Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

Consistent approximations of the zeno behaviour in affine-type switched dynamic systems

This paper proposes a new theoretic approach to a specific interaction of continuous and discrete dynamics in switched control systems known as a Zeno behaviour. We study executions of switched control systems with affine structure that admit infinitely many discrete transitions on a finite time interval. Although the real world processes do not present the corresponding behaviour, mathematical models of many engineering systems may be Zeno due to the used formal abstraction. We propose two useful approximative approaches to the Zeno dynamics, namely, an analytic technique and a variational description of this phenomenon. A generic trajectory associated with the Zeno dynamics can finally be characterized as a result of a specific projection or/and an optimization procedure applied to the original dynamic model. The obtained analytic and variational techniques provide an effective methodology for constructive approximations of the general Zeno-type behaviour. We also discuss shortly some possible applications of the proposed approximation schemes

Static self-gravitating elastic bodies in Einstein gravity

We prove that given a stress-free elastic body there exists, for sufficiently small values of the gravitational constant, a unique static solution of the Einstein equations coupled to the equations of relativistic elasticity. The solution constructed is a small deformation of the relaxed configuration. This result yields the first proof of existence of static solutions of the Einstein equations without symmetries.Comment: 29 pages. Updated to conform with published version, typos fixe

Fixed-Time Stable Proximal Dynamical System for Solving MVIPs

In this paper, a novel modified proximal dynamical system is proposed to compute the solution of a mixed variational inequality problem (MVIP) within a fixed time, where the time of convergence is finite, and is uniformly bounded for all initial conditions. Under the assumptions of strong monotonicity and Lipschitz continuity, it is shown that a solution of the modified proximal dynamical system exists, is uniquely determined and converges to the unique solution of the associated MVIP within a fixed time. As a special case for solving variational inequality problems, the modified proximal dynamical system reduces to a fixed-time stable projected dynamical system. Furthermore, the fixed-time stability of the modified projected dynamical system continues to hold, even if the assumption of strong monotonicity is relaxed to that of strong pseudomonotonicity. Connections to convex optimization problems are discussed, and commonly studied dynamical systems in the continuous-time optimization literature follow as special limiting cases of the modified proximal dynamical system proposed in this paper. Finally, it is shown that the solution obtained using the forward-Euler discretization of the proposed modified proximal dynamical system converges to an arbitrarily small neighborhood of the solution of the associated MVIP within a fixed number of time steps, independent of the initial conditions. Two numerical examples are presented to substantiate the theoretical convergence guarantees.Comment: 12 pages, 5 figure

A Wiener-Hopf Dynamical System for Mixed Equilibrium Problems

We suggest and analyze dynamical systems associated with mixed equilibrium problems by using the resolvent operator technique. We show that these systems have globally asymptotic property. The concepts and results presented in this paper extend and unify a number of previously known corresponding concepts and results in the literature

From Point Particles to Gauge Field Theories: a Differential- Geometrical approach to the Structures of the Space of Solutions

Mención Internacional en el título de doctorPrograma de Doctorado en Ingeniería Matemática por la Universidad Carlos III de MadridPresidenta: Eva Miranda.- Secretaria: María Carmela Lombardo.- Vocales: Alberto Calabri.- Marco Castrillón López.- Fernando Falceto Blecua.- Katarzyna Grabowska.- María Edith Padrón Fernández.- Narciso Román Ro