Stability and chaos control of regularized Prabhakar fractional dynamical systems without and with delay

In this paper, we studied the stabilization of nonlinear regularized Prabhakar fractional dynamical systems without and with time delay. We establish a Lyapunov stabiliy theorem for these systems and study the asymptotic stability of these systems without design a positive definite function V (without considering the fractional derivative of function V is negative). We design a linear feedback controller to control and stabilize the nonautonomous and autonomous chaotic regularized Prabhakar fractional dynamical systems without and with time delay. By means of the Lyapunov stability, we obtain the control parameters for these type of systems. We further present a numerical method to solve and analyze regularized Prabhakar fractional systems. Furthermore, by employing numerical simulation, we reveal chaotic attractors and asymptotic stability behaviors for four systems to illustrate the presented theorem.     

Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments that the preconditioned scheme combined with an explicit time integrator is unstable if the time step Δt does not satisfy the requirement to be as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to , M → 0, though producing unphysical results. A comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis is presented, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M^–2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. The theoretical results are afterwards confirmed by numerical experiments. AMS subject classification (2000) 35L65, 35C20, 76G25     

Positive definiteness in the numerical solution of Riccati differential equations

Summary. In this work we address the issue of integrating symmetric Riccati and Lyapunov matrix differential equations. In many cases -- typical in applications -- the solutions are positive definite matrices. Our goal is to study when and how this property is maintained for a numerically computed solution. There are two classes of solution methods: direct and indirect algorithms. The first class consists of the schemes resulting from direct discretization of the equations. The second class consists of algorithms which recover the solution by exploiting some special formulae that these solutions are known to satisfy. We show first that using a direct algorithm -- a one-step scheme or a strictly stable multistep scheme (explicit or implicit) -- limits the order of the numerical method to one if we want to guarantee that the computed solution stays positive definite. Then we show two ways to obtain positive definite higher order approximations by using indirect algorithms. The first is to apply a symplectic integrator to an associated Hamiltonian system. The other uses stepwise linearization. Received April 21, 1993     

Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L_∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization. AMS subject classification 65L10, 65L12, 65L15     

M-solid varieties generated by lattices

Following W. Taylor, we define an identity to be hypersatisfied by a variety V iff, whenever the operation symbols of V are replaced by arbitrary terms (of appropriate arity) in the operations of V, then the resulting identity is satisfied by V in the usual sense. Whenever the identity is hypersatisfied by a variety V, we shall say that is a hyperidentity of V, or a V hyperidentity. When the terms being substituted are restricted to a submonoid M of all the possible choices, is called an M-hyperidentity, and a variety V is M-solid if each identity is an M-hyperidentity. In this paper we examine the solid varieties whose identities are lattice M-hyperidentities. The M-solid varieties generated by the variety of lattices in this way provide new insight on the construction and representation of various known classes of non-commutative lattices. Received October 8, 1999; accepted in final form March 22, 2000.     

Quadratic forms on symmetry classes of tensors

Let V:₁,…,V_m be inner product spaces, and let L be a linear transformation on V₁ ?…?V_m which satisfies (Lz,z)=0 for every decomposable tensor z. It is known that if the field is the complex numbers, then (Lz,z)=0 for every z. This paper contains a short proof of this result, an extension of it to arbitrary symmetry classes of tensors, and an analysis of its failure when the field is the real numbers.     

Higher <Emphasis Type="Italic">K</Emphasis>-theory of group-rings of virtually infinite cyclic groups

 F.T. Farrell and L.E. Jones conjectured in [7] that Algebraic K-theory of virtually cyclic subgroups V should constitute `building blocks' for the Algebraic K-theory of an arbitrary group G. In [6], they obtained some results on lower K-theory of V. In this paper, we obtain results on higher K-theory of virtually infinite cyclic groups V in the two cases: (i) when V admits an epimorphism (with finite kernel) to the infinite cyclic group (see 2.1 and 2.2(a),(b)) and (ii) when V admits an epimorphism (with finite kernel) to the infinite dihedral group (see 3.1, 3.2, 3.3). Received: 18 April 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 19D35, 16S35, 16H05.     

Numerical investigation into the dependence of the Allen–Cahn equation on the free energy

In this paper, we study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond₂(V )) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for V and V^− 1, by which we prove that Cond\(_{2}(V)=\mathcal {O}(n^{2})\). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. A fast structure-exploiting algorithm is also designed for computing the spectral diagonalization of B. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.

     

Solution of nonstationary boundary layer problems by fastest descent method

We consider the formation of a boundary layer above a semi-infinite plate that moves with velocity V_t8=At ⁿ in a viscous incompressible fluid. The problem is solved by Meksyn's asymptotic method. First, second, and third approximation equations are obtained. A numerical calculation is performed for various n, and especially interesting results are obtained for –0.5 < n < 0. The numerical solution of the third approximation equation is consistent with other known results.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 51–55, 1988.     

Numerical solution to low rank perturbed Lyapunov equations by the sign function method

We investigate the numerical solution to a low rank perturbed Lyapunov equation A^TX + XA = W via the sign function method (SFM). The sign function method has been proposed to solve Lyapunov equations, see e.g. [1], but here we focus on a framework where the matrix A has a special structure, i. e. A = B + UCV^T , where B is a blockdiagonal matrix and UCV^T is a low rank perturbation. We show that this structure can be kept throughout the sign function iteration but the rank of the perturbation doubles per iteration. Therefore, we apply a low rank approximation to the perturbation in order to keep its numerical rank small. We compare the standard SFM with its structure preserving variant presented in this paper by means of numerical examples from viscously damped mechanical systems. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability results involving time-averaged Lyapunov function derivatives

The stability results which comprise the Direct Method of Lyapunov involve the existence of auxiliary functions (Lyapunov functions) endowed with certain definiteness properties. Although the Direct Method is very general and powerful, it has some limitations: there are dynamical systems with known stability properties for which there do not exist Lyapunov functions which satisfy the hypotheses of a Lyapunov stability theorem.In the present paper we identify a scalar switched dynamical system whose equilibrium (at the origin) has known stability properties (e.g., uniform asymptotic stability) and we prove that there does not exist a Lyapunov function which satisfies any one of the Lyapunov stability theorems (e.g., the Lyapunov theorem for uniform asymptotic stability). Using this example as motivation, we establish stability results which eliminated some of the limitations of the Direct Method alluded to. These results involve time-averaged Lyapunov function derivatives (TALFD’s). We show that these results are amenable to the analysis of the same dynamical systems for which the Direct Method fails. Furthermore, and more importantly, we prove that the stability results involving TALFD’s are less conservative than the results which comprise the Direct Method (which henceforth, we refer to as the classical Lyapunov stability results).While we confine our presentation to continuous finite-dimensional dynamical systems, the results presented herein can readily be extended to arbitrary continuous dynamical systems defined on metric spaces. Furthermore, with appropriate modifications, stability results involving TALFD’s can be generalized to discontinuous dynamical systems (DDS).     

Stochastic integral representations,stochastic derivatives and minimal variance hedging

The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Lévy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.     

On error bounds for the Gautschi-type exponential integrator applied to oscillatory second-order differential equations

Summary. This paper studies a numerical method for second-order oscillatory differential equations in which high-frequency oscillations are generated by a linear time- and/or solution-dependent part. For constant linear part, it is known that the method allows second-order error bounds independent of the product of the step-size with the frequencies and is therefore a long-time-step method. Most real-world problems are not of that kind and it is important to study more general equations. The analysis in this paper shows that one obtains second-order error bounds even in the case of a time- and/or solution-dependent linear part if the matrix is evaluated at averaged positions.Mathematics Subject Classification (2000): 65L05, 65L70Acknowledgement I am grateful to Marlis Hochbruck and Christian Lubich for helpful discussions on the subject.     

Robust pole placement controller design in LMI region for uncertain and disturbed switched systems

This paper concerns the state feedback control for continuous-time, disturbed and uncertain linear switched systems with arbitrary switching rules. The main result of this work consists in getting a LMI (Linear Matrix Inequalities) condition guaranteeing a robust pole placement according to some desired specifications. Then, external disturbance attenuation with a fixed rate according to the H_∞ criterion is ensured. This is obtained thanks to the existence of a common quadratic Lyapunov function for all sub-systems. Finally, an academic example illustrates the efficiency of the developed approach.     

Partitioning planar graphs: a?fast combinatorial approach for max-cut

The max-cut problem asks for partitioning the nodes V of a graph G=(V,E) into two sets (one of which might be empty), such that the sum of weights of edges joining nodes in different partitions is maximum. Whereas for general instances the max-cut problem is NP-hard, it is polynomially solvable for certain classes of graphs. For planar graphs, there exist several polynomial-time methods determining maximum cuts for arbitrary choice of edge weights. Typically, the problem is solved by computing a minimum-weight perfect matching in some associated graph. The most efficient known algorithms are those of Shih et al. (IEEE Trans. Comput. 39(5):694–697, 1990) and that of Berman et al. (WADS, Lecture Notes in Computer Science, vol. 1663, pp. 25–36, Springer, Berlin, 1999). The running time of the former can be bounded by O(|V|^\frac32log|V|)O(|V|^{\frac{3}{2}}\log|V|). The latter algorithm is more generally for determining T-joins in graphs. Although it has a slightly larger bound on the running time of O(|V|^\frac32(log|V|)^\frac32)a(|V|)O(|V|^{\frac{3}{2}}(\log|V|)^{\frac{3}{2}})\alpha(|V|), where α(|V|) is the inverse Ackermann function, it can solve large instances in practice.     

Symmetric collocation for unstructured nonlinear differential-algebraic equations of arbitrary index

Summary. We examine a class of symmetric collocation schemes for the solution of nonlinear boundary value problems for unstructured nonlinear systems of differential-algebraic equations with arbitrary index. We show that these schemes converge with the same orders as one would expect for ordinary differential equations. In particular, we show superconvergence for a special choice of the collocation points. We demonstrate the efficiency of the new approach with some numerical examples.Mathematics Subject Classification (2000): 65L10Revised version received November 21, 2003Supported by DFG research grant Ku964/4.Supported by DFG research grant Me790/11.     

Sets of Harmonicity for Finely Harmonic Functions

Given an open set U in R ⁿ (n3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions.     

Varietal isologism and covering groups

In this paper it will be shown that any two \bf\cal V-covering groups of a given group are V\bf\cal V-isologic with respect to the variety V\bf\cal V, which is a vast generalization of a result in B. Huppert (1967) and R. L. Griess JR (1973). We also give a criterion of existence of V\bf\cal V-covering groups for a V\bf\cal V-perfect group, and show that every automorphism of a given V\bf\cal V-perfect group G can be extended to an automorphism of the V\bf\cal V-covering G^* say, of G, this generalizes a result of J. L. Alperin and D. Gorenstein (1966), in the abelian variety.     

On Low Mach Number Preconditioning of Finite Volume Schemes

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments, as well as by analysis, that the preconditioned scheme yields a physically corrected pressure distribution and combined with an explicit time integrator it is stable if the time step Δt satisfies the requirement to be 𝒪(M ²) as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to Δt = 𝒪(M ),M → 0, though producing unphysical results. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zero cancellation for general rational matrix functions

The problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints. Firstly, the class of all factors that dislocate the zeros and feature minimal McMillan degree are derived. Further, necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being J-unitary, or J-inner, either with respect to the imaginary axis or to the unit circle. The main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial, proper or improper, rank deficient, with arbitrary poles and zeros including at infinity. A particular attention is paid to the numerically-sound construction of solutions by employing at each stage unitary transformations, reliable numerical algorithms for eigenvalue assignment and efficient Lyapunov equation solvers.