necessary conditions for optimal control problems conjugate points in this paper we introduce a definition of normality and of conjugate points for a general optimal control problem using these concepts we obtain new second order necessary conditions for optimality in the special case when the control set u is the whole space or in the classical setting of calculus of variations our conditions reduce to known results namely the jacobi condition and the existence of a solution to a certain riccati equation

state local optimality of a bang bang trajectory a hamiltonian approach we give a sufficient condition for a bang bang extremal to be a strong local optimizer for the minimum time problem with fixed endpoints we underline that the conditions imply that the optimum is local with respect to the state and not necessarily to the final time moreover it is given through a finite dimensional minimization problem hence is suited for numerical verification a geometric interpretation through the projection of the hamiltonian flow on the state space is also given

semiconcavity of the value function for the bolza control problem in this paper we investigate the regularity of the value function of the bolza control problem we propose sufficient conditions for the value function to be semiconcave or locally lipschitz when the controls are unbounded

first and second order necessary conditions for control problems with constraints second order necessary conditions are developed for an abstract nonsmooth control problem with mixed state control equality and inequality constraints as well as a constraint of the form g x u e if where f is a closed convex set of a banach space with nonempty interior the inequality constraints g s x u 0 depend on a parameter s belonging to a compact metric space s the equality constraints are split into two sets of equations k x u 0 and h x u 0 where the first equation is an abstract control equation and h is assumed to have a full rank property in u the objective function is maxtet f t x u where t is a compact metric space f is upper semicontinuous in t and lipschitz in x u the results are in terms of a function a that disappears when the parameter spaces t and s are discrete we apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints thus we obtain a generalization and extension of the existing results on this problem

coupled points in the calculus of variations and applications to periodic problems the aim of this paper is to introduce the definition of coupled points for the problems of the calculus of variations with general boundary conditions and to develop second order necessary conditions for optimality when one of the end points is fixed our necessary conditions reduce to the known ones involving conjugate points we also apply our results to the periodic problems of the calculus of variations

keeping options open an optimal control model with trajectories that reach a dnss point in positive time the so called dnss points of indifference are of interest because they give decision makers in optimal control problems a choice between following either of two or more trajectories while still achieving optimality usually they are described in terms of initial conditions so that if the system starts at a dnss point the decision maker can proceed in either of two or more directions here we present a model that has an entire curve of indifference points away from which the decision maker can move in only one direction but does so by choosing either of two trajectories that initially coincide in the state space but later diverge approaching different long run steady states

a third order boundary value problem with jumping nonlinearities abstract a third order boundary value problem with jumping nonlinearities is studied some results obtained for second order boundary value problems with jumping nonlinearities are shown to hold also for the third order case

the influence of boundary data on the number of solutions of boundary value problems with jumping nonlinearities this paper contains results concerning the number of solutions as a function of the boundary data for boundary value problems with jumping nonlinearities an example seems to indicate that boundary data has a different influence on this number than does forcing data through approximating techniques this example leads to lower bounds on solution numbers for the more general case

accelerated multigrid convergence and high reynolds recirculating flows techniques are developed for accelerating multigrid convergence in general and for advection diffusion and incompressible flow problems with small viscosity in particular it is shown by analysis that the slowing down of convergence is due mainly to poor coarse grid correction to certain error components and means for dealing with this problem are suggested analyzed and tested by numerical experiments showing very significant improvement in convergence rates at little cost

a new wavelet multigrid method the standard multigrid procedure performs poorly or may break down when used to solve certain problems such as elliptic problems with discontinuous or highly oscillatory coefficients the method discussed in this paper solves this problem by using a wavelet transform and schur complements to obtain the necessary coarse grid interpolation and restriction operators a factorized sparse approximate inverse is used to improve the efficiency of the resulting method numerical examples are presented to demonstrate the versatility of the method

krylov subspace acceleration of nonlinear multigrid with application to recirculating flows this paper deals with the combination of two solution methods multigrid and gmres siam j sci comput 14 1993 pp 856 869 the generality and parallelizabili nonlinear pdes as the preconditioner for a nonlinear krylov subspace method we use the full approximation storage fas scheme math comp 31 1977 pp 333 390 a nonlinear multigrid method the nonlinear krylov acceleration is applied also on coarse grids so that recirculating incompressible flow problems discretized with a higher order upwind scheme can be solved efficiently

the conjugate point condition for smooth control sets abstract in this paper we present for optimal control problems with smooth control constraints second order necessary conditions stated in terms of conjugate points and a riccati type equation we show explicitly how to construct a set of feasible directions and how to derive the corresponding accessory problem

global analysis of two parameter elliptic eigenvalue problems on considere le probleme aux valeurs limites non lineaire lu f u 0 x u 0 x ou l est un operateur elliptique d ordre 2 et et sont des parametres on etudie les proprietes globales du continuum de solution de ces problemes quand et varient

towards adaptive smoothed aggregation alpha sa for nonsymmetric problems applying smoothed aggregation sa multigrid to solve a nonsymmetric linear system a mathbf x mathbf b is often impeded by the lack of a minimization principle that can be used as a basis for the coarse grid correction process this paper proposes a petrov galerkin pg approach based on applying sa to either of two symmetric positive definite spd matrices sqrt a t a or sqrt aa t these matrices however are typically full and difficult to compute so it is not computationally efficient to use them directly to form a coarse grid correction the proposed approach approximates these coarse grid corrections by using sa to accurately approximate the right and left singular vectors of a that correspond to the lowest singular value these left and right singular vectors are used to construct the restriction and interpolation operators respectively a preliminary two level convergence theory is presented suggesting that more relaxation should be applied than for an spd problem additionally a nonsymmetric version of adaptive sa alpha sa is given that automatically constructs sa multigrid hierarchies using a stationary relaxation process on all levels numerical results are reported for convection diffusion problems in two dimensions with varying amounts of convection for constant variable and recirculating convection fields the results suggest that the proposed approach is algorithmically scalable for problems coming from these nonsymmetric scalar pdes with the exception of recirculating flow this paper serves as a first step for nonsymmetric alpha sa the long term goal of this effort is to develop nonsymmetric alpha sa for systems of pdes where the sa framework has proven to be well suited for adaptivity in spd problems

time scale symplectic systems without normality abstract we present a theory of the definiteness nonnegativity and positivity of a quadratic functional f over a bounded time scale the results are given in terms of a time scale symplectic system s which is a time scale linear system that generalizes and unifies the linear hamiltonian differential system and discrete symplectic system the novelty of this paper resides in removing the assumption of normality in the characterization of the positivity of f and in establishing equivalent conditions for the nonnegativity of f without any normality assumption to reach this goal a new notion of generalized focal points for conjoined bases x u of s is introduced results on the piecewise constant kernel of x t are obtained and various picone type identities are derived under the piecewise constant kernel condition the results of this paper generalize and unify recent ones in each of the discrete and continuous time setting and constitute a keystone for further development in this theory

matrix dependent prolongations and restrictions in a blackbox multigrid solver multigrid methods are studied for the solution of linear systems resulting from the 9 point discretization of a general linear second order elliptic partial differential equation in two dimensions the rate of convergence of standard multigrid methods often deteriorates when the coefficients in the differential equation are discontinuous or when dominating first order terms are present these difficulties may be overcome by choosing the prolongation and restriction operators in a special way a novel way to do this is proposed as a result a blackbox solver written in standard fortran 77 has been developed numerical experiments for several hard test problems are described and comparison is made with other algorithms the standard mg method and a method introduced by kettler a significant improvement of robustness and efficiency is found

fast multigrid solution of the advection problem with closed characteristics the numerical solution of the advection diffusion problem in the inviscid limit with closed characteristics is studied as a prelude to an efficient high reynolds number flow solver it is demonstrated by a heuristic analysis and numerical calculations that using upstream discretization with downstream relaxation ordering in a multigrid cycle with appropriate residual weighting leads to an efficient solution process upstream finite difference approximations to the advection operator are derived whose truncation terms approximate physical laplacian viscosity thus avoiding spurious solutions to the homogeneous problem when the artificial diffusivity dominates the physical viscosity a brandt and i yavneh j comput phys 93 1991 pp 128 143

sufficient conditions for the generalized problem of bolza this paper presents sufficient conditions for strong local optimality in the generalized problem of bolza these conditions represent a unification in the sense that they can be applied to both the calculus of variations and to optimal control problems as well as problems with nonsmooth data also this paper brings to light a new point of view concerning the jacobi condition in the classical calculus of variations showing that it can be considered as a condition which guarantees the existence of a canonical transformation which transforms the original hamiltonian to a locally concave convex hamiltonian

contingent cones to reachable sets of control systems high order necessary conditions for optimality for an optimal control problem are studied via properties of contingent cones to reachable sets along the optimal trajectory it is shown that the adjoint vector of pontryagin s maximum principle is normal to the set of variations of reachable sets results are applied to study optimal control problems for dynamical systems described by 1 closed loop control systems 2 nonlinear implicit systems 3 differential inclusions 4 control systems with jumps

new second order optimality conditions for variational problems with c 2 hamiltonians the generalized problem of bolza with a c2 hamiltonian is considered necessary and sufficient conditions are obtained respectively in terms of the accessory problem the existence of conjoined basis and the existence of a solution to a riccati equation with boundary conditions no strengthened legendre clebsch condition is required when applied to a general optimal control problem these results include and generalize known results

convex optimal control problems with smooth hamiltonians optimal control problems with convex costs for which hamiltonians have lipschitz continuous gradients are considered examples of such problems including extensions of the linear quadratic regulator with hard and possibly state dependent control constraints and piecewise linear quadratic penalties are given lipschitz continuous differentiability and strong convexity of the terminal cost are shown to be inherited by the value function leading to lipschitz continuity of the optimal feedback with no regularity assumptions on the limiting problem epi convergence of costs which can be equivalently described by pointwise convergence of hamiltonians is shown to guarantee epi convergence of value functions resulting schemes of approximating any concave convex hamiltonian by continuously differentiable ones are displayed auxiliary results about existence and stability of saddle points of quadratic functions over polyhedral sets are also proved tools used are based on duality theory of convex and saddle functions

some characterizations of optimal trajectories in control theory several characterizations of optimal trajectories for the classical mayer problem in optimal control are provided for this purpose the regularity of directional derivatives of the value function is studied for instance it is shown that for smooth control systems the value function v is continuously differentiable along an optimal trajectory x t 0 1 to bf r n provided v is differentiable at the initial point t 0 x t 0 then the upper semicontinuity of the optimal feedback map is deduced the problem of optimal design is addressed obtaining sufficient conditions for optimality finally it is shown that the optimal control problem may be reduced to a viability one

least squares finite element methods and algebraic multigrid solvers for linear hyperbolic pdes least squares finite element methods lsfems for scalar linear partial differential equations pdes of hyperbolic type are studied the space of admissible boundary data is identified precisely and a trace theorem and a poincare inequality are formulated the pde is restated as the minimization of a least squares functional and the well posedness of the associated weak formulation is proved finite element convergence is proved for conforming and nonconforming discontinuous lsfems that are similar to previously proposed methods but for which no rigorous convergence proofs have been given in the literature convergence properties and solution quality for discontinuous solutions are investigated in detail for finite elements of increasing polynomial degree on triangular and quadrilateral meshes and for the general case that the discontinuity is not aligned with the computational mesh our numerical studies found that higher order elements yield slightly better convergence properties when measured in terms of the number of degrees of freedom standard algebraic multigrid methods that are known to be optimal for large classes of elliptic pdes are applied without modifications to the linear systems that result from the hyperbolic lsfem formulations they are found to yield complexity that grows only slowly relative to the size of the linear systems

on the number of solutions of two classes of sturm liouville boundary value problems abstract we employ global bifurcation theory of p h rabinowitz together with sturm comparison theorem to establish multiplicity results for some classes of sturm liouville boundary value problems

lower semicontinuous solutions of hamilton jacobi bellman equations the value function of mayer s problem arising in optimal control is investigated and lower semicontinuous solutions of the associated hamilton jacobi bellman equation are defined in three equivalent ways under quite weak assumptions about the control system the value function is the unique solution moreover it is stable with respect to perturbations of the control system and the cost it coincides with the viscosity solution whenever it is continuous

conjugate points and shocks in nonlinear optimal control in this paper the authors use the method of characteristics to extend the jacobi conjugate points theory to the bolza problem arising in nonlinear optimal control this yields necessary and sufficient optimality conditions for weak and strong local minima stated in terms of the existence of a solution to a corresponding matrix riccati differential equation the same approach allows to investigate as well smoothness of the value function

aggregation based algebraic multigrid for convection diffusion equations we consider the iterative solution of large sparse linear systems arising from the upwind finite difference discretization of convection diffusion equations the system matrix is then an m matrix with nonnegative row sum and further when the convective flow has zero divergence the column sum is also nonnegative possibly up to a small correction term we investigate aggregation based algebraic multigrid methods for this class of matrices a theoretical analysis is developed for a simplified two grid scheme with one damped jacobi postsmoothing step an uncommon feature of this analysis is that it applies directly to problems with variable coefficients e g to problems with recirculating convective flow on the basis of this theory we develop an approach in which a guarantee is given on the convergence rate thanks to an aggregation algorithm that allows an explicit control of the location of the eigenvalues of the preconditioned matrix some issues that remain beyond the analysis are discussed in the

on the regularity of semipermeable surfaces in control theory with application to the optimal exit time problem part i in control theory a semipermeable surface is an in general nonsmooth oriented surface that on one hand contains solutions the so called barrier solutions of the controlled system and on the other hand may be crossed by the solutions of this system in only one direction without making any assumption on the regularity of the boundary of the semipermeable surface we show that the barrier solutions contained in this semipermeable surface satisfy the pontryagin principle that this surface is a lipschitz manifold and that it is locally the graph of a semiconcave function applying these results to the optimal exit time function from a given open set yields without any controllability assumption at the boundary of the open set that this function is semiconcave on an open dense subset of its domain

inadequacy of first order upwind difference schemes for some recirculating flows spurious numerical solutions of problems with closed sub characteristics by upwind difference schemes in particular problems of recirculating incompressible flow at high reynolds numbers are proved to be due to the anisotropy of the artificial viscosity numerical examples are presented to show that even very simple problems including basic problems in fluid dynamics are not approximated well by schemes with anisotropic artificial viscosity regardless of numerical parameters 1991 academic press inc numerical methods for solving incompressible fluid flow equations are frequently tested on problems of recirculating flow quite often results obtained by different methods differ from each other significantly reliable experimental data is usually difficult to obtain and discrepancies between numerical two dimensional solutions and experimental data may be the result of the influence of walls in the experiments the problems in solving recirculating incompressible flows at high reynolds numbers are well known several researchers have pointed out that some of the methods employed to obtain stable discretization e g upwind differencing may lead to spurious results such results have been attributed to excessive artificial viscosity in the numerical scheme to multiple solutions of the non linear set of algebraic equations that is obtained from the numerical scheme and to poor resolution by grids that are too coarse 2 4 51 but spurious results occur even when none of these reasons are valid indeed a single linear partial differential equation may exhibit such behavior when the sub characteristics are closed this is true even when the mesh size is small enough to easily resolve the solution and truncation errors are genuinely small everywhere and even when the differential solution does not depend on the

navier stokes equations in rotation form a robust multigrid solver for the velocity problem the topic of this paper is motivated by the navier stokes equations in rotation form linearization and application of an implicit time stepping scheme results in a linear stationary problem of oseen type in well known solution techniques for this problem such as the uzawa or schur complement method a subproblem consisting of a coupled nonsymmetric system of linear equations of diffusion reaction type must be solved to update the velocity vector field in this paper we analyze a standard finite element method for the discretization of this coupled system and we introduce and analyze a multigrid solver for the discrete problem both for the discretization method and the multigrid solver the question of robustness with respect to the amount of diffusion and variation in the convection field is addressed we prove stability results and discretization error bounds for the galerkin finite element method we present a convergence analysis of the multigrid method which shows the robustness of the solver results of numerical experiments are presented which illustrate the stability of the discretization method and the robustness of the multigrid solver

non galerkin multigrid based on sparsified smoothed aggregation algebraic multigrid amg methods are known to be efficient in solving linear systems arising from the discretization of partial differential equations and other related problems these methods employ a hierarchy of representations of the problem on successively coarser meshes the coarse grid operators are usually defined by petrov galerkin coarsening which is a projection of the original operator using the restriction and prolongation transfer operators therefore these transfer operators determine the sparsity pattern and operator complexity of the multigrid hierarchy in many scenarios the multigrid operators tend to become much denser as the coarsening progresses such behavior is especially problematic in parallel amg computations where it imposes an expensive communication overhead in this work we present a new algebraic technique for controlling the sparsity pattern of the operators in the amg hierarchy independently of the choice of the restriction and prolongation our method is based on

on multiple solutions of a nonlinear neumann problem abstract lower bounds on the number of solutions to a certain class of nonlinear neumann boundary value problems are obtained as a function of boundary data and of the parameters involved with the nonlinear term

coupled points in optimal control theory the concept of coupled points is introduced for an optimal control problem where both state endpoints are allowed to vary this definition leads to the extension of the theory of conjugate points to the optimal control setting under suitable controllability assumptions weaker than those previously considered it is shown that the nonexistence of coupled points in the open interval a b is a necessary condition for weak local optimality this result generalizes the ones of the same kind known from the calculus of variations in the special case when one or both state endpoints are fixed the notion of coupled points is more general than those of focal or conjugate points

coarse grid correction for nonelliptic and singular perturbation problems the suitability of coarse grid correction operators for multigrid solution of discretized nonelliptic partial differential equations is studied it is found that the requirements for the orders of intergrid transfers and the order of approximation of the coarse grid operators differ in an important way from the well known rules pertaining to elliptic operators both galerkin coarsening and coarsening schemes based on the differential operator are examined and new coarse grid correction strategies are suggested in light of this analysis

second order sensitivity relations and regularity of the value function for mayer s problem in optimal control this paper investigates the value function v of a mayer optimal control problem with the state equation given by a differential inclusion first we obtain an invariance property for the proximal and frechet subdifferentials of v along optimal trajectories then we extend the analysis to the sub and superjets of v obtaining new sensitivity relations of second order by applying sensitivity analysis to exclude the presence of conjugate points we deduce that the value function is twice differentiable along any optimal trajectory starting at a point at which v is proximally subdifferentiable we also provide sufficient conditions for the local c 2 regularity of v on neighborhoods of optimal trajectories

equivalent subgradient versions of hamiltonian and euler llagrange equations in variational analysis much effort in recent years has gone into generalizing the classical hamiltonian and euler lagrange equations of the calculus of variations so as to encompass problems in optimal control and a greater variety of integrands and constraints these generalizations in which nonsmoothness abounds and gradients are systematically replaced by subgradients have succeeded in furnishing necessary conditions for optimality that reduce to the classical ones in the classical setting but important issues have remained unsettled especially concerning the exact relationship of the subgradient versions of the hamiltonian equations versus those of the euler lagrange equations here it is shown that new tighter subgradient versions of these equations are actually equivalent to each other the theory of epi convergence of convex functions provides the technical basis for this development

hamiltonian analysis of the generalized problem of bolza on obtient des conditions necessaires en forme hamiltonienne sous des hypotheses moins exigeantes qu anterieurement en particulier sans qualification sur les contraintes on developpe le lien avec les problemes de commande optimale ainsi que l apport de ces conditions a la theorie de la regularite de la solution

oscillation theorems and rayleigh principle for linear hamiltonian and symplectic systems with general boundary conditions the aim of this paper is to establish the oscillation theorems rayleigh principle and coercivity results for linear hamiltonian and symplectic systems with general boundary conditions i e for the case of separated and jointly varying endpoints and with no controllability normality and strong observability assumptions our method is to consider the time interval as a time scale and apply suitable time scales techniques to reduce the problem with separated endpoints into a problem with dirichlet boundary conditions and the problem with jointly varying endpoints into a problem with separated endpoints these more general results on time scales then provide new results for the continuous time linear hamiltonian systems as well as for the discrete symplectic systems this paper also solves an open problem of deriving the oscillation theorem for problems with periodic boundary conditions furthermore the present work demonstrates the utility and power of the analysis on time scales in obtaining new results especially in the classical continuous and discrete time theories

on multigrid solution of high reynolds incompressible entering flows an approach is presented for effectively separating the solution process of the elliptic component of high reynolds incompressible steady entering flow for which classical multigrid techniques are well suited from that of the non elliptic part for which other methods are more effective it is shown by analysis and numerical calculations that such an approach is very effective in terms of asymptotic convergence as well as reduction of errors well below discretization level in a 1 fmg algorithm

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

pointwise second order necessary optimality conditions and second order sensitivity relations in optimal control abstract this paper is devoted to pointwise second order necessary optimality conditions for the mayer problem arising in optimal control theory we first show that with every optimal trajectory it is possible to associate a solution p of the adjoint system as in the pontryagin maximum principle and a matrix solution w of an adjoint matrix differential equation that satisfy a second order transversality condition and a second order maximality condition these conditions seem to be a natural second order extension of the maximum principle we then prove a jacobson like necessary optimality condition for general control systems and measurable optimal controls that may be only partially singular and may take values on the boundary of control constraints finally we investigate the second order sensitivity relations along optimal trajectories involving both p and w

extended jacobi sufficiency criterion for optimal control in this paper we present sufficient conditions for strong local minimality in optimal control we do not require the dynamic f to be linear nor the control set u to be open moreover these conditions extend a known sufficiency criterion in the calculus of variations involving the jacobi condition