ON THE DISTURBANCE ATTENUATION FOR INPUT DELAY SYSTEMS

We consider boundary value problems with periodic boundary conditions for first-order linear systems of impulsive ordinary differential equations with unknown right-hand sides and jumps of solutions at the impulse points entering into the statement of these problems which are assumed to be subjected to some quadratic restrictions. From indirect noisy observations of their solutions on a finite system of intervals, we obtain the optimal, in certain sense, estimates of images of their right-hand sides under linear continuous operators. Under the condition that the unknown correlation functions of noises in observations belong to some special sets, it is established that such estimates and estimation errors are expressed explicitly via solutions of special periodic boundary value problems for linear impulsive systems of ordinary differential equations.

Analytic study on linear systems of fractional differential equations

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

Introduction. Kron [1] has developed a technique for tearing large, sparse linear systems of algebraic equations into smaller systems, then putting the solutions of these smaller systems together to form the solution of the original problem. This can result in a great savings over solving the whole system in one piece. The trick is to find where to tear so that on the one hand the remaining pieces are small and easily solvable, but on the other hand not so much violence is done to the equations so that the task of putting their solutions back together becomes too difncult. Kron's pro? cedure for solving the pieces and putting their solutions together is based upon representing systems of equations as electric networks. He makes his choice of where to tear from his insight into these networks and the physical processes the equations represent. In the hands of someone such as Kron who has an understanding of the methods and good insight into the prob? lem, this technique has had outstanding success. The widespread use of Kron's technique has perhaps been limited severely for lack of practitioners skilled in this art of determining effective ways of tearing specific problems. Many problems are too large to submit to such insight. It is the purpose of this paper to contribute to the development of procedures which can be performed on a computer for analyzing the structures of the systems of equations themselves as an aid in choosing how to break them up for easier solution. The methods developed here require only that we consider what variables occur in each equation, but not how they appear. Thus, the tech? niques are not limited to linear systems.

A method of establishing equivalent linear systems for classes of third-order nonlinear systems is presented along with the sufficient conditions on the non-linearities of the systems which will permit the determination of such equivalent systems. This equivalence is established through the use of differential transformation techniques involving transformations of both the dependent and independent variables.

A new technique for solving the large system of linear algebraic equations associated with implicit differencing of multidimensio nal partial differential equations is presented. The coefficient matrix of the equations is factored, and then approximations to certain terms in the matrix are obtained from series expansions. The resulting system of equations is solved easily. The method is developed and demonstrated using a simple representative two-dimensional equation. Very good results are obtained when one direction is dominant. MPLICIT finite-difference schemes for the solution of multidimensional partial differential equations are usually stable and therefore applicable to a large class of problems. However, they are difficult to implement and may require an excessive amount of computer storage and time. The long computing time arises from the need to solve the large system of linear algebraic equations that result from the differencing. The computing time can be reduced significantly by approximating the coefficient matrix of the linear equations with a matrix that produces a system of equations that are relatively easy to solve. Among such methods are the alternating direction method (ADI)l used by Beam and Warming2 and Stone's strongly implicit method,3 which has been tested by Linetal.4 In this paper, a new technique for solving the large system of linear algebraic equations associated with implict differencing of multidimensional partial differential equations is presented. This method, called the pseudo-elimination method (PE), is shown to be faster than Stone's method for certain problems. The method is directly applicable to linear and linearized nonlinear systems of parabolic or elliptic partial differential equations. In order to discuss the method, a simple linear partial differential equation will be used; however, it should be kept in mind that the PE method is applicable to much more complicated problems. The question of whether the method will work when applied to difficult problems is not addressed. The scope of this paper is limited to presenting the method and illustrating, via a simple problem, that the method has some merit and deserves further study.

CHAPTER NINE - SYSTEMS OF EQUATIONS AND INEQUALITIES

This chapter is devoted to the study of L p Lyapunov-type inequalities for linear systems of ordinary differential equations with different boundary conditions (which include the case of Neumann, Dirichlet, periodic, and antiperiodic boundary value problems) and for any constant p ≥ 1. Elliptic problems are also considered. As in the scalar case, the results obtained in the linear case are combined with Schauder fixed point theorem to provide several results about the existence and uniqueness of solutions for resonant nonlinear systems. In addition, we study the stable boundedness of linear periodic conservative systems. The proof uses in a fundamental way the nontrivial relation (proved in Chap. 2) between the best Lyapunov constants and the minimum value of some especial constrained or unconstrained minimization problems (depending on the considered problems are resonant or nonresonant, respectively).

Numerical use of {[formula omitted]}-approximation method thermoelastic anisotropic thin shell theory equations represented in a special form

Applications of the Lanczos method

In this paper, we deal with residual vector generation for fault detection problems in linear systems via unknown input observer (UIO) when the so-called observer matching condition is not satisfied. Based on the relative degree between unknown input and output, a vector of the auxiliary output is introduced so that the observer matching condition is satisfied with respect to the vector. Auxiliary outputs are related to the derivatives of measured signals. However, differentiation leads to excessive amplification of measurement noise. A dynamically equivalent configuration of linear systems is developed using successive integrations to avoid differentiation. As such, auxiliary outputs are constructed without differentiation. Then, the equivalent dynamic system and its corresponding auxiliary outputs are used to generate the residual vector via an exponentially converging UIO. Fault detection in the generated residual vector is also investigated. Finally, the effectiveness of the proposed method is shown via numerical simulation.

Uniform decoupling by restricted static state feedback for linear time-varying analytic systems

The vast majority of the occurrences of condition numbers in the study of linear systems of equations involve the normwise condition number κ(A). Almost invariably, the use of κ(A) is enough to provide a satisfying explanation of the phenomena observed in practice.The case of triangular systems of linear equations provides, in contrast, an example in which κ(A) turns out to be inadequate. Practitioners long observed that triangular systems of equations are generally solved to high accuracy in spite of being, in general, ill-conditioned. Thus, for instance, J.H. Wilkinson wrote: “In practice one almost invariably finds that if L is ill-conditioned, so that ∥L∥∥L −1∥≫1, then the computed solution of Lx=b (or the computed inverse) is far more accurate than [what forward stability analysis] would suggest.”A first goal in this chapter is to give a precise meaning to the feeling that triangular matrices are, in general, ill-conditioned. We prove that if \(L\in \mathbb {R}^{n\times n}\) is a lower-triangular matrix whose entries are independent standard Gaussian random variables, then \(\mathbb {E}(\log \kappa(L))=\varOmega(n)\). This yields an expected loss of precision satisfying $$\mathbb {E}\bigl(\mathsf {LoP}\bigl(L^{-1}b\bigr)\bigr)= \mathcal {O}(n). $$ Were the loss of precision in the solution of triangular systems to conform to this bound, we would not be able to accurately find these solutions. The reason we actually do find them can be briefly stated. The error analysis of triangular systems reveals that we may use a componentwise condition number Cw(L,b) instead of the normwise condition number. The second goal of this chapter is to prove that when L and \(b\in \mathbb {R}^{n}\) are drawn from standard Gaussian distributions, then we have \(\mathbb {E}(\log \mathsf {Cw}(L,b))=\mathcal {O}(\log n)\). This bound, together with some backward error analysis, yields bounds for \(\mathbb {E}(\mathsf {LoP}(L^{-1}b))\) that are much smaller than the one above, as well as closer to the loss of precision observed in practice.KeywordsTriangular Linear SystemNormwise Condition NumberComponentwise ConditionIndependent Standard Gaussian Random VariablesBackward Error AnalysisThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A discrete maximum principle is presented for linear algebraic systems with coefficients of arbitrary signs and the number of unknowns exceeding (in the general case) the number of equations. The presentation uses the formal language of linear algebra (without any notions from the theory of difference schemes). This formal algebraic approach allows one to reveal easily the existence or absence of data for realization of an extended maximum principle in any algorithm represented in the form of a system of linear algebraic equations. The theorems formulated here directly imply the absolute stability of solutions to systems of equations. Variants of homogeneous algebraic systems and systems of equations with a right-hand side are considered. Theoretical results are illustrated by examples of actual problems.

The construction of highresolution differenceschemes for hyperbolic equations has been addressedin numerous works (see [1, 2] and the referencestherein). Several methods are available for improvingthe accuracy of schemes in space. They can be basedon multipoint stencils; differential consequences ofthe original equations; compact approximations ofderivatives [3]; and combinations of grid functionsobtained on different grids, for example, the Richardson method [4, 5]. There are also approaches thatcombine these methods. For example, in the wellknown work [6] concerning gasdynamic computations, a fourthorder compact approximation and differential consequences of the original equation wereused to construct a fourthorder accurate scheme on atwopoint stencil for a thirdorder scalar nonlineardifferential equation. An analysis of the literature shows that [6] is apparently the first to construct a fourthorder accuratescheme on a twopoint stencil in space. Importantly,in the computation of highgradient flows on the basisof hyperbolic conservation laws [1, 2], twopoint compact schemes seem more promising than threepointones [3] because the former have the following important properties. First, twopoint stencil schemes preserve the order of accuracy in the transition from auniform to a nonuniform grid. Second, interpolationacross a discontinuity in compact twopoint schemescan be avoided by placing a grid node at the discontinuity point. The major elements of the technique of [6] for constructing difference schemes include the introductionof derivatives as additional unknown functions inorder to reduce highorder equations to systems offirstorder equations, the use of the integrointerpolation method, and the Simpson and Maclaurin quadrature rules. Note that in this technique replacing theSimpson and Maclaurin formulas by the trapezoidalrule yields secondorder accurate twopoint schemes.In [6, 7], a secondorder scheme on a twopointminimum stencil for hyperbolic equations wasobtained via the introduction of the derivative of theunknown function as an additional independent variable. In [8] the same technique was used to construct afourthorder scheme on a twopoint stencil for theheat equation. In the recent paper [9], higher orderaccurate schemes on a twopoint stencil were called,for brevity, bicompact. Below, we will follow this (inour view apt) terminology.In [10] the bicompact scheme of [6] for a thirdorder scalar differential equation was modified andextended to systems of secondorder nonlinear differential equations. This modified scheme was used in[11, 12] to solve inner and outer problems in viscousgas dynamics.In contrast to [6–12], for deriving bicompactschemes, we use, as auxiliary functions, primitives ofthe original unknown functions rather than their spatial derivatives. Importantly, the auxiliary functionsthus introduced are once more differentiable than theoriginal unknown functions. For example, if a gasdynamic variable is a piecewise continuous function,then its primitive is a piecewise smooth function. In this work, we construct unconditionally stablefourthorder accurate implicit bicompact schemes forlinear and nonlinear equations and systems of equations of the hyperbolic type written in conservativeform. Based on these schemes, the solution to a mixedCauchy problem [13] is determined by the runningcalculation method. The properties of these schemesare examined. Test computations confirm the accuracy and high quality of the resulting solutions.Below, we construct and analyze bicompactschemes for linear and nonlinear hyperbolic equationsand systems of equations wri tten in conservative form.