long time versus steady state optimal control this paper analyzes the convergence of optimal control problems for an evolution equation in a finite time horizon 0 t toward the limit steady state ones as t to infty we focus on linear problems we first consider linear time independent finite dimensional systems and show that the optimal controls and states exponentially converge in the transient time as t tends to infinity to the ones of the corresponding steady state model for this to occur suitable observability assumptions need to be imposed we then extend the results to infinite dimensional systems including the linear heat and wave equations with distributed controls

source and boundary implementation in vector and scalar dgtd we summarize the boundary and source implementation for the several formulations of the discontinuous galerkin time domain method dgtd since dgtd with zeroth order scalar basis functions using the upwind flux coincides with the finite volume time domain fvtd many of the concepts developed for fvtd can be ported to dgtd in any of its different formulations scalar vector basis upwind centered flux numerical examples illustrate the different alternatives

on necessity and robustness of dissipativity in economic model predictive control in this paper we study a dissipativity property which was recently used in several results on economic model predictive control to ensure optimal operation of a system at steady state as well as stability in particular we first investigate whether this dissipativity property is not only sufficient but also necessary for optimal steady state operation in the most general case this is not true nevertheless under an additional controllability assumption we show that dissipativity is in fact necessary second we provide a robustness analysis of the dissipativity property with respect to changes in the constraint set which can result in a change in the considered supply rate

turnpike properties and strict dissipativity for discrete time linear quadratic optimal control problems we analyze the connection between turnpike behaviors and strict dissipativity properties for discrete time finite dimensional linear quadratic optimal control problems we first use strict dissipativity as a sufficient condition for the turnpike property next we characterize strict dissipativity and the newly introduced property of strict pre dissipativity in terms of the system matrices related to the linear quadratic problem these characterizations then lead to new necessary conditions for the turnpike properties under consideration and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities one of the key novelties which distinguishes the results in the present paper from earlier ones on linear quadratic optimal control problems is the consideration of state and input constraints

on turnpike and dissipativity properties of continuous time optimal control problems this paper investigates the relations between three different properties which are of importance in optimal control problems dissipativity of the underlying dynamics with respect to a specific supply rate optimal operation at steady state and the turnpike property we show in a continuous time setting that if along optimal trajectories a strict dissipation inequality is satisfied then this implies optimal operation at this steady state and the existence of a turnpike at the same steady state finally we establish novel converse turnpike results i e we show that the existence of a turnpike at a steady state implies optimal operation at this steady state and dissipativity with respect to this steady state we draw upon a numerical example to illustrate our findings

a hybrid pstd adi cfdtd method for mixed scale electromagnetic problems we propose a hybrid technique combining the pseudospectral time domain pstd method with the alternating direction implicit conformal finite difference time domain adi cfdtd method to solve 3 d mixed scale problems in computational electromagnetics a mixed scale problem contains both electrically large and relatively homogeneous regions and electrically small fine details thus poses a significant computational challenge to any single computational method if it is utilized alone in particular the adi cfdtd method is an unconditionally stable time domain method with second order spatial accuracy and allows the time step to be increased beyond the courant friedrichs levy limit it is suitable for electrically small problem structure details much smaller than a wavelength but is inefficient and suffers from large numerical errors for electrically large scale regions the pstd method on the other hand is accurate and efficient for regions with large relatively homogeneous materials but loses its efficiency for electrically small structures the hybrid pstd adi cfdtd method overcomes these disadvantages and is potentially more useful than the individual solvers the implementation details and numerical accuracy of this hybrid method are examined numerical examples demonstrate the advantages of the hybrid pstd adi cfdtd method

exponential sensitivity and turnpike analysis for linear quadratic optimal control of general evolution equations abstract we analyze the sensitivity of linear quadratic optimal control problems governed by general evolution equations with bounded or admissible control operator we show that if the problem is stabilizable and detectable the solution of the extremal equation can be bounded by the right hand side including initial data with the bound being independent of the time horizon consequently the influence of perturbations of the extremal equations decays exponentially in time this property can for example be used to construct efficient space and time discretizations for a model predictive control scheme furthermore a turnpike property for unbounded but admissible control of general semigroups can be deduced

a schwarz generalized eigen oscillation spectral element method gesem for 2 d high frequency electromagnetic scattering in dispersive inhomogeneous media in this paper we propose a parallel schwarz generalized eigen oscillation spectral element method gesem for 2 d complex helmholtz equations in high frequency wave scattering in dispersive inhomogeneous media this method is based on the spectral expansion of complex generalized eigen oscillations for the electromagnetic fields and the schwarz non overlapping domain decomposition iteration method the gesem takes advantages of a special real orthogonality property of the complex eigen oscillations and a new radiation interface condition for the system of equations for the spectral expansion coefficients numerical results validate the high resolution and the flexibility of the method for various materials

optimal neumann control for the 1d wave equation finite horizon infinite horizon boundary tracking terms and the turnpike property we consider a vibrating string that is fixed at one end with neumann control action at the other end we investigate the optimal control problem of steering this system from given initial data to rest in time t by minimizing an objective functional that is the convex sum of the l 2 norm of the control and of a boundary neumann tracking term we provide an explicit solution of this optimal control problem showing that if the weight of the tracking term is positive then the optimal control action is concentrated at the beginning and at the end of the time interval and in between it decays exponentially we show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term this implies that if the time t is large the optimal trajectory approximately consists of three arcs where the first and the third short time arcs are transient arcs and in the middle arc the optimal control and the corresponding state are exponentially close to 0 this is an example for a turnpike phenomenon for a problem of optimal boundary control if t infinite horizon time problem then only the exponentially decaying component of the control remains and the norms of the optimal control action and of the optimal state decay exponentially in time in contrast to this situation if the weight of the tracking term is zero and only the control cost is minimized then the optimal control is distributed uniformly along the whole interval 0 t and coincides with the control given by the hilbert uniqueness method

periodic optimal control dissipativity and mpc recent research has established the importance of strict dissipativity for proving stability of economic mpc in the case of an optimal steady state in many cases though steady state operation is not economically optimal and periodic operation of the system yields a better performance in this technical note we propose ways of extending the notion of strict dissipativity for periodic systems we prove that optimal p periodic operation and mpc stability directly follow similarly to the steady state case which can be seen as a special case of the proposed framework finally we illustrate the theoretical results with several simple examples

analysis and design of unconstrained nonlinear mpc schemes for finite and infinite dimensional systems we present a technique for computing stability and performance bounds for unconstrained nonlinear model predictive control mpc schemes the technique relies on controllability properties of the system under consideration and the computation can be formulated as an optimization problem whose complexity is independent of the state space dimension based on the insight obtained from the numerical solution of this problem we derive design guidelines for nonlinear mpc schemes which guarantee stability of the closed loop for small optimization horizons these guidelines are illustrated by a finite and an infinite dimensional example

a p adaptive method for electromagnetic wave propagation summary the discontinuous galerkin dg finite element method is used for the numerical solution of the three dimensional maxwell equations control of errors in the numerical level for the divergence free constraint of the magnetic field can be obtained through the use of divergence free vector bases in this work the so called perfectly hyperbolic ph formulation of the maxwell equations is employed to retain both divergence free magnetic field and in the presence of charges to satisfy the gauss constraint for the electric field at the numerical level for both approaches it is found that higher order approximations have favorable effect on the preservation of the divergence constraints and that the ph formulations retains these errors to a lower level it is shown that high order accuracy in space and time is achieved in unstructured meshes using implicit time marching for nonuniform meshes local resolution refinement is employed using p type adaptivity to ensure accurate electromagnetic wave propagation thus the potential of the method to reach the required higher resolution in anisotropic meshes and obtain accurate electromagnetic wave propagation with reduced computational effort is demonstrated this article is protected by copyright all rights reserved

turnpike theorem for an infinite horizon optimal control problem with time delay an optimal control problem for systems described by a special class of nonlinear differential equations with time delay is considered the cost functional adopted could be considered as an analogue of the terminal functional defined over an infinite time horizon the existence of optimal solutions as well as the asymptotic stability of optimal trajectories that is the turnpike property are established under some quite mild restrictions on the nonlinearities of the functions involved in the description of the problem such mild restrictions on the nonlinearities allowed us to apply these results to a blood cell production model

a pseudo spectral scheme for the incompressible navier stokes equations using unstructured nodal elements a pseudo spectral scheme for solving the incompressible navier stokes equations using unstructured nodal triangles is proposed efficient algorithms are developed with numerical evidence that indicates optimal rates of convergence can be achieved navier stokes simulations of kovasznay shear layer roll up and row past a cylinder are included to show comparisons between the different nodal sets considered and an alternative modal approach c 2000 academic press

a lyapunov function for economic optimizing model predictive control standard model predictive control mpc yields an asymptotically stable steady state solution using the following procedure given a dynamic model a steady state of interest is selected a stage cost is defined that measures deviation from this selected steady state the controller cost function is a summation of this stage cost over a time horizon and the optimal cost is shown to be a lyapunov function for the closed loop system in this technical note the stage cost is an arbitrary economic objective which may not depend on a steady state and the optimal cost is not a lyapunov function for the closed loop system for a class of nonlinear systems and economic stage costs this technical note constructs a suitable lyapunov function and the optimal steady state solution of the economic stage cost is an asymptotically stable solution of the closed loop system under economic mpc both finite and infinite horizons are treated the class of nonlinear systems is defined by satisfaction of a strong duality property of the steady state problem this class includes linear systems with convex stage costs generalizing previous stability results and providing a lyapunov function for economic mpc or mpc with an unreachable setpoint and a linear model a nonlinear chemical reactor example is provided illustrating these points

optimal control problems over large time intervals abstract the paper focuses on problems of optimal control over a large time interval with end states prescribed for a wide class of systems and performance indices an approximate solution is characterized by piecing together two solutions for two infinite time problems these solutions exhibit similar transient behaviour

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

an exponential turnpike theorem for dissipative discrete time optimal control problems we investigate the exponential turnpike property for finite horizon undiscounted discrete time optimal control problems without any terminal constraints considering a class of strictly dissipative systems we derive a boundedness condition for an auxiliary optimal value function which implies the exponential turnpike property two theorems illustrate how this boundedness condition can be concluded from structural properties like controllability and stabilizability of the control system under consideration

the existence of catching up optimal solutions for a class of infinite horizon control problems with time delay the optimal control of a system whose states are governed by a nonlinear autonomous volterra integrodifferential equation with unbounded time interval is considered specifically it is assumed that the delay occurs only in the state variable the results obtained extend those of brock and haurie math oper res 1 1976 pp 337 346 and leizarowitz math oper res 10 1985 pp 450 461 in particular it is shown that under appropriate hypotheses catching up optimal solutions asymptotically approach a unique optimal steady state and thus enjoy the so called turnpike property found in the economics literature by combining this result with an associated optimal control problem the desired existence result is obtained furthermore it is remarked that in addition to extending these earlier works to the time delay case the results presented below utilize convexity seminormality and growth hypotheses that in some cases are weaker than those encountered in the above mentioned papers

the solutions to some operator equations abstract in this paper we study the solvability of the operator equation axb bx a c in the general setting of the adjointable operators between hilbert c modules based on the generalized inverses of the associated operators we propose the necessary and sufficient conditions for the existence of a solution to this equation and obtain the general expression of the solution in the solvable case we apply the results to the study of the real positive and positive solutions to the operator equation axb c in the case that the underlying space is finite dimensional or the range of b is contained in that of a we propose new necessary and sufficient conditions for the existence of a positive solution to the operator equation axb c and derive new formula in each case for the general positive solution to this operator equation

dispersion analysis of finite difference and discontinuous galerkin schemes for maxwell s equations in linear lorentz media abstract in this paper we consider maxwell s equations in linear dispersive media described by a single pole lorentz model for electronic polarization we study two classes of commonly used spatial discretizations finite difference methods fd with arbitrary even order accuracy in space and high spatial order discontinuous galerkin dg finite element methods both types of spatial discretizations are coupled with second order semi implicit leap frog and implicit trapezoidal temporal schemes by performing detailed dispersion analysis for the semi discrete and fully discrete schemes we obtain rigorous quantification of the dispersion error for lorentz dispersive dielectrics in particular comparisons of dispersion error can be made taking into account the model parameters and mesh sizes in the design of the two types of schemes this work is a continuation of our previous research on energy stable numerical schemes for nonlinear dispersive optical media 6 7 the results for the numerical dispersion analysis of the reduced linear model considered in the present paper can guide us in the optimal choice of discretization parameters for the more complicated and nonlinear models the numerical dispersion analysis of the fully discrete fd and dg schemes for the dispersive maxwell model considered in this paper clearly indicate the dependence of the numerical dispersion errors on spatial and temporal discretizations their order of accuracy mesh discretization parameters and model parameters the results obtained here cannot be arrived at by considering discretizations of maxwell s equations in free space in particular our results contrast the advantages and disadvantages of using high order fd or dg schemes and leap frog or trapezoidal time integrators over different frequency ranges using a variety of measures of numerical dispersion errors finally we highlight the limitations of the second order accurate temporal discretizations considered

a new high order dispersive fdtd method for drude material with complex interfaces in this paper motivated by the needs of tracking the transient change in the regularity of the electromagnetic fields across a drude interface we propose a new maxwell drude formulation for transverse magnetic problems with inhomogeneous drude dispersive materials based on the auxiliary differential equation approach the proposed formulation couples the wave equation for the electric component with maxwell s equations for the magnetic components a new finite difference time domain fdtd algorithm is introduced for solving the proposed maxwell drude system in which the time dependent jump conditions across the drude interface are enforced through the matched interface and boundary mib method the proposed fdtd method achieves a second order of accuracy in solving drude interfaces with fluctuating curvatures and non smooth corners based on a simple yee lattice while it can be generalized to up to sixth order in dealing with a straight drude interface therefore the proposed fdtd method is more accurate and cost efficient than the classical fdtd methods for drude material with complex interfaces

discontinuous galerkin methods for dispersive and lossy maxwell s equations and pml boundary conditions in this paper we will present a unified formulation of discontinuous galerkin method dgm for maxwell s equations in linear dispersive and lossy materials of debye type and in the artificial perfectly matched layer pml regions an auxiliary differential equation ade method is used to handle the frequency dependent constitutive relations with the help of auxiliary polarization currents in the computational and pml regions the numerical flux for the dispersive lossy maxwell s equations with the auxiliary polarization current variables is derived various numerical results are provided to validate the proposed formulation

analysis of unconstrained nonlinear mpc schemes with time varying control horizon for discrete time nonlinear systems satisfying an exponential or finite time controllability assumption we present an analytical formula for a suboptimality estimate for model predictive control schemes without stabilizing terminal constraints based on our formula we perform a detailed analysis of the impact of the optimization horizon and the possibly time varying control horizon on stability and performance of the closed loop

a new efficient 3d discontinuous galerkin time domain dgtd method for large and multiscale electromagnetic simulations a new discontinuous galerkin time domain dgtd method for solving the 3d time dependent maxwell s equations via the electric field intensity e and magnetic flux density b fields is proposed for the first time it uses curl conforming and divergence conforming basis functions for e and b respectively with the same order of interpolation in this way higher accuracy is achieved at lower memory consumption than the conventional approach based on the field variables e and h the centered flux and riemann solver are both used to treat interfaces with non conforming meshes and both explicit runge kutta method and implicit crank nicholson method are implemented for time integration numerical examples for realistic cases will be presented to verify that the proposed method is a non spurious and efficient dgtd scheme

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

a high order accurate scheme for maxwell s equations with a generalized dispersive material model abstract a high order accurate scheme for solving the time domain maxwell s equations with a generalized dispersive material model is described the equations for the electric field are solved in second order form and a general dispersion model is treated with the addition of one or more polarization vectors which obey a set of auxiliary differential equations ade numerical methods are developed for both second order and fourth order accuracy in space and time the equations are discretized using finite differences and advanced in time with a single stage three level space time scheme which remains stable up to the usual explicit cfl restriction as proven using mode analysis because the equations are treated in their second order form there is no need for grid staggering and instead a collocated grid is used composite overlapping grids are used to treat complex geometries with boundary conforming grids and a high order upwind dissipation is added to ensure robust and stable approximations on overlapping grids numerical results in two and three space dimensions confirm the accuracy and stability of the new schemes

development of discontinuous galerkin methods for maxwell s equations in metamaterials and perfectly matched layers the discontinuous galerkin method has proved to be an accurate and efficient way to numerically solve many differential equations in this paper we extend this method to solve the time dependent maxwell s equations when metamaterials and perfectly matched layers are involved numerical results are presented to demonstrate that our method is not only simple to implement but also quite effective in solving maxwell s equations in complex media

locally divergence free discontinuous galerkin methods for the maxwell equations in this paper we develop the locally divergence free discontinuous galerkin method for numerically solving the maxwell equations the distinctive feature of the method is the use of approximate solutions that are exactly divergence free inside each element as a consequence this method has a smaller computational cost than that of the discontinuous galerkin method with standard piecewise polynomial spaces we show that in spite of this fact it produces approximations of the same accuracy we also show that this method is more efficient than the discontinuous galerkin method using globally divergence free piecewise polynomial bases finally a post processing technique is used to recover 2k 1 th order of accuracy when piecewise polynomials of degree k are used

sensitivity analysis of optimal control for a class of parabolic pdes motivated by model predictive control we analyze the sensitivity of the extremal equations that arise from the first order optimality conditions for time dependent optimization problems more specifically we consider parabolic pdes wi

mathematical analysis of ziolkowski s pml model with application for wave propagation in metamaterials abstract in this paper we investigate one perfectly matched layer pml model proposed by ziolkowski in 1999 various schemes for solving this pml model have been developed and have been shown to be effective in absorbing outgoing waves when the wave propagation in unbounded domain problem is reduced to a bounded domain problem however a rigorous analysis of this model is lacking in this paper we establish the stability of this pml model and propose a fully discrete finite element scheme to solve this model with edge elements discrete stability and optimal error estimate of this scheme are proved numerical results justifying the analysis and demonstrating the effectiveness of this pml model are presented

computation of electromagnetic scattering with a non conforming discontinuous spectral element method in this paper we solve electromagnetic scattering problems by approximating maxwell s equations in the time domain with a high order quadrilateral discontinuous spectral element method dsem the method is a collocation form of the discontinuous galerkin method for hyperbolic systems where the solution is approximated by a tensor product legendre expansion and inner products are replaced with gauss legendre quadratures to increase flexibility of the method we use a mortar element method to couple element faces mortars provide a means for coupling element faces along which the polynomial orders differ which allows the flexibility to choose the approximation order within an element by considering only local resolution requirements mortars also permit local subdivision of a mesh by connecting element faces that do not share a full side we present evidence showing that the convergence of the non conforming approximations is spectral along with examples of their use copyright 2001 john wiley sons ltd

on the relation between strict dissipativity and turnpike properties abstract for discrete time nonlinear systems we study the relation between strict dissipativity and so called turnpike like behavior in optimal control under appropriate controllability assumptions we provide several equivalence statements involving these two properties the relation of strict dissipativity to an exponential variant of the turnpike property is also studied