balanced truncation model reduction for a class of descriptor systems with application to the oseen equations we discuss the computation of balanced truncation model reduction for a class of descriptor systems which include the semidiscrete oseen equations with time independent advection and the linearized navier stokes equations linearized around a steady state the purpose of this paper is twofold first we show how to apply standard balanced truncation model reduction techniques which apply to dynamical systems given by ordinary differential equations to this class of descriptor systems this is accomplished by eliminating the algebraic equation using a projection the second objective of this paper is to demonstrate how the important class of adi smith type methods for the approximate computation of reduced order models using balanced truncation can be applied without explicitly computing the aforementioned projection instead we utilize the solution of saddle point problems we demonstrate the effectiveness of the technique in the computation of reduced order models for semidiscrete oseen equations

an equilibrium existence theorem bewley 14 theorem l proved an infinite dimensional equilibrium existence theorem which is a significant extension of the classical finite dimensional theorem of arrow and debreu 11 the assumptions on technology and preferences are natural and applicable in a wide variety of cases the proof is based on a limit argument that makes direct use of the existence of equilibrium in the finite dimensional case this paper establishes the existence of equilibrium under assumptions which are essentially the same as those given by bewley with the additional assumption that the preference orderings of consumers are representable by real valued utility functions this approach is related to the welfare approach of negishi 9 and arrow and hahn 2 chap 51 in the finite dimensional case and simplifies the approach originally adopted by bewley 5 in addition to making clear the role played by each of the assumptions in establishing the existence of equilibrium this approach has the merit of constructing directly a certain real valued function that is maximised at an equilibrium a result that provides a powerful tool in the analysis of qualitative properties of an equilibrium a model of resource allocation in continuous time over an infinite horizon that may be viewed as an application of the model that follows is given in 181

analysis of the solution of the sylvester equation using low rank adi with exact shifts abstract the solution to a general sylvester equation a x x b g f with a low rank right hand side is analyzed quantitatively through the low rank alternating directional implicit method lr adi with exact shifts new bounds and perturbation bounds on x are obtained a distinguished feature of these bounds is that they reflect the interplay between the eigenvalue decompositions of a and b and the right hand side factors g and f numerical examples suggest that because of this inclusion of details new perturbation bounds are much sharper than the existing ones

krylov subspace based model reduction for a class of bilinear descriptor systems we consider model order reduction for bilinear descriptor systems using an interpolatory projection framework such nonlinear descriptor systems can be represented by a series of generalized linear descriptor systems also called subsystems by utilizing the volterra wiener approach rugh 1981 standard projection techniques for bilinear systems utilize the generalized transfer functions of these subsystems to construct an interpolating approximation however the resulting reduced order system may not match the polynomial parts of the generalized transfer functions this may result in an unbounded error in terms of h 2 or h norms in this paper we derive an explicit expression for the polynomial part of each subsystem by assuming a special structure of the bilinear system which reduces to an index 1 linear descriptor system or differential algebraic equation dae if the bilinear terms are zero this allows us to propose an interpolatory technique for bilinear daes which not only achieves interpolation but also retains the polynomial parts of the bilinear systems the approach extends the interpolatory technique for index 1 linear daes beattie and gugercin 2009 to bilinear daes numerical examples are used to illustrate the theoretical results

a differential algebraic riccati equation for applications in flow control we investigate existence and structure of solutions to quadratic control problems with semiexplicit differential algebraic constraints as they appear in the modeling of flows we introduce a decoupled representation of the state and the formal adjoint equations to identify the conditions for the existence of solutions we derive a differential algebraic riccati equation formulated in the original coefficients for the efficient numerical computation of the optimal solution

polynomials and operator orderings it is shown that there is an exact one to one correspondence between all possible sets of polynomials both orthogonal and nonorthogonal and rules for operator orderings operator orderings that are hermitian give polynomials with definite parity most of the standard classical orthogonal polynomials are associated with operator orderings that are not particularly simple however there is a special one parameter class of hermitian operator orderings that corresponds to a class of elegant but little known orthogonal polynomials called continuous hahn polynomials

discounted cost linear quadratic gaussian control for descriptor systems we consider the linear quadratic gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon based on recent results from the deterministic frame

lower closure problems with weak convergence conditions in a new perspective a transparent approach is taken towards the class of lower closure problems with weak convergence conditions for the derivatives a deparametrization procedure is formulated for the abstract lower closure problem of this class as well as for its variant in a control problem of lagrange type it is shown that if one follows this procedure the solution of the lower closure problem merely lies in proving that a certain modified lagrangian is a normal integrand a lower closure result is obtained that generalizes in itself all comparable results of 2a 6e f 7a 21 for control problems of lagrange type with uniform boundedness conditions on the controls a special approximate lagrangian can be formulated by which the deparametrization procedure yields results that are superior to those obtained previously 2b 7b c an important novelty in deriving the measurability properties of lagrangians is also introduced here it consists of the employment of the kunugui novikov stchegolkov proj

model reduction by moment matching for linear and nonlinear systems the model reduction problem for single input single output linear and nonlinear systems is addressed using the notion of moment a re visitation of the linear theory allows to obtain novel results for linear systems and to develop a nonlinear enhancement of the notion of moment this in turn is used to pose and solve the model reduction problem by moment matching for nonlinear systems to develop a notion of frequency response for nonlinear systems and to solve model reduction problems in the presence of constraints on the reduced model connections between the proposed results projection methods the covariance extension problem and interpolation theory are presented finally the theory is illustrated by means of simple worked out examples and case studies

the pontryagin maximum principle and transversality conditions for a class of optimal control problems with infinite time horizons this paper suggests some further developments in the theory of first order necessary optimality conditions for problems of optimal control with infinite time horizons we describe an approximation technique involving auxiliary finite horizon optimal control problems and use it to prove new versions of the pontryagin maximum principle special attention is paid to the behavior of the adjoint variables and the hamiltonian typical cases in which standard transversality conditions hold at infinity are described several significant earlier results are generalized

shadow prices in infinite horizon optimal control problems with dominating discounts we consider a nonlinear optimal control problem in which an integrated discounted utility index is maximized over an infinite time interval the problem statement is motivated by various optimization problems arising in economics assuming that the discount parameter dominates the growth rates in the state variables and in the gradient of the current utility we develop a version of the pontryagin maximum principle providing a complete set of necessary optimality conditions and also suggesting an analytic expression for the values of the adjoint variables often viewed as shadow prices in the economic literature we illustrate the proposed methodology by applying it to the problem of optimal capital accumulation for a stylized model of an enterprise

model order reduction with preservation of passivity non expansivity and markov moments a new model order reduction technique is presented which preserves passivity and non expansivity it is a projection based method which exploits the solution of linear matrix inequalities to generate a descriptor state space format which preserves positive realness and bounded realness in the case of both non singular and singular systems solving the linear matrix inequality can be replaced by equivalently solving an algebraic riccati equation which is known to be a more efficient approach a new algebraic riccati equation and a frequency inversion technique are also presented to specifically deal with the important singular case the preservation of markov moments is also guaranteed by the judicious choice of a projection matrix three pertinent examples comparing the present approach with positive real balanced truncation show the strength and accuracy of the present approach

steady state matching and model reduction for systems of differential algebraic equations the problem of model reduction for nonlinear differential algebraic systems is addressed using the notions of moment and of steady state response these notions are formally introduced for this class of systems and families of nonlinear differential algebraic reduced order models achieving moment matching with additional properties are presented stronger results for the special class of linear singular systems are provided two simple examples illustrate the proposed technique

structure preserving tangential interpolation for model reduction of port hamiltonian systems port hamiltonian systems result from port based network modeling of physical systems and are an important example of passive state space systems in this paper we develop a framework for model reduction of large scale multi input multi output port hamiltonian systems via tangential rational interpolation the resulting reduced model is a rational tangential interpolant that retains the port hamiltonian structure hence it remains passive we introduce an h 2 inspired algorithm for effective choice of interpolation points and tangent directions and present several numerical examples illustrating its effectiveness

linear quadratic gaussian balancing for model reduction of differential algebraic systems we generalise the model reduction method of linear quadratic gaussian balanced truncation to linear time invariant systems governed by differential algebraic equations the presented method relies on the solution of generalised algebraic riccati equations a simple a priori error bound in the gap metric is given

continuous hahn polynomials and the heisenberg algebra continuous hahn polynomials have surfaced in a number of somewhat obscure physical applications for example they have emerged in the description of two photon processes in hydrogen hard hexagon statistical mechanical models and clebsch gordan expansions for unitary representations of the lorentz group so 3 1 in this paper it is shown that there is a simple and elegant way to construct these polynomials using the heisenberg algebra

underdetermined systems of partial differential equations this paper examines underdetermined systems of partial differential equations in which the independent variables may be classical c numbers or even quantum operators one can view an underdetermined system as expressing the kinematic constraints on a set of dynamical variables that generate a lie algebra the arbitrariness in the general solution reflects the freedom to specify the dynamics of such a system

chained aggregation of singular systems extends the notion of aggregation to singular systems se a necessary and sufficient conditions for complete aggregability are given which reduce in the state space case e i to aoki s conditions a geometric quotient space analysis is given which uses a new result on the pencil induced in the quotient space by a regular matrix pencil a generalized chained aggregation algorithm is used to put the system into a hessenberg form which provides a reduced order model for the output dynamics it is shown that as in the state space situation chained aggregation corresponds to removal of the unobservable subspace it is also pointed out that the usual state space chained aggregation algorithm is very efficient and numerically unstable specializing the singular system chained aggregation algorithm to the state space case yields a better and more streamlined approach

index aware model order reduction for linear index 2 daes with constant coefficients a model order reduction method for index 2 differential algebraic equations daes is introduced which is based on the intrinsic differential equations and on the remaining algebraic constraints this extends the method introduced in a previous paper for index 1 daes this procedure is implemented numerically and the results show numerical evidence of its robustness over the traditional methods

singularities and moments of nonlinear systems the notions of eigenvalue pole and moment at a pole of a continuous time nonlinear time invariant system are studied eigenvalues and poles are first characterized in terms of invariant subspaces tools from geometric control theory are used to define nonlinear enhancements of these notions and to study their relationship with the solution of certain partial differential equations cascade decompositions and steady state impulse responses the theory is illustrated by means of worked out examples and its significance is demonstrated by solving the model reduction problem by moment matching at poles for nonlinear systems

involution analysis of the partial differential equations characterizing hamiltonian vector fields in a recent article certain underdetermined linear systems of partial differential equations connected with lie poisson structures have been studied they were constructed via power series solutions of the evolution equation for a given hamiltonian we extend the results to arbitrary poisson manifolds correct an error in the case of degenerate poisson structures and show that these linear systems simply characterize hamiltonian vector fields our basic tool is the formal theory of differential equations with its central concept of an involutive system

a unifying note on fatou s lemma in several dimensions a general version of fatou s lemma in several dimensions is presented it subsumes the fatou lemmas given by schmeidler schmeidler d 1970 fatou s lemma in several dimensions proc amer math soc 24 300 306 hildenbrand hildenbrand w 1974 core and equilibria of a large economy princeton university press princeton cesari suryanarayana cesari l m b suryanarayana 1978 an existence theorem for pareto problems nonlinear anal 2 225 233 and artstein artstein z 1979 a note on fatou s lemma in several dimensions j math econom 6 277 282 also it is equivalent to an abstract variational existence result that extends and generalizes results by aumann perles aumann r j m perles 1965 a variational problem arising in economics j math anal appl 11 488 503 berliocchi lasry berliocchi h j m lasry 1973 integrands normales et mesures parametrees en calcul des variations bull soc math france101 129 184 artstein artstein z 1974 on a variational problem j math anal appl 45 404 415 and balder balder e j 1979 on a useful compactification for optimal control problems j math anal appl 72 391 398 in several respects

model order reduction techniques with a posteriori error control for nonlinear robust optimization governed by partial differential equations we consider a nonlinear optimization problem governed by partial differential equations with uncertain parameters it is addressed by a robust worst case formulation the resulting optimization problem is of bilevel structure and is difficult to treat numerically we propose an approximate robust formulation that employs linear and quadratic approximations to speed up the computation reduced order models based on proper orthogonal decomposition in combination with a posteriori error estimators are developed the proposed strategy is then applied to the optimal placement of a permanent magnet in the rotor of a synchronous machine with moving rotor numerical results are presented to validate the presented approach

discounted value representations of temporal preferences this paper examines the proposition that temporal preferences may be represented in terms of discount functions an analytical framework is developed within which preferences are defined over a set of programs or prospects over time necessary and sufficient conditions for generalized discounting discounting as well as exponential discounting are established for a class of program sets over discrete and continuous time

overtaking optimal solutions for convex lagrange problems with time delay abstract in this paper we study infinite horizon convex problems of lagrange in which hereditary effects in the state are present the objective functional in these problems is an improper integral defined over the time interval 0 dictated by applications arising in mathematical economics we do not assumea priorithat these improper integrals converge and therefore we consider overtaking optimal solutions this concept arose in the economics literature in the 1960s we extend known results for the analogous nondelay problem providing sufficient conditions for the existence of an overtaking optimal solution our proof rests on the assumption that a particular delay differential inclusion enjoys a strong asymptotic stability property called property s examples are given to indicate the applicability of these results

sylvester equations and projection based model reduction in this paper we establish a connection between krylov subspace techniques for multipoint pade interpolation and the use of sylvester equations for constructing reduced order models we also briefly point out that this connection partly extends to adi type techniques and to the smith iteration for computing approximate solutions of lyapunov equations

positive real and bounded real balancing for model reduction of descriptor systems we present an extension of the positive real and bounded real balanced truncation model reduction methods to large scale descriptor systems these methods are based on balancing the solutions of the projected lur e matrix equations important properties of these methods are that respectively passivity and contractivity are preserved in the reduced order models and that there exist approximation error bounds we also discuss the numerical solution of the projected lur e equations numerical examples are given

existence results and finite horizon approximates for infinite horizon optimization problems the paper deals with infinite horizon optimization problems the existence of optimal solutions is obtained as a consequence of an asymptotic growth condition we also exhibit finite horizon approximates that yield upper and lower bounds for the optimal values and whose optimal solutions converge to the long term optimal trajectories

discrete time quantum mechanics abstract this paper summarizes a research program that has been underway for a decade the objective is to find a fast and accurate scheme for solving quantum problems which does not involve a monte carlo algorithm we use an alternative strategy based on the method of finite elements we are able to formulate fully consistent quantum mechanical systems directly on a lattice in terms of operator difference equations one advantage of this discretized formulation of quantum mechanics is that the ambiguities associated with operator ordering are eliminated furthermore the scheme provides an easy way in which to obtain the energy levels of the theory numerically a generalized version of this discretization scheme can be applied to quantum field theory problems the difficulties normally associated with fermion doubling are eliminated also one can incorporate local gauge invariance in the finite element formulation results for some field theory models are summarized in particular we review the calculation of the anomaly in two dimensional quantum electrodynamics the schwinger model finally we discuss nonabelian gauge theories

invariant variational problems on linear frame bundles the hamiltonian structure of variational problems defined by the natural basis jki of diff m invariant lagrangians on the 1 jet bundle of linear frames of a m dimensional manifold m is described diffeomorphism invariance on j1 fm and its infinitesimal counterpart i e invariance under the natural representation of vector fields of m are analysed the lagrangians jki are proved to be the basic tools required to factor diff m g invariance into diff m invariance and g invariance the densities jki jki 1 m where the i are the components of the canonical form are shown to define two types of variational problem according to whether i j k or i j k the field equations for their extremals are deduced these equations are examples of underdetermined non linear systems of partial differential equations extremals defining a lie algebra structure are characterized the functions in the real linear space spanned by jki are the only lagrangians on fm admitting a hamiltonian formalism of order zero infinitesimal symmetries and noether invariants of the densities jki are studied in detail in particular it is proved that the noether invariant of every vertical symmetry vanishes hence only the horizontal symmetries appear in the hamiltonian structure the equations of the jacobi fields along an extremal are explicitly obtained the pre symplectic structure attached to jki is defined to be an alternate bilinear map 2 s on the space of jacobi fields along an extremal s with values in closed m 1 forms on m and its kernel is related to vertical infinitesimal symmetries of the lagrangian for m 3 4 the equations of the extremals are integrated explicitly we thus obtain normal forms which when transformed by an arbitrary diffeomorphism yield the general solution to the field equations

efficient solution of large scale saddle point systems arising in riccati based boundary feedback stabilization of incompressible stokes flow we investigate numerical methods for solving large scale saddle point systems which arise during the feedback control of flow problems we focus on the instationary stokes equations that describe instationary incompressible flows for moderate viscosities after a mixed finite element discretization we get a differential algebraic system of differential index two j weickert navier stokes equations as a differential algebraic system preprint sfb393 96 08 department of mathematics chemnitz university of technology chemnitz germany 1996 to reduce this index we follow the analytic ideas of j p raymond siam j control optim 45 2006 pp 790 828 coupled with the projection idea of m heinkenschloss d c sorensen and k sun siam j sci comput 30 2008 pp 1038 1063 avoiding this explicit projection leads to solving a series of large scale saddle point systems in this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners b

a survey of projection based model reduction methods for parametric dynamical systems numerical simulation of large scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena however the inherent large scale nature of the models often leads to unmanageable demands on computational resources model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate yet accurately represent the original large scale system behavior model reduction of linear nonparametric dynamical systems has reached a considerable level of maturity as reflected by several survey papers and books however parametric model reduction has emerged only more recently as an important and vibrant research area with several recent advances making a survey paper timely thus this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods parametric model reduction targets the broad class of problems for wh

model reduction of descriptor systems by interpolatory projection methods in this paper we investigate an interpolatory projection framework for model reduction of descriptor systems with a simple numerical example we first illustrate that directly applying the subspace conditions from the standard state space settings to descriptor systems generically leads to unbounded mathcal h 2 or mathcal h infty errors due to the mismatch of the polynomial parts of the full and reduced order transfer functions we then develop modified interpolatory subspace conditions based on the deflating subspaces that guarantee a bounded error for the special cases of index 1 and index 2 descriptor systems we also show how to avoid computing these deflating subspaces explicitly while still enforcing interpolation the question of how to choose interpolation points optimally naturally arises as in the standard state space setting we answer this question in the framework of the mathcal h 2 norm by extending the iterative rational krylov algorithm to descriptor systems

families of moment matching based low order approximations for linear systems the problem of finding a framework for linear moment matching based approximation of linear systems is discussed it is shown that it is possible to define five equivalent notions of moments i e one from a complex s domain point of view two using krylov projections and two from a time domain perspective based on these notions classes of parameterized reduced order models that achieve moment matching are obtained we analyze the controllability and observability properties of the models belonging to each of the classes of reduced order models we find the subclasses of models of orders lower than the number of matched moments i e we find the sets of parameters that allow for pole zero cancellations to occur furthermore we compute the subclass of minimal order model s the minimal order is equal to half of the number of matched moments i e generically the largest number of possible pole zero cancellations is half of the number of matched moments finally we present classes of reduced order models that match larger numbers of moments based on interconnections between reduced order models

different interpretations of the improper integral objective in an infinite horizon control problem abstract we provide an example of a convex infinite horizon problem with a linear objective functional where the different interpretations of the improper integral 0 f t x t u t d t in either lebesgue or riemann sense lead to different but finite optimal values

lower semicontinuity of integral functionals with nonconvex integrands by relaxation compactification a new approach to the lower semicontinuity of integral functionals is presented by a topological embedding of the control and state spaces in the hilbert cube and a simultaneous relaxation of the control functions a powerful approach emerges whose main features include i a generalized convexity condition is imposed upon the integrand of which the classical convexity condition is a special case ii in the embedded setting the integrand can be supposed lipschitz continuous in control and state arguments without loss of generality iii convergence in measure of the trajectories metamorphoses into l 1 norm convergence in the embedded setting

optimal control problems involving optimal economic growth with infinite horizon abstract we examine the existence of optimal solutions for a very general optimal economic growth problem with unbounded planning time horizon a convexity hypothesis about a certain orientor field plays a crucial role in the proof then we conduct sensitivity analysis and determine the relation between optimal pairs and the initial capital accumulation finally we see what happens if the convexity hypothesis we have made no longer exists this leads us to the introduction of the relaxed system and its comparison with the original one

optimal solvers for pde constrained optimization optimization problems with constraints which require the solution of a partial differential equation arise widely in many areas of the sciences and engineering particularly in problems of design the solution of such pde constrained optimization problems is usually a major computational task here we consider simple problems of this type distributed control problems in which the 2 and 3 dimensional poisson problem is the pde the large dimensional linear systems which result from discretization and which need to be solved are of saddle point type we introduce two optimal preconditioners for these systems which lead to convergence of symmetric krylov subspace iterative methods in a number of iterations which does not increase with the dimension of the discrete problem these preconditioners are block structured and involve standard multigrid cycles the optimality of the preconditioned iterative solver is proved theoretically and verified computationally in several test cases the theoretical proof indicates that these approaches may have much broader applicability for other pdes

interpolatory h h model reduction abstract we introduce an approach to h model reduction that is founded on ideas originating in realization theory interpolatory h 2 optimal model reduction and complex chebyshev approximation within this new framework we are able to formulate a method that remains effective in large scale settings with the main cost dominated by sparse linear solves by employing loewner data driven partial realizations within each optimization cycle computationally demanding h norm calculations can be completely avoided several numerical examples illustrate that our approach will produce high fidelity reduced models consistently exhibiting better h performance than those produced by balanced truncation these models often are as good as and occasionally better than those models produced by optimal hankel norm approximation in all cases reduced models are produced at far lower cost than is possible either with balanced truncation or with optimal hankel norm approximation

an existence result for optimal economic growth problems abstract an existence result for optimal control problems of lagrange type with unbounded time domain is derived very directly from a corresponding result for problems with bounded time domain this subsumes the main existence result of r f baum j optim theory appl 19 1976 89 116 and has the existence results for optimal economic growth problems of s i takekuma j math econom 7 1980 193 208 and m j p magill econometrica 49 1981 679 711 j math anal appl 82 1981 66 74 as simple corollaries in addition a new notion of uniform integrability is used which coincides with the classical notion if the time domain is bounded