabolic equations ii a posteriori error estimation and adaptive approach abstract we extend in this paper the analysis of a posteriori estimates of the space discretization error presented in a previous paper 3 for time independent space meshes in the context of the model problem studied there results are given relating the effectivity of the error estimator to properties of the solution space meshes and manner in which the meshes change a procedure based upon this theory is presented for the adaptive construction of time dependent meshes the results of some computational experiments show that this procedure is practically very effective and suggest that it can be used to control the space discretization error in more general problems author

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

an analysis of a superconvergence result for a singularly perturbed boundary value problem we give a new proof that the el mistikawy and werle finite difference scheme is uniformly second order accurate for a nonselfadjoint singularly perturbed boundary value problem to do this we use exponential finite elements and a discretized green s function the proof is direct gives the nodal errors explicitly in integral form and involves much less computation than in previous proofs of the result

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

reachability of a set of facets for linear affine systems with n 1 inputs this note provides new necessary and sufficient conditions for an n dimensional linear affine system with n 1 inputs to reach an exit facet or set of exit facets of a simplex the conditions reduce the original np hard necessary and sufficient conditions to a set of at most n lp problems

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

on reachability and controllability of switched boolean control networks this paper investigates the reachability and controllability of switched boolean control networks bcns by using the semi tensor product method first a kind of switching input state transfer graph is defined based on which and the algebraic expression of switched bcns we thus propose a new kind of switching input state incidence matrix that contains the complete information of the switching input state transfer graph second by the switching input state incidence matrix several necessary and sufficient conditions are obtained for the reachability and controllability of switched bcns finally an algorithm is established to find a switching sequence and a control scheme to realize the reachability of switched bcns in the shortest time the study of an illustrative example shows that the new results obtained in this paper are very effective in investigating the reachability and controllability of switched bcns

synthesis for constrained nonlinear systems using hybridization and robust controllers on simplices in this technical note we propose an approach to controller synthesis for a class of constrained nonlinear systems it is based on the use of a hybridization that is a hybrid abstraction of the nonlinear dynamics this abstraction is defined on a triangulation of the state space where on each simplex of the triangulation the nonlinear dynamics is conservatively approximated by an affine system subject to disturbances except for the disturbances this hybridization can be seen as a piecewise affine hybrid system on simplices for which appealing control synthesis techniques have been developed in the past decade we extend these techniques to handle systems subject to disturbances by synthesizing and coordinating local robust affine controllers defined on the simplices of the triangulation we show that the resulting hybrid controller can be used to control successfully the original constrained nonlinear system our approach though conservative can be fully automated and is computationally tractable to show its effectiveness in practical applications we apply our method to control a pendulum mounted on a cart

superconvergence of the gradient in piecewise linear finite element approximation to a parabolic problem some recent results concerning maximum norm superconvergence of the gradient in piecewise linear finite element approximations of an elliptic problem are carried over to a parabolic problem both the standard semidiscrete in space galerkin method and the lumped mass modification are analyzed for both smooth and nonsmooth data situations and the crank nicolson discretizations in time of these procedures are considered as examples of completely discrete schemes

time varying affine feedback for reach control on simplices abstract we study the reach control problem for affine systems on simplices and the focus is on cases when it is known that the problem is not solvable by continuous state feedback in previous work we used the reach control indices to construct a discontinuous piecewise affine control that solves the problem here we investigate the extent to which time varying feedbacks can be used a simple time varying affine feedback is proposed and shown to solve the problem

a posteriori error estimators in the finite element method in the paper we develop a structured approach to thea posteriori estimation of the error in the approximation obtained via the finite element method this aids the classification of existing estimators as well as allowing new estimators to be proposed for new situations a class of abstract estimators for finite elements of orderp 1 in re n n 2 3 based on exploiting the superconvergence phenomenon are analyzed

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

shafer hermite pade approximants for functions with exponentially small imaginary part with application to equatorial waves with critical latitude quadratic shafer approximants and their generalization to higher degree polynomials called hermite padeapproximants have been successfully used in quantum mechanics for calculation of exponentially small escape rates in this paper we test quadratic shafer approximants in representing growth rates typical for equatorial atmosphere one of the characteristic features of the equatorial kelvin wave the dominant mode in equatorial dynamics is its exceptionally small linear growth rate for example kelvin wave evolving on the zonal basic state with small linear shear e has growth rate o exp 1 e 2 in contrast to i e o exp 1 e common to similar problems in quantum mechanics it is interesting to know how well hermite pade approximants handle this more computationally expensive problem first we apply the quadratic shafer approximants to calculate the imaginary part of the gauss stieltjes function defined asg s e lim d 0 0 exp t 2t 1 i d edton its branch cut the imaginary part of g s e can be shown to bei g s e pexp 1 e 2 which is of the same order of magnitude as kelvin in shear growth rate next we use this technique to sum the divergent rayleigh schrodinger perturbation series of the hermite with pole equationu y y e11 ey l y 2u 0 which is a simple model for equatorial waves in shear the hermite with pole equation has been previously studied numerically and analytically in j p boyd a nataroc stud appl math 101 1998 43 we compare shafer approximants against numerical integration and find that for small e shafer approximants are more efficient primarily because with rational coefficients one does not need multiple precision at the main stage of the calculation although the higher order approximants are usually more accurate the overall improvement of accuracy is not monotonic due to the appearance of nearly coincident zeros in the approximants

superconvergent derivatives a taylor series analysis an analysis based on the use of taylor series expansions is developed to determine accuracy estimates for derivatives in one and two dimensions computed by differentiation of a finite element interpolant or approximation the analysis clarifies some issues concerning special points at which the derivatives are believed to be exceptionally accurate with higher convergence rates superconvergence moreover it leads directly to a class of post processing strategies for the derivatives and offers a more direct constructive approach to the subject

an obstruction to solvability of the reach control problem using affine feedback this paper studies the reach control problem rcp using affine feedback on simplices the contributions of this paper are threefold first we identify a new obstruction to solvability of the rcp using affine feedback and provide necessary and sufficient conditions for occurrence of such an obstruction second for two input systems these conditions are formulated in terms of scalar linear inequalities third computationally efficient necessary conditions are proposed for checking the obstruction for multi input systems as feasibility programs in terms of linear inequalities in contrast to the previous work in the literature no assumption is imposed on the set of possible equilibria so the results are applicable to the general rcp

constrained regulation of linear continuous time dynamical systems abstract in this paper the linear constrained regulation problem lcrp for continuous time dynamical systems is studied the first part of the paper deals with the existence of linear state feedback control laws that transfer asymptotically to the origin all initial states belonging to a given polyhedral subset of the state space while respecting linear constraints on both state and control vectors then the lcrp is formulated as an optimization problem which can be solved by applying linear programming techniques

control to facet by piecewise affine output feedback the control to facet problem plays an important role in the design of feedback controllers for piecewise affine hybrid systems on polytopes in the literature necessary conditions and sufficient conditions for solvability by static state feedback exist in this paper we extend these results to the case of continuous piecewise affine static output feedback for the construction of a controller a triangulation of the output polytope is made which satisfies additional conditions to guarantee compatibility with the induced subdivision of the state polytope in the state feedback case the use of this special type of triangulation is not required

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

flow functions control flow functions and the reach control problem the paper studies the reach control problem rcp to make trajectories of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope we introduce the notion of a flow function which provides the analog of a lyapunov function for the equilibrium stability problem a flow function comprises a scalar function that decreases along closed loop trajectories and its existence is a necessary and sufficient condition for closed loop trajectories to exit the polytope it provides an analysis tool for determining if a specific instance of rcp is solved without the need for calculating the state trajectories of the closed loop system results include a variant of the lasalle principle tailored to rcp an open problem is to identify suitable classes of flow functions we explore functions of the form v x max v i x and we give evidence that these functions arise naturally when rcp is solved using continuous piecewise affine feedbacks next we introduce the notion of a control flow function it is shown that the artstein sontag theorem of control lyapunov functions has direct analogies to rcp via control flow functions

superconvergent approximations to the solution of a boundary integral equation on polygonal domains the solution to the interior dirichlet problem for laplace s equation in a two dimensional domain is the double layer potential of a double layer distribution satisfying a second kind boundary integral equation this may be solved numerically by galerkin s method using piecewise polynomials of degree r the iterated galerkin solution may then be calculated which gives an order 2r 2 approximation if the domain is smooth however if the domain is polygonal the corners cause singularities which degrade the order of convergence here it is shown that a suitable grading of the mesh near the corners almost restores the rate of convergence when the error is measured in the uniform norm we then describe a second means of calculating a higher order approximation from the galerkin solution this is cheaper to calculate than the iterated galerkin solution but maintains the same order of convergence

superconvergence phenomena in the finite element method abstract we give a brief survey of superconvergence phenomena in finding a numerical solution of differential equations by finite elements several new results and open problems are introduced