Linear stationary control systems over a Boolean semiring: Geometric properties and the isomorphism problem

In the present paper, we consider linear stationary dynamical systems over a Boolean semiring B. We analyze the complete observability, identifiability, reachability, and controllability of such systems. We define the notion of a “graph of modules” of completely controllable, completely reachable Boolean linear stationary systems by analogy with the spaces of modules in the case of systems over fields. We give a graph-theoretic interpretation of systems of this class. We solve the isomorphism problem in this class of systems.     

Refinement equations with nonnegative coefficients

In this paper we analyze solutions of the n-scale functional equation Ф(x) = Σ_k∈ℤ ^Pk Ф(nx−k), where n≥2 is an integer, the coefficients {P_k} are nonnegative and Σpk = 1. We construct a sharp criterion for the existence of absolutely continuous solutions of bounded variation. This criterion implies several results concerning the problem of integrable solutions of n-scale refinement equations and the problem of absolutely continuity of distribution function of one random series. Further we obtain a complete classification of refinement equations with positive coefficients (in the case of finitely many terms) with respect to the existence of continuous or integrable compactly supported solutions.     

Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization

We consider the wave equation on an interval of length 1 with an interior damping at ξ. It is well-known that this system is well-posed in the energy space and that its natural energy is dissipative. Moreover, as it was proved in Ammari et al. (Asymptot Anal 28(3–4):215–240, 2001), the exponential decay property of its solution is equivalent to an observability estimate for the corresponding conservative system. In this case, the observability estimate holds if and only if ξ is a rational number with an irreducible fraction x = \fracpq,\xi=\frac{p}{q}, where p is odd, and therefore under this condition, this system is exponentially stable in the energy space. In this work, we are interested in the finite difference space semi-discretization of the above system. As for other problems (Zuazua, SIAM Rev 47(2):197–243, 2005; Tcheugoué Tébou and Zuazua, Adv Comput Math 26:337–365, 2007), we can expect that the exponential decay of this scheme does not hold in general due to high frequency spurious modes. We first show that this is indeed the case. Secondly we show that a filtering of high frequency modes allows to restore a quasi exponential decay of the discrete energy. This last result is based on a uniform interior observability estimate for filtered solutions of the corresponding conservative semi-discrete system.     

Linear stationary control systems over the Boolean semiring and their graphs of modules

We consider linear stationary dynamical systems over the Boolean semiring. We analyze the properties of complete observability, identifiability, attainability, and controllability of a system. We define the notion of the “graph of modules” of totally controllable totally attainable Boolean linear stationary systems by analogy with spaces of modules in the case of systems over fields. The above-mentioned graph is described in the simplest case of one-dimensional inputs and outputs. We prove the weak connectedness of this oriented graph.     

Multitime Controllability,Observability and Bang-Bang Principle

Control problems for multitime first-order PDE arise in many different contexts and ways. The obstruction of complete integrability conditions (path independent curvilinear integrals) has determined the mathematicians to study such problems only in the discrete context, though thus they loose the geometrical character which is proper to the continuous approach. In this paper, we study controllability, observability and bang-bang properties of multitime completely integrable autonomous linear PDE systems, overcoming the existent mathematical prejudices regarding the importance of a multitime evolution of m-flow type. Our geometrical arguments show that each basic theorem has a correspondent in the case of a single-time linear controlled ODE system. The main results include controllability criteria, equivalence between controllability of a PDE system and observability of the dual PDE system, geometry of the control set, extremality and multitime bang-bang principle. All of these show that the passing from controlled single-time evolution (1-flow) to the controlled multitime evolution (m-flow) is not trivial. Changing the geometrical language, the case of nonholonomic evolution can be recovered easily from our theory.     

Impulsive non‐autonomous dynamical systems and impulsive cocycle attractors

In this work, we define the notions of ‘impulsive non‐autonomous dynamical systems’ and ‘impulsive cocycle attractors’. Such notions generalize (we will see that not in the most direct way) the notions of autonomous dynamical systems and impulsive global attractors in the current published literature. We also establish conditions to ensure the existence of an impulsive cocycle attractor for a given impulsive non‐autonomous dynamical system, which are analogous to the continuous case. Moreover, we prove the existence of such attractor for a non‐autonomous 2D Navier–Stokes equation with impulses, using energy estimates. Copyright © 2016 John Wiley & Sons, Ltd.     

Observability of continuous Petri nets with infinite server semantics

Timed continuous Petri net (contPN) systems with infinite server semantics are nonlinear systems, particularly a subclass of piecewise linear (PWL) systems. This paper addresses several problems regarding the state observability of these systems. We assume that the initial marking/state is not known and by measuring the marking of some places we want to estimate all the others. First, a study of the different linear systems corresponding to a continuous Petri net system is performed. It is shown that in some cases, some of them are redundant, and so can be disregarded. The notion of distinguishable modes is introduced which helps us in giving a necessary and sufficient criterion for the observability in infinitesimal time. Structural observability, i.e., observability for all possible values of firing rates of transitions, is studied and it is proved that in some cases it can be reduced to a linear problem, even if the system is nonlinear. Using results from linear structured systems, the concept of weak structural or generic observability is considered.     

Nonlinear second-order multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vectorp-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.     

On observability and stability in infinite-dimensional linear systems

It seems to be generally known (Ref. 1) that, in the case of finite-dimensional systems, the state stability properties of the system may be inferred from the output stability properties if the system is completely observable.In the case of infinite observable systems, the notion of observability generalizes in two distinct ways, weak and strong notions. The above-mentioned stability properties remain true only for the case of strong observability.This research was partially supported by the Bat-Sheva de Rothschild Fund for Advancement of Science and Technology.     

Asymptotic properties of continuous solutions of systems of nonlinear difference equations with continuous argument

We obtain conditions for the existence of continuous (for t ≥ T > 0) solutions for a wide class of systems of nonlinear difference equations with continuous argument and analyze their properties as t → +∞.     

The <Emphasis Type="Italic">R</Emphasis>-observability and <Emphasis Type="Italic">R</Emphasis>-controllability of linear differential-algebraic systems

We study the R-controllability (the controllability within the attainability set) and the R-observability of time-varying linear differential-algebraic equations (DAE). We analyze DAE under assumptions guaranteeing the existence of a structural form (which is called “equivalent”) with separated “differential” and “algebraic” subsystems. We prove that the existence of this form guarantees the solvability of the corresponding conjugate system, and construct the corresponding “equivalent form” for the conjugate DAE. We obtain conditions for the R-controllability and R-observability, in particular, in terms of controllability and observability matrices. We prove theorems that establish certain connections between these properties.     

Global exact boundary observability for first‐order quasilinear hyperbolic systems of diagonal form

In this paper, by means of a constructive method based on the theory of the existence and the uniqueness of the C¹ solution to the Cauchy problem and the Goursat problem, the global exact boundary observability for the first‐order quasilinear hyperbolic systems of diagonal form with linearly degenerate characteristics is obtained. In the case that the system has no zero characteristics, we realize the two‐sided and one‐sided global exact boundary observability by the boundary observed values and obtain the observability inequality. Copyright © 2012 John Wiley & Sons, Ltd.     

Necessary and Sufficient Conditions for the Interchange Between Infimum and the Symbol of Integration

We make a study of various notions of decomposability for subsets of measurable functions in relation with the interchange results between infimum and integration. For this we introduce the notions of serial decomposability and of decomposability relatively to an integrand. A characterization of closed serially decomposable subsets of the Lebesgue spaces L _p is given. The second notion of decomposability introduced is characteristic for the interchange property studied. Many examples are presented. The links are made with R. T. Rockafellar’s decomposability, F. Hiai, H. Umegaki’s decomposability, G. Bouchitté and M. Valadier’s stability and normal decomposability introduced by O. Anza Hafsa and J.-P. Mandallena. As applications we obtain exact lower bounds for minimization problems of integral functionals on normally decomposable spaces (spaces of continuous functions for example), and for the minimization of a class of functionals of the Calculus of Variations.     

Observability with bearing-only observations and smoothness of the attainable set

The attainable set of a linear control system can have both smooth and nonsmooth boundary. This smoothness property is known to be used to classify such systems. One approach, suggested by A. I. Ovseevich in the case of a smooth control set, is based on connecting smoothness of the attainable set with spherical observability of the dual system. This paper generalizes these results to the case of nonsmooth control sets. The corresponding property of the spherical observability notion can be treated as observability with several bearing-only observations. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 8, pp. 119–130, 2005.     

Exact Controllability of Semilinear Evolution Systems and Its Application

In this paper, we obtain several abstract results concerning the exact controllability of semilinear evolution systems. First, we prove the null local exact controllability of semilinear first-order systems by means of the contraction mapping principle; in this case, we do not assume any compactness. Next, we derive the global and/or local exact controllability of semilinear second-order systems by means of the Schauder fixed-point theorem; in this case, we assume only the embedding of the related spaces having some compactness, which is reasonable for many concrete problems. Our main result shows that the observability of the dual of the linearized system implies the exact controllability of the original semilinear system. Finally, we apply our abstract results to the exact controllability of the semilinear wave equation.     

Operators in pre-Riesz spaces: moduli and homomorphisms

We focus on two topics that are related to moduli of elements in partially ordered vector spaces. First, we relate operators that preserve moduli to generalized notions of lattice homomorphisms, such as Riesz homomorphisms, Riesz* homomorphisms, and positive disjointness preserving operators. We also consider complete Riesz homomorphisms, which generalize order continuous lattice homomorphisms. Second, we characterize elements with a modulus by means of disjoint elements and apply this result to obtain moduli of functionals and operators in various settings. On spaces of continuous functions, we identify those differences of Riesz* homomorphisms that have a modulus. Many of our results for pre-Riesz spaces of continuous functions lead to results on order unit spaces, where the functional representation is used.

     

Intrinsic metrics and Lipschitz functions

We study the notions of measurable metric and Lipschitz function which were introduced by N. Weaver ([12]), in the framework of Dirichlet spaces. To this respect, we bring some precisions and complements to [15], notably concerning links with the notion of intrinsic metric ([2]). In the particular case of an abstract Wiener space, we establish the relationship between these notions and that of H-metric ([5]) and μ-a.e. H-Lipschitz continuous function ([4]).     

Decompositions of continuity and complete continuity

We introduce the notions of δ-t-sets, δ_β-t-sets, δ-B-continuity and δ_β -B-continuity and obtain decompositions of continuity and complete continuity.