Lipschitz Matrix Research Articles

On a coupled system of fractional $ (p, q) $-differential equation with Lipschitzian matrix in generalized metric space

<abstract><p>This work is concerned with the study of the existing solution for the fractional $ (p, q) $-difference equation under first order $ (p, q) $-difference boundary conditions in generalized metric space. To achieve the solution, we combine some contraction techniques in fixed point theory with the numerical techniques of the Lipschitz matrix and vector norms. To do this, we first associate a matrix to a desired boundary value problem. Then we present sufficient conditions for the convergence of this matrix to zero. Also, we design some algorithms to use the computer for calculate the eigenvalues of such matrices and different values of $ (p, q) $-Gamma function. Finally, by presenting two numerical examples, we examine the performance and correctness of the proposed method. Some tables and figures are provided to better understand the issues.</p></abstract>

A Lipschitz Matrix for Parameter Reduction in Computational Science

We introduce the Lipschitz matrix: a generalization of the scalar Lipschitz constant for functions with many inputs. Among the Lipschitz matrices compatible a particular function, we choose the smallest such matrix in the Frobenius norm to encode the structure of this function. The Lipschitz matrix then provides a function-dependent metric on the input space. Altering this metric to reflect a particular function improves the performance of many tasks in computational science. Compared to the Lipschitz constant, the Lipschitz matrix reduces the worst-case cost of approximation, integration, and optimization; if the Lipschitz matrix is low-rank, this cost no longer depends on the dimension of the input, but instead on the rank of the Lipschitz matrix defeating the curse of dimensionality. Both the Lipschitz constant and matrix define uncertainty away from point queries of the function and by using the Lipschitz matrix we can reduce uncertainty. If we build a minimax space-filling design of experiments in the Lipschitz matrix metric, we can further reduce this uncertainty. When the Lipschitz matrix is approximately low-rank, we can perform parameter reduction by constructing a ridge approximation whose active subspace is the span of the dominant eigenvectors of the Lipschitz matrix. In summary, the Lipschitz matrix provides a new tool for analyzing and performing parameter reduction in complex models arising in computational science.

Robust Observer-Based Fault Estimation for Lipschitz Nonlinear Systems with State-Coupled Disturbance: A Dissipativity Approach

In this paper, a robust fault diagnosis method is proposed to estimate the states and faults simultaneously for a special class of Lipschitz nonlinear systems. In the model of system, fault is considered as a linear and additive function and disturbance as a nonlinear function coupled with the system states. An augmented system is constructed by forming a vector composed of states and faults, and a Luenberger observer is designed for fault estimation. The nonlinear function of state-coupled disturbance is replaced by a Lipschitz matrix. In order to reduce the conservatism of the problem, this matrix depends on both the states and disturbances. This approach attempts to attenuate the effects of the state-coupled disturbances using the dissipativity theory such that the existing problem finally leads to solving convex optimization problems. The necessary conditions for the existence of such an observer are expressed. Finally the performance of the proposed method is simulated on a robot, and the results indicate good performance of the proposed method.

Model Predictive Control of Nonlinear Discrete Time Systems with Guaranteed Stability

AbstractThis paper presents the design of a new robust model predictive control algorithm for nonlinear systems represented by a linear model with unstructured uncertainty. The linear model is obtained by linearizing the nonlinear system at an operating point and the difference between the nonlinear and linear model is considered as a Lipschitz nonlinear function. The controller is designed for the linear model, which fulfills the stabilization condition for the nonlinear term. Unlike previous studies that have not considered a valid Lipschitz matrix of nonlinear term in the design process, we propose an algorithm in this paper in which it is considered. Therefore, the closed loop stability of the nonlinear system is guaranteed. A novel SOS optimization problem to determine design parameters is introduced, which leads to improved closed‐loop performance in comparison to a trial and error tuning procedure. Furthermore, an algorithm is presented to enlarge the region of attraction for the nonlinear closed‐loop system. Stability is improved by checking some additional conditions if which the system may be unstable if not considered. The validity of the proposed algorithm is confirmed by examples.

Perturbations of vectorial coverings and systems of equations in metric spaces

E. R. Avakov, A. V. Arutyunov, S. E. Zhukovskiĭ, and E. S. Zhukovskiĭ studied the problem of Lipschitz perturbations of conditional coverings of metric spaces. Here we propose some extension of the concept of conditional covering to vector-valued mappings; i.e., the mappings acting in products of metric spaces. The idea is that, to describe a mapping, we replace the covering constant by the matrix of covering coefficients of the components of the vector-valued mapping with respect to the corresponding arguments. We obtain a statement on the preservation of the property of conditional and unconditional vectorial coverings under Lipschitz perturbations; the main assumption is that the spectral radius of the product of the covering matrix and the Lipschitz matrix is less than one. In the scalar case this assumption is equivalent to the traditional requirement that the covering constant be greater than the Lipschitz constant. The statement can be used to study various simultaneous equations. As applications we consider: some statements on the solvability of simultaneous operator equations of a particular form arising in the problems on n-fold coincidence points and n-fold fixed points; as well as some conditions for the existence of periodic solutions to a concrete implicit difference equation.

Construction of ${\mathcal L}^{{p}}$ -strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients and singular drifts on general open sets with absorption at the boundary. In this application elliptic regularity results imply the desired regularity assumptions.

Robust Adaptive Fault Estimation for a Class of Nonlinear Systems Subject to Multiplicative Faults

In this paper, a fault estimation problem for a class of nonlinear systems subject to multiplicative faults and unknown disturbances is investigated. Multiplicative faults usually mixed with system states and inputs can cause additional complexity in the design of fault estimator due to parameter changes within process. Especially for the nonlinear system corrupted with unknown disturbances, it is not an easy work to distinguish the real fault factor from the mixed term. Under the nonlinear Lipschitz condition, the proposed robust adaptive fault estimation approach not only estimates the multiplicative faults and system states simultaneously, but also extracts the real effect of the faults. Meanwhile, the effect of disturbances is restricted to an L 2 gain performance criteria which can be formulated into the basic feasibility problem of a linear matrix inequality (LMI). In order to reduce the conservatism of the proposed method, a relaxing Lipschitz matrix is introduced. Finally, an illustrative example is applied to verify the efficiency of the proposed robust adaptive estimation scheme.

On interval operators obtained by splitting the Lipschitz matrix

Interval operators of a function strip obtained by splitting the given Lipschitz matrix A = B − C are discussed. Using these operators we can construct a sequence of intervals which converges to a limit interval, if the set of solutions of a given function strip is not empty. Under certain conditions the limit interval is a fixed interval of the corresponding interval operator. The different fixed intervals and convergence factors for some splittings are determined.

Optimal Enclosure of a Generalized Zero Set of a Function Strip

An interval operator of a function strip with a special Lipschitz matrix is discussed. Using this operator we can construct a sequence of intervals which converges to a fixed interval, if the set of solutions is not empty. The fixed interval optimally encloses the fixed point set.

Optimal Inclusion of a Solution Set

An interval operator of a function strip with inverse-nonnegative Lipschitz matrix is discussed. Using this operator we can construct a sequence of intervals which converges to a limit interval, if the set of solutions of a given function strip is not empty. The limit interval optimally encloses the set of solutions.

Interval operators of a function of which the Lipschitz matrix is an interval M-matrix

There are several interval iterations by applying interval operators which supply an interval sequence including all solutions of an equationf(x)=0 in a given intervalX0 and which require the knowledge of an interval Lipschitz matrixL off. In this paper statements are made about existence and convergence in case thatL is an intervalM-matrix.

A remark about the convergence of interval sequences

By some interval iterations the condition σ(r)<1 will be assumed for the convergence of the corresponding interval sequence by which σ denotes the spectral radius ofr, andr is a nonnegative Lipschitz matrix. In this paper the theorems are extended for the case σ (r)=1.