Narrow Results

Is gauge-invariant complete decomposition of the nucleon spin possible? Although it is a difficult theoretical question which has not reached a complete consensus yet, a general agreement now is that there are at least two physically inequivalent gauge-invariant decompositions (I) and (II) of the nucleon. In these two decompositions, the intrinsic spin parts of quarks and gluons are just common. What discriminate these two decompositions are the orbital angular momentum parts. The orbital angular momenta of quarks and gluons appearing in the decomposition (I) are the so-called "mechanical" orbital angular momenta, while those appearing in the decomposition (II) are the generalized (gauge-invariant) "canonical" ones. By this reason, these decompositions are also called the "mechanical" and "canonical" decompositions of the nucleon spin, respectively. A crucially important question is which decomposition is more favorable from the observational viewpoint. The main objective of this concise review is to try to answer this question with careful consideration of recent intensive researches on this problem.

In this paper, by means of case studies we discuss an observable feature of 3D continuous-time autonomous chaotic systems through scalar output and its time derivatives. We observe that the Lorenz system, the Rössler system, the Chua circuit and the Chen system are all observable based on scalar output and its derivatives. This leads to our conjecture that chaotic motion described by 3D continuous-time autonomous dynamical system is observable based on a scalar output and its first- and second-order derivatives. Finally, we present some mathematical analysis and put up some theoretical questions for future studies.

We establish a quantitative observation inequality for the Schrödinger and the Heisenberg equations on $R^d$, uniform in the Planck constant $\hslash \in [0,1]$. The proof is based on the pseudometric introduced in F. Golse and T. Paul, The Schrödinger equation in the mean-field and semiclassical regime, Arch. Ration. Mech. Anal. 223 (2017) 57–94. This inequality involves only effective constants which are computed explicitly in their dependence in $\hslash$ and all parameters involved.

This paper studies the observability of the initial value of continuous-time dynamic fuzzy control systems. The sufficient and necessary conditions for the initial value of fuzzy control system to be observable are given. A concept called strong observability is introduced and a sufficient condition for the initial value of system to be strongly observable is discussed. The fuzzy logic description of input-output relations using IF-THEN rule base is presented.

When are quantum filters asymptotically independent of the initial state? We show that this is the case for absolutely continuous initial states when the quantum stochastic model satisfies an observability condition. When the initial system is finite dimensional, this condition can be verified explicitly in terms of a rank condition on the coefficients of the associated quantum stochastic differential equation.

Reference to the notions of verifiability and observability is widespread in contract theory. This paper is a contribution towards a formalization, and related characterizations, of these two notions. In particular we first define them, through knowledge operators, and then provide characterization results in terms of the relevant state spaces. Since, when referring to a contract, observability typically pertains to parties while verifiability to the court, we define them differently. A main finding of the paper is that for proper contract verifiability to obtain the court must imagine true states and have information processing abilities as good as the parties.

The purpose of this work is to introduce the concept of fuzzy regular languages (FRL) accepted by fuzzy finite automata, and try to introduce the categorical look of fuzzy languages where the codomain of membership functions are taken as a complete residuated lattice, instead of $[0,1]$. Also, we have introduced pumping lemma for FRL, which is used to establish a necessary and sufficient condition for a given fuzzy languages to be non-constant.

In this paper the concept of a nonlinear verticum-type observation system is introduced. These systems are composed from several "subsystems" connected sequentially in a particular way: a part of the state variables of each "subsystem" also appears in the next "subsystem" as an "exogenous variable" which can also be interpreted as a control generated by an "exosystem". Therefore, these "subsystems" are not observation systems, but formally can be considered as control-observation systems. The problem of observability of such systems can be reduced to rank conditions on the "subsystems". Indeed, under the condition of Lyapunov stability of an equilibrium of the "large", verticum-type system, it is shown that the Kalman rank condition on the linearization of the "subsystems" implies the observability of the original, nonlinear verticum-type system.

For an illustration of the above linearization result, a stage-structured fishery model with reserve area is considered. Observability for this system is obtained by applying the above linearization and decomposition approach. Furthermore, it is also shown that, applying an appropriate observer design method to each subsystem, from the observation of the biomass densities of the adult (harvested) stage, in both areas, the biomass densities of the pre-recruit stage can be efficiently estimated.

The present paper studies the properties of observability and controllability using Bond Graph (BG) approach. It is especially applied to a pseudo BG model that describes the thermal transfers in a household refrigerator compartment exposed to an outdoor natural cold airflow. This latter is spread out in an installation, which consists of a cavity covering the side wall of the appliance and connected to inlet and outlet ducts through two openings. The study results show that the pseudo BG model is observable and controllable in time-varying (TV) systems using the concept of duality.

The paper is concerned with the problem of reconstructing inaccessible parts of the state x of a finite - dimensional system from observed output y: Find an estimate for x(t) given y{τ), τ ≤ t. By an observer we mean a control system with input y whose state (or parts of it) provide an estimate which is asymptotically accurate. Under the assumption that x evolves according to a differential equation and that y is part of x concrete proposals for observers will be made in two situations. The results are new at least in the general non-linear and non-autonomous case. They will be obtained with the help of the advanced invariant manifold technique which has been developed in [1] for the purposes of control theory.

We report on a series of works done in collaboration with Y. Privat and E. Zuazua, concerning the problem of optimizing the shape and location of sensors and actuators for systems whose evolution is driven by a linear partial differential equation. This problem is frequently encountered in applications where one wants to optimally design sensors in order to maximize the quality of the reconstruction of solutions by using only partial observations, or to optimally design actuators in order to control a given process with minimal efforts. For example, we model and solve the following informal question: what is the optimal shape and location of a thermometer?

Note that we want to optimize not only the placement but also the shape of the observation or control subdomain over the class of all possible measurable subsets of the domain having a prescribed Lebesgue measure. By probabilistic considerations we model this optimal design problem as the one of maximizing a spectral functional interpreted as a randomized observability constant, which models optimal observabnility for random initial data.

Solving this problem strongly depends on the operator in the PDE model and requires fine knowledge on the asymptotic properties of eigenfunctions of that operator. For parabolic equations like heat, Stokes or anomalous diffusion equations, we prove the existence and uniqueness of a best domain, proved to be regular enough, and whose algorithmic construction depends in general on a finite number of modes. In contrast, for wave or Schrödinger equations, relaxation may occur and our analysis reveals intimate relations with quantum chaos, more precisely with quantum ergodicity properties of the Laplacian eigenfunctions.

The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Car-leman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equations.