When do Trajectories have Bounded Sensitivity to Cumulative Perturbations?

We investigate sensitivity to cumulative perturbations for a few dynamical system classes of practical interest. A system is said to have bounded sensitivity to cumulative perturbations (bounded sensitivity, for short) if an additive disturbance leads to a change in the state trajectory that is bounded by a constant multiple of the size of the cumulative disturbance. As our main result, we show that there exist dynamical systems in the form of (negative) gradient field of a convex function that have unbounded sensitivity. We show that the result holds even when the convex potential function is piecewise linear. This resolves a question raised in [1], wherein it was shown that the (negative) (sub)gradient field of a piecewise linear and convex function has bounded sensitivity if the number of linear pieces is finite. Our results establish that the finiteness assumption is indeed necessary. Among our other results, we provide a necessary and sufficient condition for a linear dynamical system to have bounded sensitivity to cumulative perturbations. We also establish that the bounded sensitivity property is preserved, when a dynamical system with bounded sensitivity undergoes certain transformations. These transformations include convolution, time discretization, and spreading of a system (a transformation that captures approximate solutions of a system)

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)

The implicit objective of the biennial "international - Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST) is to foster collaboration between international scientific teams by disseminating ideas through both specific oral/poster presentations and free discussions. For its second edition, the iTWIST workshop took place in the medieval and picturesque town of Namur in Belgium, from Wednesday August 27th till Friday August 29th, 2014. The workshop was conveniently located in "The Arsenal" building within walking distance of both hotels and town center. iTWIST'14 has gathered about 70 international participants and has featured 9 invited talks, 10 oral presentations, and 14 posters on the following themes, all related to the theory, application and generalization of the "sparsity paradigm": Sparsity-driven data sensing and processing; Union of low dimensional subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph sensing/processing; Blind inverse problems and dictionary learning; Sparsity and computational neuroscience; Information theory, geometry and randomness; Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?; Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website: http://sites.google.com/site/itwist1

Computational Techniques for Reachability Analysis of Max-Plus-Linear Systems ⋆

Abstract This work discusses a computational approach to reachability analysis of Max-Plus-Linear (MPL) systems, a class of discreteevent systems widely used in synchronization and scheduling applications. Given a set of initial states, we characterize and compute its "reach tube," namely the collection of set of reachable states (regarded step-wise as "reach sets"). By an alternative characterization of the MPL dynamics, we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of difference-bound matrices, and further derive worst-case bounds on the complexity of these operations. The approach is also extended to backward reachability analysis. The concepts and results are elucidated by a running example, and we further illustrate the performance of the approach by a numerical benchmark: the technique comfortably handles twenty-dimensional MPL systems (i.e., with twenty continuous state variables), and as such it outperforms the state-of-the-art alternative approaches in the literature

Análise de alcançabilidade em sistemas max plus incertos

Orientadores: Rafael Santos Mendes, Laurent Hardouin, Mehdi LhommeauTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Os Sistemas a Eventos Discretos (SEDs) constituem uma classe de sistemas caracterizada por apresentar espaço de estados discreto e dinâmica dirigida única e exclusivamente pela ocorrência de eventos. SEDs sujeitos aos problemas de sincronização e de temporização podem ser descritos em termos de equações lineares usando a álgebra max-plus. A análise de alcançabilidade visa o cálculo do conjunto de todos os estados que podem ser alcançados a partir de um conjunto de estados iniciais através do modelo do sistema. A análise de alcançabilidade de sistemas Max Plus Lineares (MPL) pode ser tratada por meio da decomposição do sistema MPL em sistemas PWA (Piece-Wise Affine) e de sua correspondente representação por DBM (Difference-Bound Matrices). A principal contribuição desta tese é a proposta de uma metodologia similar para resolver o problema de análise de alcançabilidade em sistemas MPL sujeitos a ruídos limitados, chamados de sistemas MPL incertos ou sistemas uMPL (uncertain Max Plus Linear Systems). Primeiramente, apresentamos uma metodologia para particionar o espaço de estados de um sistema uMPL em componentes que podem ser completamente representados por DBM. Em seguida, estendemos a análise de alcançabilidade de sistemas MPL para sistemas uMPL. Além disso, a metodologia desenvolvida é usada para resolver o problema de análise de alcançabilidade condicional, o qual esta estritamente relacionado ao cálculo do suporte da função de probabilidade de densidade envolvida no problema de filtragem estocásticaAbstract: Discrete Event Dynamic Systems (DEDS) are discrete-state systems whose dynamics are entirely driven by the occurrence of asynchronous events over time. Linear equations in the max-plus algebra can be used to describe DEDS subjected to synchronization and time delay phenomena. The reachability analysis concerns the computation of all states that can be reached by a dynamical system from an initial set of states. The reachability analysis problem of Max Plus Linear (MPL) systems has been properly solved by characterizing the MPL systems as a combination of Piece-Wise Affine (PWA) systems and then representing each component of the PWA system as Difference-Bound Matrices (DBM). The main contribution of this thesis is to present a similar procedure to solve the reachability analysis problem of MPL systems subjected to bounded noise, disturbances and/or modeling errors, called uncertain MPL (uMPL) systems. First, we present a procedure to partition the state space of an uMPL system into components that can be completely represented by DBM. Then we extend the reachability analysis of MPL systems to uMPL systems. Moreover, the results on reachability analysis of uMPL systems are used to solve the conditional reachability problem, which is closely related to the support calculation of the probability density function involved in the stochastic filtering problemDoutoradoAutomaçãoDoutor em Engenharia Elétrica164765/2013-199999.002340/2015-01CNPQCAPE

Penalty methods for the solution of generalized Nash equilibrium problems and hemivariational inequalities with VI constraints

In this thesis we propose penalty methods for the solution of Generalized Nash Equilibrium Problems (GNEPs) and we consider centralized and distributed algorithms for the solution of Hemivariational Inequalities (HVIs) where the feasible set is given by the intersection of a closed convex set with the solution set of a lower-level monotone Variational Inequality (VI)