Abstract. A novel method for approximating structured singular values (also known as µ-values) is proposed and investigated. These quantities constitute an important tool in the stability analysis of uncertain linear control systems as well as in structured eigenvalue perturbation theory. Our approach consists of an inner-outer iteration. In the outer iteration, a Newton method is used to adjust the perturbation level. The inner iteration solves a gradient system associated with an optimization problem on the manifold induced by the structure. Numerical results and comparison with the well-known Matlab function mussv, implemented in the Matlab Control Toolbox, illustrate the behavior of the method.Key words. Structured singular value, µ-value, spectral value set, block diagonal perturbations, stability radius, differential equation, low-rank matrix manifold. AMS subject classifications. 15A18, 65K051. Introduction. The structured singular value (SSV) [14] is an important and versatile tool in control, as it allows to address a central problem in the analysis and synthesis of control systems: To quantify the stability of a closed-loop linear time-invariant systems subject to structured perturbations. The class of structures addressed by the SSV is very general and allows to cover all types of parametric uncertainties that can be incorporated into the control system via real or complex linear fractional transformations. We refer to [1,3,4,8,9,10,14,17,20] and the references therein for examples and applications of the SSV.The versatility of the SSV comes at the expense of being notoriously hard, in fact NP hard [2], to compute. Algorithms used in practice thus aim at providing upper and lower bounds, often resulting in a coarse estimate of the exact value. An upper bound of the SSV provides sufficient conditions to guarantee robust stability, while a lower bound provides sufficient conditions for instability and often also allows to determine structured perturbations that destabilize the closed loop linear system.The widely used function mussv in the Matlab Control Toolbox computes an upper bound of the SSV using diagonal balancing / LMI techniques [19,5]. The lower bound is computed by a generalization of the power method developed in [18,15]. This algorithm resembles a mixture of the power methods for computing the spectral radius and the largest singular value, which is not surprising, since the SSV can be viewed as a generalization of both. When the algorithm converges, a lower bound of the SSV results and this is always an equilibrium point of the iteration. However, in contrast to the standard power method, there are, in general, several stable equilibrium points and not all of them correspond to the SSV. In turn, one cannot guarantee convergence to the exact value but only to a lower bound. We remark that, despite this drawback, mussv is a very reliable and powerful routine, which reflects the state of the art in the approximation of the SSV.

In this paper, we study the Fokas–Lenells (F-L) equation in the presence of perturbation terms and we construct exact solutions. The modified extended direct algebraic method (MEDAM) is utilized, and soliton solutions, solitary wave solutions and elliptic function solutions are achieved. The physical meaning of the geometrical shapes for some of the obtained results is investigated for various choices of the free parameters that appear in the results. The stability of the model is investigated by using the standard linear stability analysis, which verifies that the exact solutions are stable. The achieved solutions and computational work show that the current method is powerful and influential.

In this article, the computation of µ-values known as Structured SingularValues SSV for the companion matrices is presented. The comparison of lower bounds with the well-known MATLAB routine mussv is investigated. The Structured Singular Values provides important tools to analyze the stability and instability analysis of closed loop time invariant systems in the linear control theory as well as in structured eigenvalue perturbation theory.

<abstract><p>In this article, we investigate some necessary and sufficient conditions required for the existence of solutions for mABC-fractional differential equations (mABC-FDEs) with initial conditions; additionally, a numerical scheme based on the the Lagrange's interpolation polynomial is established and applied to a dynamical system for the applications. We also study the uniqueness and Hyers-Ulam stability for the solutions of the presumed mABC-FDEs system. Such a system has not been studied for the mentioned mABC-operator and this work generalizes most of the results studied for the ABC operator. This study will provide a base to a large number of dynamical problems for the existence, uniqueness and numerical simulations. The results are compared with the classical results graphically to check the accuracy and applicability of the scheme.</p></abstract>