A discrete approximation framework for hereditary systems

A discrete approximation framework for initial-value problems involving certain classes of linear functional differential equations (FDE) of the retarded type is constructed. An equivalence between the FDE and abstract evolution equations (AEE) in an appropriately chosen Hilbert space is established. This equivalence is then employed in the development of discrete approximation schemes in which the infinite-dimensional AEE is replaced by a finite-dimensional system of difference equations. Convergence and rates of convergence are demonstrated via the properties of rational functions with operator arguments and both classical and recent results from linear semigroup theory. Two examples of families of approximation schemes which are included in the general framework and which may be implemented directly on high-speed computing machines are developed. A numerical study of examples which illustrates the application and feasibility of the approximation techniques in a variety of problems together with a summary and analysis of the numerical results are also included.     

Application of LQR techniques to the adaptive control of quasilinear parabolic PDEs

LQR problems for linear parabolic PDEs have been studied in detail in the literature for the past 3 to 4 decades. The solvability of feedback control problems for a large class of problems is well understood. In recent years numerical methods for the approximation of the corresponding Riccati operators have been developed. These methods are able to calculate the feedback operator directly and thus can compute the solutions to linear problems efficiently. Here we study the applicability of such techniques to the control of quasilinear equations via local linearization in an adaptive control setting. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Equivalent Conditions for the Solvability of Nonstandard and Singular LQ-Problems with Application to Parabolic Equations

We consider an optimal control problem with indefinite cost for an abstract model, which covers, in particular, parabolic systems in a general bounded domain. Necessary and sufficient conditions are given for the synthesis of the optimal control, which is given in terms of the Riccati operator arising from a nonstandard Riccati equation. The theory extends also a finite-dimensional frequency theorem to the infinite-dimensional setting. Applications include the heat equation with Dirichlet and Neumann controls, as well as the strongly damped Euler–Bernoulli and Kirchhoff equations with the control in various boundary conditions.     

Persistence of Invariant Manifolds for Nonlinear PDEs

We prove that under certain stability and smoothing properties of the semi-groups generated by the partial differential equations that we consider, manifolds left invariant by these flows persist under C ¹ perturbation. In particular, we extend well-known finite-dimensional results to the setting of an infinite-dimensional Hilbert manifold with a semi-group that leaves a submanifold invariant. We then study the persistence of global unstable manifolds of hyperbolic fixed points, and as an application consider the two-dimensional Navier–Stokes equation under a fully discrete approximation.Finally, we apply our theory to the persistence of inertial manifolds for those PDEs that possess them.     

On the approximation of spectra of linear operators on Hilbert spaces

We present several new techniques for approximating spectra of linear operators (not necessarily bounded) on an infinite-dimensional, separable Hilbert space. Our approach is to take well-known techniques from finite-dimensional matrix analysis and show how they can be generalized to an infinite-dimensional setting to provide approximations of spectra of elements in a large class of operators. We conclude by proposing a solution to the general problem of approximating the spectrum of an arbitrary bounded operator by introducing the n-pseudospectrum and argue how that can be used as an approximation to the spectrum.     

A theory of discretization for nonlinear evolution inequalities applied to parabolic Signorini problems

We present a discretization theory for a class of nonlinear evolution inequalities that encompasses time dependent monotone operator equations and parabolic variational inequalities. This discretization theory combines a backward Euler scheme for time discretization and the Galerkin method for space discretization. We include set convergence of convex subsets in the sense of Glowinski-Mosco-Stummel to allow a nonconforming approximation of unilateral constraints. As an application we treat parabolic Signorini problems involving the p-Laplacian, where we use standard piecewise polynomial finite elements for space discretization. Without imposing any regularity assumption for the solution we establish various norm convergence results for piecewise linear as well piecewise quadratic trial functions, which in the latter case leads to a nonconforming approximation scheme. Entrata in Redazione il 16 marzo 1998, in versione riveduta il 15 febbraio 1999.     

Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space

In an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.     

Pseudomonotone Variational Inequalities: Convergence of Proximal Methods

In this paper, we study the convergence of proximal methods for solving pseudomonotone (in the sense of Karamardian) variational inequalities. The main result is given in the finite-dimensional case, but we show that we still obtain convergence in an infinite-dimensional Hilbert space under a strong pseudomonotonicity or a pseudo-Dunn assumption on the operator involved in the variational inequality problem.     

Spectral representations and density operators for infinite-dimensional homogeneous random fields

A theory of spectral representations and spectral density operators of infinite-dimensional homogeneous random fields is established. Some results concerning the form of the spectral representation are given in the general infinite-dimensional case, while the results pertaining to the density operator are confined to Hilbert space valued fields. The concept of a purely non-deterministic (p.n.d.) field is defined, and necessary and sufficient conditions for the property of p.n.d. are obtained in terms of the spectral density operator. The theory is developed using some isomorphisms induced by families of self-adjoint operators in the linear second order space associated with the field. The method seems to lead to more direct results also in the random process case, and it sheds new light on concepts such as multiplicity of the field and rank of the spectral density operator.     

On Landweber iteration for nonlinear ill-posed problems in Hilbert scales

Summary. In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index for exact data and in terms of the noise level for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented confirming the theoretical ones. Received May 15, 1998 / Revised version received January 29, 1999 / Published online December 6, 1999     

A relaxed self-adaptive CQ algorithm for the multiple-sets split feasibility problem

The multiple-sets split feasibility problem (MSFP) is to find a point belongs to the intersection of a family of closed convex sets in one space, such that its image under a linear transformation belongs to the intersection of another family of closed convex sets in the image space. Many iterative methods can be employed to solve the MSFP. Jinling Zhao et al. proposed a modification for the CQ algorithm and a relaxation scheme for this modification to solve the MSFP. The strong convergence of these algorithms are guaranteed in finite-dimensional Hilbert spaces. Recently López et al. proposed a relaxed CQ algorithm for solving split feasibility problem, this algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, this algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed self-adaptive CQ algorithm for solving the MSFP where closed convex sets are level sets of some convex functions such that the strong convergence is guaranteed in the framework of infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results.     

Finite-dimensional linear approximations of solutions to general irregular nonlinear operator equations and equations with quadratic operators

A general scheme for improving approximate solutions to irregular nonlinear operator equations in Hilbert spaces is proposed and analyzed in the presence of errors. A modification of this scheme designed for equations with quadratic operators is also examined. The technique of universal linear approximations of irregular equations is combined with the projection onto finite-dimensional subspaces of a special form. It is shown that, for finite-dimensional quadratic problems, the proposed scheme provides information about the global geometric properties of the intersections of quadrics.     

Estimates for the rate of strong approximation in Hilbert space

We obtain infinite-dimensional corollaries of our recent results. We show that the finite-dimensional results imply meaningful estimates for the accuracy of strong Gaussian approximation of sums of independent identically distributed Hilbert space-valued random vectors with finite power moments. We establish that the accuracy of approximation depends substantially on the decay rate of the sequence of eigenvalues of the covariance operator of the summands.     

Linear rank and corank preserving maps on (H) and an application to *-semigroup isomorphisms of operator ideals

We characterize linear rank-k nonincreasing, rank-k preserving, and corank-k preserving maps on B(H), the algebra of all bounded linear operators on the Hilbert space H. This unifies and extends finite-dimensional results and results on linear rank-1 non-increasing and rank-1 preserving maps in the infinite-dimensional case. We conclude with an application to *-semigroup isomorphisms of operator ideals.     

Approximation of variational eigenvalue problems

A variational eigenvalue problem in an infinite-dimensional Hilbert space is approximated by a problem in a finite-dimensional subspace. We analyze the convergence and accuracy of the approximate solutions. The general results are illustrated by a scheme of the finite element method with numerical integration for a one-dimensional second-order differential eigenvalue problem. For this approximation, we obtain optimal estimates for the accuracy of the approximate solutions.     

Absolute stability criteria for infinite-dimensional discrete Lur'e systems with application to loaded distortionless electric RLCG-transmission line

We study absolute stability of a class of discrete Lur'e control system on an infinite-dimensional Hilbert space. Counterparts of the circle criterion and the Szegö criterion are derived using, respectively, quadratic and non-quadratic Lyapunov functionals. A link with existing finite-dimensional theory of absolute stability is shown. The results are illustrated by an example of the loaded distortionless electric RLCG-transmission line with a nonlinear static feedback. Its stability was previously investigated using the circle criterion for continuous infinite-dimensional systems with unbounded control and observation in the frame of systems in factor form.     

Semigroup approach for the approximation of a control problem with unbounded dynamics

In this paper, we approximate a control problem in an infinite-dimensional Hilbert space by means of a sequence of discrete problems. In the differential equation which describes the dynamics, a Lipschitz perturbation of an unbounded linear operator appears. We prove a convergence result of the approximation value functions to the value function of the original problem.This research was supported by Ministero dell'Università e della Ricerca Scientifica e Technologica and by Consiglio Nazionale delle Ricerche, Rome, Italy.     

Stabilization of a coupled transport-diffusion system with boundary input

We consider the problem of stabilizing a coupled transport-diffusion system with boundary input. The system is described by two linear transport-diffusion equations and is not asymptotically stable. In order to stabilize the system with boundary input, sensor influence functions are assumed to be located at interior of the domain. First, we formulate the system as an evolution equation with unbounded output operators in a Hilbert space, using variable transformation. Next, we derive a reduced-order model with a finite-dimensional state variable for the infinite-dimensional system. Then, a stabilizing controller is constructed for the reduced-order model under an additional assumption. It is shown that the finite-dimensional controller together with a residual mode filter plays a role of a finite-dimensional stabilizing controller for the original infinite-dimensional system, if the order of the residual mode filter is chosen sufficiently large. Finally, the validity of the design method is demonstrated through a numerical simulation.     

对边简支矩形薄板方程的算子半群方法

考虑弹性理论中对边简支矩形薄板方程,用算子半群方法求解问题.首先,将方程转换成抽象Cauchy问题.其次,构造空间框架并证明对应的算子矩阵生成压缩半群.最后,经Fourier变换,采用一致连续半群做逼近,进而给出对边简支矩形薄板方程的解析解.该方法自然蕴含着解的存在唯一性.     

Galerkin approximations with embedded boundary conditions for retarded delay differential equations

Finite-dimensional approximations are developed for retarded delay differential equations (DDEs). The DDE system is equivalently posed as an initial-boundary value problem consisting of hyperbolic partial differential equations (PDEs). By exploiting the equivalence of partial derivatives in space and time, we develop a new PDE representation for the DDEs that is devoid of boundary conditions. The resulting boundary condition-free PDEs are discretized using the Galerkin method with Legendre polynomials as the basis functions, whereupon we obtain a system of ordinary differential equations (ODEs) that is a finite-dimensional approximation of the original DDE system. We present several numerical examples comparing the solution obtained using the approximate ODEs to the direct numerical simulation of the original non-linear DDEs. Stability charts developed using our method are compared to existing results for linear DDEs. The presented results clearly demonstrate that the equivalent boundary condition-free PDE formulation accurately captures the dynamic behaviour of the original DDE system and facilitates the application of control theory developed for systems governed by ODEs.