l_ infty and l_ 2 low gain feedback their properties characterizations and applications in constrained control low gain feedback has found several applications in constrained control robust control and nonlinear control in this paper we first generalize the existing low gain design methods by introducing the notion of l vanishment and by providing a full characterization of feedback gains that achieve such a property we observe that l low gain feedback can lead to energy peaking namely the control energy required by l low gain feedback increases toward infinity as the low gain parameter decreases to zero motivated by this observation we consider the notion of l2 vanishment and establish several of its characterizations based on which a new design approach referred to as the l2 low gain feedback approach for linear systems is developed different from the l low gain feedback the l2 low gain feedback is instrumental in the control of systems with control energy constraints as an application of l2 low gain feedback the problem of semiglobal stabilization of linear systems with control energy constraints is solved in this paper the notions of l and l2 vanishment also allow us to establish a systematic approach to the design of l and l2 low gain feedback the advantage of this new design approach is that it results in a family of control laws including those resulting from the existing design methods

global stabilization and restricted tracking with bounded feedback for multiple oscillator systems this paper is concerned with the problems of global stabilization and restricted tracking for a multiple oscillator system by bounded feedback a nonlinear feedback law consisting of nested saturation functions is proposed to achieve global stabilization to improve the transient performance of the closed loop system the parameters in this feedback law can be tuned to assign the poles of the closed loop system when none of the saturation elements in the control law is in effect the proposed feedback law is also shown to achieve restricted tracking simulation results are given to show the effectiveness of the proposed feedback law

implicit lyapunov functions and isochrones of linear systems lyapunov s direct method is based on a lyapunov function that explicitly depends on the system s state vector this article describes implicit lyapunov functions and corresponding stability theorems that are counterparts of the explicit case we also define isochrones of linear systems and describe their relationship to implicit lyapunov functions determining settling times is considered as an application

an analysis and design method for a class of nonlinear systems with nested saturations abstractthis paper is concerned with the analysis and design of a class of nonlinear systems subject to nested saturations the proposed controller incorporates both an extended state observer eso which is utilised to estimate the nonlinear dynamics of the plant as well as a set of observer based feedbacks we first present analysis results for systems with nonlinear esos and show that local stabilisation can be achieved in a region including the origin then in the case that the eso is in linear form leso the conditions for determining the estimate of the domain of attraction of the resulting closed loop system are formulated as a convex optimisation problem a linear matrix inequality based algorithm is then established to design the feedback gains and the leso gain an illustrative example is given to show the effectiveness of the proposed approach

technical communique a fast nonlinear control method for linear systems with input saturation we present a novel fast saturating nonlinear feedback law for single input systems with linear dynamics and input saturation it is fast in the sense that it yields a better performance than a saturating linear control law the control law is based on implicit soft variable structure control a convex optimization procedure for the controller synthesis based on linear matrix inequalities lmis is derived at the price of some conservatism as an example we consider the control of a submarine

brief an analysis and design method for linear systems subject to actuator saturation and disturbance we present a method for estimating the domain of attraction of the origin for a system under a saturated linear feedback a simple condition is derived in terms of an auxiliary feedback matrix for determining if a given ellipsoid is contractively invariant this condition is shown to be less conservative than the existing conditions which are based on the circle criterion or the vertex analysis moreover the condition can be expressed as linear matrix inequalities lmis in terms of all the varying parameters and hence can easily be used for controller synthesis this condition is then extended to determine the invariant sets for systems with persistent disturbances lmi based methods are developed for constructing feedback laws that achieve disturbance rejection with guaranteed stability requirements the effectiveness of the developed methods is illustrated with examples

variational multiscale element free galerkin vmefg and local discontinuous galerkin ldg methods for solving two dimensional brusselator reaction diffusion system with and without cross diffusion abstract the finite element method fem is one of the basic methods for solving deterministic and stochastic partial differential equations this method is proposed in the 19 decade and after years several modifications for this well known technique such as mortal fem discontinuous galerkin fem extended fem least squares fem spectral fem mixed fem immersed fem adaptive fem etc have been proposed also for improving the accuracy and for considering complex domain shape functions of the moving least squares approximation have been replaced with the traditional shape function of the finite element technique the name of this improved method is element free galerkin efg method which is classified in the category of meshless methods the efg method has been applied for solving many problems in engineering in the current paper we select two numerical procedures that are extracted from fem the discontinuous galerkin approach is a useful technique for solving problems that their exact solutions have discontinuity it should be noted that many enriched ideas have been explained for improving accuracy of efg method in the current paper the local discontinuous galerkin and variational multiscale efg methods are applied for obtaining numerical solution of two dimensional brusselator system with and without cross diffusion the brusselator system is a theoretical model for a type of autocatalytic reaction up to best of authors knowledge the accuracy of the obtained numerical solutions using local discontinuous galerkin method is very related to the used time discrete scheme therefore fourth order exponential time differencing method has been employed for discretizing the time variable also to achieve a full discretization scheme the local discontinuous galerkin finite element method is used moreover several test problems are given that show the acceptable accuracy and efficiency of the proposed schemes

a real distinct poles exponential time differencing scheme for reaction diffusion systems a second order exponential time differencing etd method for reaction diffusion systems which uses a real distinct poles discretization method for the underlying matrix exponentials is developed the method is established to be stable and second order convergent it is demonstrated to be robust for problems involving non smooth initial and boundary conditions and steep solution gradients we discuss several advantages over competing second order etd schemes

remediation of contaminated groundwater by meshless local weak forms in this paper four meshless local weak form methods such as direct meshless local petrovgalerkin method dmlpg meshless local petrovgalerkin method mlpg local weak radial point interpolation method lwrpim and local weak moving kriging method lwmk are applied to find the numerical solution of coupled non linear advectiondiffusionreaction system arising in the prevention of groundwater contamination problem a comparison between these methods is done from the perspective of accuracy and computational efficiency an efficient fourth order exponential time differencing rungekutta method is utilized for the time discretization the computational efficiency is the most significant advantage of the dmlpg method in comparison with the other meshless local weak form methods because dmlpg reduces the computational costs significantly this is due to the fact that dmlpg shifts the numerical integrations over low degree polynomials rather than over complicated shape functions the main aim of this paper is to show that the meshless local weak form methods can be used for solving the system of non linear partial differential equations especially coupled non linear advectiondiffusionreaction system the numerical results confirm the good efficiency of the proposed methods for solving our model

linear lmi based external anti windup augmentation for stable linear systems we study linear anti windup augmentation for linear control systems with saturated linear plants in the special case when the anti windup compensator can only modify the input and the output of the windup prone linear controller we also measure the arising performance in terms of the finite l 2 gain from exogenous inputs to selected performance outputs our main results are a system theoretic feasibility characterization for fixed order anti windup design and a linear matrix inequality lmi formulation for optimal static and plant order anti windup design interpretations of lower bounds on the achievable performance are also given the effectiveness of the design procedure is demonstrated on a simulation example

rogue waves dissipation and downshifting abstract we investigate the effects of dissipation on the development of rogue waves and downshifting by adding nonlinear and linear damping terms to the one dimensional dysthe equation significantly rogue waves do not develop after the downshifting becomes permanent thus in our experiments permanent downshifting serves as an indicator that damping is sufficient to prevent the further development of rogue waves using the inverse spectral theory of the nls equation simulations of the damped dysthe equation for sea states characterized by jonswap spectrum consistently show that rogue wave events are well predicted by proximity to homoclinic data as measured by the spectral splitting distance the cut off distance c u t o f f decreases as the strength of the damping increases indicating that for stronger damping the jonswap initial data must be closer to homoclinic data for rogue waves to occur

incomplete iterations in multistep backward difference methods for parabolic problems with smooth and nonsmooth data backward difference methods for the discretization of parabolic boundary value problems are considered in this paper in particular we analyze the case when the backward difference equations are only solved approximately by a preconditioned iteration we provide an analysis which shows that these methods remain stable and accurate if a suitable number of iterations often independent of the spatial discretization and time step size are used results are provided for the smooth as well as nonsmooth initial data cases finally the results of numerical experiments illustrating the algorithms performance on model problems are given

fourth order time stepping for stiff pdes a modification of the exponential time differencing fourth order runge kutta method for solving stiff nonlinear pdes is presented that solves the problem of numerical instability in the scheme as proposed by cox and matthews and generalizes the method to nondiagonal operators a comparison is made of the performance of this modified exponential time differencing etd scheme against the competing methods of implicit explicit differencing integrating factors time splitting and fornberg and driscoll s sliders for the kdv kuramoto sivashinsky burgers and allen cahn equations in one space dimension implementation of the method is illustrated by short matlab programs for two of the equations it is found that for these applications with fixed time steps the modified etd scheme is the best

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

design of state feedback control for polynomial systems with quadratic performance criterion and control input constraints abstract this paper presents a novel convex optimization approach to design state feedback control for polynomial systems design criteria are comprised of a quadratic cost function and bounded magnitudes of control inputs specifically we formulate a control synthesis of closed loop systems operated in a given bounded domain characterized by a semi algebraic set we consider an extended class of rational lyapunov functions and derive an upper bound of the cost function together with a state feedback control law by exploiting bounds on the control input magnitudes the controller design condition can be cast as a parameter dependent linear matrix inequality pdlmi which is convex optimization and can be efficiently solved by sum of squares sos technique in addition we derive a sufficient condition to compute a lower bound of the cost function when choosing polynomial structure of the solution candidate the lower bound can also be written as pdlmi numerical examples are provided to illustrate the effectiveness of the proposed design

technical communique a convex characterization of dynamically constrained anti windup controllers based on a recent description of deadzone nonlinearities via local sector conditions a new lmi characterization is proposed to compute full order continuous time anti windup controllers with pole constraints more precisely an upper bound is introduced on the real part of the controller poles to avoid slow dynamics which often leads to poor time domain performance as is demonstrated in a short applicative part the introduction of such a bound allows us to efficiently handle the trade off between stability domain enlargement and time domain response relevance

high order time stepping fourier spectral method for multi dimensional space fractional reaction diffusion equations abstract this paper introduces a high order time stepping scheme that is based on using fourier spectral in space and a fourth order diagonal pade approximation to the matrix exponential function for solving multi dimensional space fractional reaction diffusion equations the resulting time stepping scheme is developed based on an exponential time differencing approach such that it alleviates solving a large non linear system at each time step while maintaining the stability of the scheme the non locality of the fractional operator in some other numerical schemes for these equations leads to full and dense matrices this scheme is able to overcome such computational inefficiency due to the full diagonal representation of the fractional operator it also attains spectral convergence for multiple spatial dimensions the stability of the scheme is discussed through the investigation of the amplification symbol and plotting its stability regions which provides an indication of the stability of the method the convergence analysis is performed empirically to show that the scheme is fourth order accurate in time as expected numerical experiments on reaction diffusion systems with application to pattern formation are discussed to show the effect of the fractional order in space fractional reaction diffusion equations and to validate the effectiveness of the scheme

design of polynomial control laws for polynomial systems subject to actuator saturation this paper presents results for the design of polynomial control laws for polynomial systems in global and regional contexts the proposed stabilization conditions are based on inequalities which are affine in both the lyapunov function coefficients and the controller gains input saturations are incorporated to the stability analysis and the design of polynomial controllers using a generalization of a sector condition the polynomial constraints of the stability stabilization conditions are relaxed to be sum of squares and formulated as semi definite programs

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

higher order exponential time differencing scheme for system of coupled nonlinear schrodinger equations the coupled nonlinear schrodinger equations are highly used in modeling the various phenomena in nonlinear fiber optics like propagation of pulses efficient and reliable numerical schemes are required for analysis of these models and for improvement of the fiber communication system in this paper we introduce a new version of the cox and matthews third order exponential time differencing runge kutta etd3rk scheme based on the 1 2 pade approximation to the exponential function in addition we present its local extrapolation form to improve temporal accuracy to the fourth order the developed scheme and its extrapolation are seen to be strongly stable which have ability to damp spurious oscillations caused by high frequency components in the solution a computationally efficient algorithm of the new scheme based on a partial fraction splitting technique is presented in order to investigate the performance of the novel scheme we considered the system of two and four coupled nonlinear schrodinger equations and performed several numerical experiments on them the numerical experiments showed that the developed numerical scheme provide an efficient and reliable way for computing long range solitary solutions given by coupled nonlinear schrodinger equations and conserved the conserved quantities mass and energy exactly to at least five decimal places

stability and performance for saturated systems via quadratic and nonquadratic lyapunov functions in this paper we develop a systematic lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration the only assumptions we make about the system are well posedness of the algebraic loop and local stability problems to be considered include the estimation of the domain of attraction the reachable set under a class of bounded energy disturbances and the nonlinear l2 gain the regional analysis is established through an effective treatment of the algebraic loop and the saturation deadzone function this treatment yields two forms of differential inclusions a polytopic differential inclusion pdi and a norm bounded differential inclusion ndi that contain the original system adjustable parameters are incorporated into the differential inclusions to reflect the regional property the main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain lyapunov function where the pdi or the ndi is valid with quadratic lyapunov functions conditions for stability and performances are derived as linear matrix inequalities lmis to obtain less conservative conditions we use a pair of conjugate non quadratic lyapunov functions the convex hull quadratic function and the max quadratic function these functions yield bilinear matrix inequalities bmis as conditions for stability and guaranteed performance level the bmi conditions cover the corresponding lmi conditions as special cases hence the bmi results are guaranteed to be as good as the lmi results in most examples the bmi results are significantly better than the lmi results

serk2v3 solving mildly stiff nonlinear partial differential equations abstract multidimensional nonlinear parabolic partial differential equations pdes appear in a large variety of disciplines usually the scientific literature advises against the use of explicit ode solvers for the integration of stiff problems however in this manuscript we would like to show how some explicit methods can be very efficient for specific problems stabilized explicit runge kutta methods serk are a class of explicit methods with extended stability domains along the negative real axis it is necessary to evaluate the function s times but the stability region is o s 2 hence the computational cost is o s times lower than for traditional explicit algorithms in this way moderately stiff problems can be integrated by the use of simple explicit evaluations serk schemes can easily be applied to many different classes of problems with large dimensions and additionally they have low memory demand since these methods are explicit they do not require algebra routines to solve large nonlinear systems associated to odes and therefore are especially well suited for the method of lines mol discretizations of parabolic multi dimensional pdes additionally the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way that is with minimal number of additional stages hence adaptation of the length step is possible at practically no extra cost in this work we study several strategies to change the time step for non smooth functions and finally propose a new technique this procedure improves the numerical results especially when the data is non smooth we also derive two algorithms to estimate the spectral radius in the new code a nonlinear power method and another procedure based on the gershgorin theorem although serk algorithms are only second order schemes in this manuscript we propose to utilize them after higher order discretizations in space to obtain higher accurate solutions efficiently and usually faster since these higher order schemes allow the use of a smaller number of nodes some numerical experiments shown in the paper support these conclusions

controller design for a class of nonlinear systems with input saturation using convex optimization abstract in this paper we propose a new design strategy for nonlinear systems with input saturation the resulting nonlinear controllers are locally asymptotically stabilizing the origin the proposed methodology is based on exact feedback linearization which is used to reformulate the nonlinear system as a linear system having state dependent input saturation linear saturating state feedback controllers and soft variable structure controllers are developed based on this system formulation the resulting convex optimization problems can be written in terms of linear matrix inequalities and sum of squares conditions for which efficient solvers exist polynomial approximation based on legendre polynomials is used to extend the methodology to a more general class of nonlinear systems to demonstrate the benefit of this design method a stabilizing controller for a single link manipulator with flexible joint is developed

an antiwindup approach to enlarging domain of attraction for linear systems subject to actuator saturation the article considers linear systems subject to actuator saturation an antiwindup technique is used to enlarge the domain of attraction of the closed loop system under an a priori designed linear dynamic feedback law the design of the antiwindup compensation gain is formulated and solved as an iterative optimization problem with lmi constraints numerical examples are used to demonstrate the effectiveness of the proposed design technique

exponential integrators for large systems of differential equations we study the numerical integration of large stiff systems of differential equations by methods that use matrix vector products with the exponential or a related function of the jacobian for large problems these can be approximated by krylov subspace methods which typically converge faster than those for the solution of the linear systems arising in standard stiff integrators the exponential methods also offer favorable properties in the integration of differential equations whose jacobian has large imaginary eigenvalues we derive methods up to order 4 which are exact for linear constant coefficient equations the implementation of the methods is discussed numerical experiments with reaction diffusion problems and a time dependent schrodinger equation are included

exponential fitting bdf algorithms explicit and implicit 0 stable methods we present bdf type formulas capable of the exact integration with only round off errors of differential equations the solutions of which belong to the space generated by the linear combinations of 1 eax xeax plots of their 0 stability regions in terms of are provided we will see that the explicit method is 0 stable with many parameters h plots of their regions absolute stability that include all the negative real axis are provided numerical examples show the efficiency of the proposed codes specially when we are integrating stiff problems with the explicit method

time discretization and stability regions for dissipative dispersive kuramoto sivashinsky equation arising in turbulent gas flow over laminar liquid abstract in this study spatio temporal discretization for semilinear dissipative partial differential equations type is introduced analyzed and implemented the model studied here is the dispersively kuramoto sivashinsky equation with an additional term representing the dispersive term arising in turbulent gas flow over laminar liquid tseluiko and kalliadasis 2011 this additional term is multiplied by a parameter that represents the influence of the turbulent gas flow our objective is to examine the effect of this additional term on the dynamics of the kuramoto sivashinsky equation characterized by its chaotic behavior this is achieved by combining the exponential time differencing crank nicolson etd cn scheme derived by kleefed et al 2012 and fourier pseudospectral schemes for temporal and spatial stepping respectively the method is known to be stable and second order convergent in addition a theoretical study and a plot of stability regions of etd cn were performed showing its effectiveness for the stiff problem studied here

a parametric periodic lyapunov equation with application in semi global stabilization of discrete time periodic systems subject to actuator saturation this paper is concerned with semi global stabilization of discrete time linear periodic systems subject to actuator saturation provided that the open loop characteristic multipliers are within the closed unit circle a low gain feedback design approach is proposed to solve the problem by state feedback our approach is based on the solution to a parametric discrete time periodic lyapunov equation the proposed approaches not only generalize the corresponding results for time invariant systems to periodic systems but also reveal some important intrinsic properties of this class of periodic matrix equations a numerical example is worked out to illustrate the effectiveness of the proposed approaches

numerical study of three dimensional turing patterns using a meshless method based on moving kriging element free galerkin efg approach in this paper a numerical procedure is presented for solving a class of three dimensional turing system first we discrete the spatial direction using element free galerkin efg method based on the shape functions of moving kriging interpolation then to achieve a high order accuracy we use the fourth order exponential time differencing runge kutta etdrk4 method using this discretization for the temporal dimension we obtain an explicit scheme and do not need to solve nonlinear system of equations the efg method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the efg method test and trial functions are moving least squares approximation mls shape functions since the shape functions of moving least squares mls approximation do not have kronecker delta property we cannot implement the essential boundary condition directly also building shape functions of mls approximation is a time consuming procedure because of the mentioned reasons we employ the shape functions of moving kriging interpolation technique which have the mentioned property and less cpu time is required for building them for testing this method on three dimensional pdes we select some equations and system of pdes such as allen cahn gray scott ginzburg landau brusselator models predator prey model with additional food supply to predator several test problems are solved and numerical simulations are reported which confirm the efficiency of the proposed scheme

exponential time differencing for stiff systems we develop a class of numerical methods for stiff systems based on the method of exponential time differencing we describe schemes with second and higher order accuracy introduce new runge kutta versions of these schemes and extend the method to show how it may be applied to systems whose linear part is nondiagonal we test the method against other common schemes including integrating factor and linearly implicit methods and show how it is more accurate in a number of applications we apply the method to both dissipative and dispersive partial differential equations after illustrating its behavior using forced ordinary differential equations with stiff linear parts

a variable θ method for parabolic problems of nonsmooth data abstract parabolic initial boundary value problems with nonsmooth data show either rapid transitions or reduced smoothness in its solution for those problems specific numerical methods are required to avoid spurious oscillations as well as unrealistic smoothing of steep changes in the numerical solution this article investigates characteristics of the method and introduces a variable method as a synergistic combination of the crank nicolson cn method 1 2 and the implicit method 1 it suppresses spurious oscillations by evolving the solution implicitly at points where the solution shows a certain portent of oscillations or reduced smoothness and maintains as a similar accuracy as the cn method with smooth data an effective strategy is suggested for the detection of points where the solution may introduce spurious oscillations the wobble set the resulting variable method is analyzed for its accuracy and stability various numerical examples are given to verify its effectiveness

hardness results for multicast cost sharing we continue the study of multicast cost sharing from the viewpoints of both computational complexity and economic mechanism design we provide fundamental lower bounds on the network complexity of group strategyproof budget balanced mechanisms we also extend a classical impossibility result in game theory to show that no strategyproof mechanism can be both approximately efficient and approximately budget balanced our results show that one important and natural case of multicast cost sharing is an example of a canonical hard problem in distributed algorithmic mechanism design in this sense they represent progress toward the development of a complexity theory of internet computation

brief paper anti windup synthesis for linear control systems with input saturation achieving regional nonlinear performance in this paper we present lmi based synthesis tools for regional stability and performance of linear anti windup compensators for linear control systems we consider both static and dynamic compensators algorithms are developed that minimize the upper bound on the regional l 2 gain for exogenous inputs with l 2 norm bounded by a given value and that minimize this upper bound with a guaranteed reachable set or domain of attraction based on the structure of the optimization problems it is shown that for systems whose plants have poles in the closed left half plane plant order dynamic anti windup can achieve semiglobal exponential stability and finite l 2 gain for exogenous inputs with l 2 norm bounded by any finite value the problems are studied in a general setting where the only requirement on the linear control system is well posedness and internal stability the effectiveness of the proposed techniques is illustrated with an example

smoothing schemes for reaction diffusion systems with nonsmooth data cox and matthews s m cox p c matthews exponential time differencing for stiff systems j comput phys 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic equations kassam and trefethen a k kassam ll n trefethen fourth order time stepping for stiff pdes siam j sci comput 26 2005 1214 1233 have shown that these schemes can suffer from numerical instability and they proposed a modified form of the fourth order etdrk4 scheme they use complex contour integration to implement these schemes in a way that avoids inaccuracies when inverting matrix polynomials but this approach creates new difficulties in choosing and evaluating the contour for larger problems neither treatment addresses problems with nonsmooth data where spurious oscillations can swamp the numerical approximations if one does not treat the problem carefully such problems with irregular initial data or mismatched initial and boundary conditions are important in various applications including computational chemistry and financial engineering we introduce a new version of the fourth order cox matthews kassam trefethen etdrk4 scheme designed to eliminate the remaining computational difficulties this new scheme utilizes an exponential time differencing runge kutta etdrk scheme using a diagonal pade approximation of matrix exponential functions while to deal with the problem of nonsmooth data we use several steps of an etdrk scheme using a sub diagonal pade formula the new algorithm improves computational efficiency with respect to evaluation of the high degree polynomial functions of matrices having an advantage of splitting the matrix polynomial inversion problem into a sum of linear problems that can be solved in parallel in this approach it is only required that several backward euler linear problems be solved in serial or parallel numerical experiments are described to support the new scheme

on the stability of exponential fitting bdf algorithms we present bdf type formulas capable of the exact integration with only round off errors of differential equations whose solutions are linear combinations of an exponential with parameter l and ordinary polynomials for l 0 new formulas reduces to the classical bdf formulas plots of their 0 stability regions in terms of l are provided plots of their regions absolute stability that include all the negative real axis are provided numerical examples shows the efficiency of the proposed codes specially when we are integrating stiff oscillatory problems

group strategyproof cost sharing mechanisms for makespan and other scheduling problems classical results in economics show that no truthful mechanism can achieve budget balance and efficiency simultaneously roughgarden and sundararajan recently proposed an alternative efficiency measure which was subsequently used to exhibit that many previously known cost sharing mechanisms approximate both budget balance and efficiency in this work we investigate cost sharing mechanisms for combinatorial optimization problems using this novel efficiency measure with a particular focus on scheduling problems our contribution is threefold first for a large class of optimization problems that satisfy a certain cost stability property we prove that no budget balanced moulin mechanism can approximate efficiency better than w logn where n denotes the number of players in the universe second we present a group strategyproof cost sharing mechanism for the minimum makespan scheduling problem that is tight with respect to budget balance and efficiency finally we show a general lower bound on the budget balance factor for cost sharing methods which can be used to prove a lower bound of w n on the budget balance factor for completion and flow time scheduling objectives

extrapolated stabilized explicit rungekutta methods extrapolated stabilized explicit rungekutta methods eserk are proposed to solve multi dimensional nonlinear partial differential equations pdes in such methods it is necessary to evaluate the function nt times per step but the stability region is o nt2 hence the computational cost is o nt times lower than for a traditional explicit algorithm in that way stiff problems can be integrated by the use of simple explicit evaluations in which case implicit methods usually had to be used therefore they are especially well suited for the method of lines mol discretizations of parabolic nonlinear multi dimensional pdes in this work first s stages first order methods with extended stability along the negative real axis are obtained they have slightly shorter stability regions than other traditional first order stabilized explicit rungekutta algorithms also called rungekuttachebyshev codes later they are used to derive nt stages second and fourth order schemes using richardson extrapolation the stability regions of these fourth order codes include the interval 0 01nt2 0 nt being the number of total functions evaluations which are shorter than stability regions of rock4 methods for example however the new algorithms neither suffer from propagation of errors as other rungekuttachebyshev codes as rock4 or dumka nor internal instabilities additionally many other types of higher order and also lower order methods can be obtained easily in a similar way these methods also allow adaptation of the length step with no extra cost hence the stability domain is adapted precisely to the spectrum of the problem at the current time of integration in an optimal way i e with minimal number of additional stages we compare the new techniques with other well known algorithms with good results in very stiff diffusion or reactiondiffusion multi dimensional nonlinear equations

nonlinear control synthesis by convex optimization a stability criterion for nonlinear systems recently derived by the third author can be viewed as a dual to lyapunov s second theorem the criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space and flows along the system trajectories toward the equilibrium the new criterion has a remarkable convexity property which in this note is used for controller synthesis via convex optimization recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used

a compact fourth order l stable scheme for reaction diffusion systems with nonsmooth data exponential time differencing runge kutta etdrk schemes based on diagonal pade approximations for the numerical solution of reaction diffusion systems containing nonsmooth data have the disadvantage of producing poor numerical results when the time steps are too large relative to the spatial steps on the other hand methods based on sub diagonal pade approximations do not suffer from this bottleneck in this paper we present a novel modified version of cox and matthews fourth order etdrk scheme based on sub diagonal pade approximation to matrix exponential functions in combination with compact fourth order finite difference scheme in space for direct integration of reaction diffusion problems with nonsmooth data for an efficient implementation of the scheme a partial fraction splitting technique is utilized in which it is required to solve several backward euler type linear systems at each time step moreover the design of the algorithm offers parallel implementation so we implement the proposed algorithm in parallel on two processors utilizing matlabmpi a parallel message passing version of matlab and obtain that the parallel version is computationally more efficient than the existing schemes considered in this paper we investigate the amplification factor of the scheme and plot its boundaries of stability regions which give an indication of the stability of the scheme calculation of the local truncation error and an empirical convergence analysis demonstrates the fourth order accuracy of the proposed scheme accuracy computational efficiency and reliability of the new scheme are demonstrated with numerical examples and comparing it with existing schemes

eserk5 a fifth order extrapolated stabilized explicit runge kutta method abstract a new algorithm is developed and analyzed for multi dimensional non linear parabolic partial differential equations pdes which are semi discretized in the spatial variables leading to a system of ordinary differential equations odes it is based on fifth order extrapolated stabilized explicit runge kutta schemes eserk they are explicit methods and therefore it is not necessary to employ complicated software for linear or non linear system of equations additionally they have extended stability regions along the negative real semi axis hence they can be considered to solve stiff problems coming from very common diffusion or reaction diffusion problems previously only lower order codes up to fourth order have been constructed and made available in the scientific literature however at the same time higher order codes were demonstrated to be very efficient to solve equations where it is necessary to have a high precision or they have transient zones that are very severe and where functions change very fast the new schemes allow changing the step length very easily and with a very small computational cost thus a variable step length with variable number of stages algorithm is constructed and compared with good numerical results in relation to other well known ode solvers

explicit controller formulas for lmi based h synthesis the set of h controllers with closed loop performance c can be implicitly parametrized by the solutions r s of a system of linear matrix inequalities lmi the matrices r and s play a role analogous to that of the riccati solutions x and y in classical riccati based h control useful applications include lmi based h synthesis mixed h 2h design and h design with a pole placement constraint this paper is concerned with the reliable computation of h controllers given a solution r s of the characteristic system of lmis explicit formulas are derived for both the regular and singular cases remarkably these formulas are extensions of the usual central controller formulas where the lmi solutions r and s replace the riccati solutions x and y simple and numerically appealing new formulas for discrete time h controllers are also derived

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

spectral variational integrators for semi discrete hamiltonian wave equations abstract in this paper we present a highly accurate hamiltonian structure preserving numerical method for simulating hamiltonian wave equations this method is obtained by applying spectral variational integrators svi to the system of hamiltonian odes which are derived from the spatial semi discretization of the hamiltonian pde the spatial variable is discretized by using high order symmetric finite differences an efficient implementation of svi for high dimensional systems of odes is presented

brief paper an improved treatment of saturation nonlinearity with its application to control of systems subject to nested saturation this paper is concerned with the treatment of saturation nonlinearity existing commonly in practical control systems an improved treatment of such nonlinearity is proposed which includes the well known linear differential inclusion method for handling saturation nonlinearity as a special case to show the effectiveness of this new treatment the problem of analysis and design of linear systems subject to nested saturation function is revisited the proposed results not only complement the existing work on this problem but also are less conservative than the existing methods four numerical examples are worked out to illustrate the applicability of the established approaches

anti windup synthesis using riccati equations the aim of this paper is to give a novel solution to the full order anti windup aw compensation problem for stable systems with input saturation the solution is obtained by completing the square in three steps and requires the solution to a single bounded real riccati equation characterized by the open loop plant s norm the riccati equation plays the role of the lmis usually found in anti windup synthesis but in addition to its numerical advantages it yields a family of anti windup compensators with the same performance this family of compensators is parameterized by a matrix which is intimately linked with both the poles of the anti windup compensator and the robustness properties of the closed loop saturated system thus this matrix allows a robust anti windup problem to be solved in a straightforward and intuitive manner the effectiveness of the proposed technique is demonstrated on a simple example

efficient l stable method for parabolic problems with application to pricing american options under stochastic volatility efficient l stable numerical method for semilinear parabolic problems with nonsmooth initial data is proposed and implemented to solve heston s stochastic volatility model based pde for pricing american options under stochastic volatility the proposed new method is also used to solve two asset american options pricing problem cox and matthews s m cox p c matthews exponential time differencing for stiff systems journal of computational physics 176 2002 430 455 developed a class of exponential time differencing runge kutta schemes etdrk for nonlinear parabolic problems kassam and trefethen a k kassam l n trefethen fourth order time stepping for stiff pdes siam journal on scientific computing 26 4 2005 1214 1233 showed that while computing certain functions involved in the cox matthews schemes severe cancelation errors can occur which affect the accuracy and stability of the schemes kassam and trefethen proposed complex contour integration technique to implement these schemes in a way that avoids these cancelation errors but this approach creates new difficulties in choosing and evaluating the contour integrals for larger problems we modify the etdrk schemes using positivity preserving pade approximations of the matrix exponential functions and construct computationally efficient parallel version using splitting technique as a result of this approach it is required only to solve several backward euler linear problems in serial or parallel

brief paper estimating the domain of attraction for non polynomial systems via lmi optimizations this paper proposes a strategy for estimating the domain of attraction da for non polynomial systems via lyapunov functions lfs the idea consists of converting the non polynomial optimization arising for a chosen lf in a polynomial one which can be solved via lmi optimizations this is achieved by constructing an uncertain polynomial linearly affected by parameters constrained in a polytope which allows us to take into account the worst case remainders in truncated taylor expansions moreover a condition is provided for ensuring asymptotical convergence to the largest estimate achievable with the chosen lf and another condition is provided for establishing whether such an estimate has been found the proposed strategy can readily be exploited with variable lfs in order to search for optimal estimates lastly it is worth remarking that no other method is available to estimate the da for non polynomial systems via lmis

brief paper simultaneous linear and anti windup controller synthesis using multiobjective convex optimization we present a method for the synthesis of a control law for input constrained linear systems that incorporates both a traditional linear output feedback controller as well as a static anti windup compensator unlike traditional two step anti windup controller designs in which the linear controller and anti windup compensator are designed sequentially our method synthesizes all controller parameters simultaneously this one step design retains the anti windup structure thus providing structurally a priori compensation for saturation we derive sufficient conditions for guaranteeing global quadratic stability and for satisfying multiple possibly conflicting performance objectives on the constrained and unconstrained closed loop dynamics the resulting synthesis problem is recast as an optimization over linear matrix inequalities lmis we demonstrate the proposed method on a benchmark problem

on improving the performance with bounded continuous feedback laws we present controller design methods to smooth the discontinuity resulting from a piecewise linear control law which was proposed to improve the convergence performance for systems with input constraints the continuous control laws designed in this paper are explicit functions of the state and are easily implementable we also show that the convergence performance can be further improved by using a saturated high gain feedback law the efficiency of the proposed methods is illustrated with the puma 560 robot model

lmi techniques for optimization over polynomials in control a survey numerous tasks in control systems involve optimization problems over polynomials and unfortunately these problems are in general nonconvex in order to cope with this difficulty linear matrix inequality lmi techniques have been introduced because they allow one to obtain bounds to the sought solution by solving convex optimization problems and because the conservatism of these bounds can be decreased in general by suitably increasing the size of the problems this survey aims to provide the reader with a significant overview of the lmi techniques that are used in control systems for tackling optimization problems over polynomials describing approaches such as decomposition in sum of squares positivstellensatz theory of moments polya s theorem and matrix dilation moreover it aims to provide a collection of the essential problems in control systems where these lmi techniques are used such as stability and performance investigations in nonlinear systems uncertain systems time delay systems and genetic regulatory networks it is expected that this survey may be a concise useful reference for all readers

using inverse nonlinearities in robust output feedback guaranteed cost control of nonlinear systems this paper extends the result of an earlier paper on constructive robust nonlinear output feedback controller design which is based on the use of iqcs and minimax lqg control in this paper in addition to copies of the nonlinearities we include copies of the inverses of the nonlinearities in the controller consequently more iqcs are exploited in the design procedure which gives a smaller bound on the closed loop cost function than the earlier result

a fast numerical method to price american options under the bates model we consider the problem of pricing american options in the framework of a well known stochastic volatility model with jumps the bates model according to this model the price of an american option can be obtained as the solution of a linear complementarity problem governed by a partial integro differential equation in this paper a numerical method for solving such a problem is proposed in particular first of all using a bermudan approximation and a richardson extrapolation technique the linear complementarity problem is reduced to a set of standard linear partial differential problems see for example ballestra and sgarra 2010 chang et al 2007 2012 then these problems are solved using an ad hoc pseudospectral method which efficiently combines the chebyshev polynomial approximation an implicit explicit time stepping and an operator splitting technique numerical experiments are presented showing that the novel algorithm is very accurate and fast and significantly outperforms other methods that have recently been proposed for pricing american options under the bates model

a hybrid fem for solving the allen cahn equation abstract we present an unconditionally stable hybrid finite element method for solving the allen cahn equation which describes the temporal evolution of a non conserved phase field during the antiphase domain coarsening in a binary mixture its various modified forms have been applied to image analysis motion by mean curvature crystal growth topology optimization and two phase fluid flows the hybrid method is based on the operator splitting method the equation is split into a heat equation and a nonlinear equation an implicit finite element method is applied to solve the diffusion equation and then the nonlinear equation is solved analytically various numerical experiments are presented to confirm the accuracy and efficiency of the method our simulation results are consistent with previous theoretical and numerical results

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

brief multivariable anti windup controller synthesis using linear matrix inequalities we present a general formulation of the problem of multi variable anti windup bumpless transfer awbt controller synthesis which provides sufficient conditions for stability covering a large class of typical input nonlinearities such as saturation deadzone and switching override nonlinearities the resulting synthesis method demonstrates graceful performance degradation whenever input nonlinearities are active through minimization of a weighted l 2 gain the main contribution of the paper is that the awbt controller synthesis using static compensation is cast as a convex optimization over linear matrix inequalities

an analysis and design method for linear systems under nested saturation this paper considers a linear system under nested saturation nested saturation arises for example when the actuator is subject to magnitude and rate saturation simultaneously a condition is derived in terms of a set of auxiliary feedback gains for determining if a given ellipsoid is contractively invariant moreover this condition is shown to be equivalent to linear matrix inequalities lmis in the actual and auxiliary feedback gains as a result the estimation of the domain of attraction for a given set of feedback gains can be formulated as an optimization problem with lmi constraints by viewing the feedback gains as extra free parameters the optimization problem can be used for controller design

on smoothing of the crank nicolson scheme and higher order schemes for pricing barrier options most option pricing problems have nonsmooth payoffs or discontinuous derivatives at the exercise price discrete barrier options have not only nonsmooth payoffs but also time dependent discontinuities in pricing barrier options certain aspects are triggered if the asset price becomes too high or too low standard smoothing schemes used to solve problems with nonsmooth payoff do not work well for discrete barrier options because of discontinuities introduced in the time domain when each barrier is applied moreover these unwanted oscillations become worse when estimating the hedging parameters e g delta and gamma we have an improved smoothing strategy for the crank nicolson method which is unique in achieving optimal order convergence for barrier option problems numerical experiments are discussed for one asset and two asset problems time evolution graphs are obtained for one asset problems to show how option prices change with respect to time this smoothing strategy is then extended to higher order methods using diagonal m m pade main schemes under a smoothing strategy of using as damping schemes the 0 2m 1 subdiagonal pade schemes

cross monotonic cost sharing methods for connected facility location games we present cost sharing methods for connected facility location games that are cross monotonic and competitive and that recover a constant fraction of the cost of the constructed solution the novelty of this paper is that we use randomized algorithms and that we share the expected cost among the participating users as a consequence our cost sharing methods are simple and achieve attractive approximation ratios we also provide a primal dual cost sharing method for the connected facility location game with opening costs

a second order exponential time differencing scheme for non linear reaction diffusion systems with dimensional splitting abstract a second order l stable exponential time differencing etd method is developed by combining an etd scheme with approximating the matrix exponentials by rational functions having real distinct poles rdp together with a dimensional splitting integrating factor technique a variety of non linear reaction diffusion equations in two and three dimensions with either dirichlet neumann or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second order implicit explicit schemes an additional performance boost is gained through further use of basic parallelization techniques

stabilized explicit runge kutta methods for multi asset american options american derivatives have become very popular instruments in financial markets however they are more complicated to price than european options since at each time level we have to determine not only the option value but also whether or not it should be exercised several procedures have been proposed to dissolve these difficulties but they usually involve the solution of nonlinear partial differential equations pdes in the case of multi dimensional problems solving these equations is a very challenging task in this paper we propose stabilized explicit runge kutta serk methods to solve this kind of problems they can easily be applied to many different classes of problems with large dimensions and they have low memory demand since these methods are explicit they do not require algebra routines to solve large nonlinear systems associated to odes as for example lapack and blas packages or multigrid or iterative methods applied together with newton type algorithms and are especially well suited for the method of lines mol discretizations of parabolic pdes

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

higher order time stepping methods for time dependent reaction diffusion equations arising in biology abstract this paper demonstrates the use of higher order methods to solve some time dependent stiff pdes in the past the most popular numerical methods for solving system of reaction diffusion equations was based on the combination of low order finite difference method with low order time stepping method we extend in this report the compatibility of fourth order finite difference scheme in space coupled with fourth order time stepping methods such as imexlm4 imexpc4 imexrk4 and etdrk4 b in time for direct integration of reaction diffusion equations in one space dimension some interesting numerical anomaly phenomenons associated with steady state solutions of the examples chosen from the literature are well presented to address the naturally arising points and queries our findings have led to the understanding of pattern formation such as spiral waves and patchy structures as well as some spatio temporal dynamical structures

the locally extrapolated exponential time differencing lod scheme for multidimensional reaction diffusion systems this paper introduces the local extrapolation of first order locally one dimensional exponential time differencing scheme for numerical solution of multidimensional nonlinear reaction diffusion systems this novel scheme has the benefit of solving multidimensional problems in locally one dimensional fashion by implementing sequences of tridiagonal matrix solvers instead of solving a banded system the storage size needed for solving systems in higher dimensions with this scheme is similar to that needed for one spatial dimension systems the stability monotonicity and convergence of the locally extrapolated exponential time differencing scheme have been examined stability analysis shows that the scheme is strongly stable l stable and is particularly beneficial to nonlinear partial differential equations with irregular initial data or discontinuity involving initial and boundary conditions due to its ability to damp spurious oscillations caused by high frequency components in the solution the performance of the novel scheme has been investigated by testing it on a two dimensional schnakenberg model two and three dimensional brusselator models and a three dimensional enzyme kinetics of michaelis menten type reaction diffusion model numerical experiments demonstrate the efficiency accuracy and reliability of the scheme locally extrapolated lod exponential time differencing scheme is developed multidimensional nonlinear reaction diffusion systems are considered monotonicity and stability are analyzed the order of convergence is examined

brief paper output feedback design for saturated linear plants using deadzone loops in this paper we present an lmi based synthesis approach on output feedback design for input saturated linear systems by using deadzone loops algorithms are developed for minimizing the upper bound on the regional l 2 gain for exogenous inputs with l 2 norm bounded by a given value and for minimizing this upper bound with a guaranteed reachable set or domain of attraction the proposed synthesis approach will always lead to regionally stabilizing controllers if the plant is exponentially unstable to semi global results if the plant is non exponentially unstable and to global results if the plant is already exponentially stable where the only requirement on the linear plant is detectability and stabilizability the effectiveness of the proposed techniques is illustrated with one example