Observability of linear differential systems defined in differential rings

We study the observability of linear differential systems in sufficiently arbitrary differential rings; observability is treated in the classical sense, i.e., as the injectivity of the “initial condition-output vector” mapping. A number of observability conditions is obtained. We define observability by finitely many measurements, where a measurement is the value of a homomorphism into the ring of constants on the output variable. We show that observability is equivalent to observability by finitely many measurements.     

Observability of nonlinear systems

New necessary and sufficient conditions are presented for the observability of systems described by nonlinear ordinary differential equations with nonlinear observations. The conditions are based on extension of the necessary and sufficient conditions for observability of time-varying linear systems to the linearized trajectory of the nonlinear system. The result is that the local observability of any initial condition can be readily determined, and the observability of the entire initial domain can be computed. The observability of constant parameters appearing in the differential equations is also considered. Examples are presented to illustrate the theory.This research was supported by the National Science Foundation, Grant No. NSF GK-10136.     

一维线性波动方程耦合组的精确边界同步能观性

作者介绍了多种精确同步能观性,并对一维波动方程耦合组在多种边界条件下分别实现了精确边界同步能观性,分组精确边界同步能观性以及分组精确边界零能观性与同步能观性.     

Boundary observability for the finite-difference space semi-discretizations of the 2-d wave equation in the square

We extend our previous results on the boundary observability of the finite-difference space semidiscretizations of the 1-d wave equation to 2-d in the square. As in the 1-d case, we prove that the constants on the boundary observability inequality blow-up as the mesh-size tends to zero. However, we prove a uniform observability inequality in a subspace of solutions generated by the low frequencies. The dimension of these subspaces grows as the mesh size tends to zero and eventually, in the limit, covers the whole energy space. Our result is sharp in the sense that the uniformity of the observability inequality is lost when the dimension of the subspaces grows faster. Our method of proof combines discrete multiplier techniques, Fourier series developments and compactness-uniqueness arguments.     

Observability of hybrid discrete-continuous systems

We study the statement and solvability of observability problems in linear stationary hybrid discrete-continuous dynamical systems. Necessary and sufficient observability conditions expressed directly via the system parameters are derived. We consider linear observability problems and the dual controllability and reachability problems. The problem of computing the minimum number of inputs for which the system has a given observability is discussed. An example illustrating the results is presented.     

分布参数系统能控能观性问题的统一处理及应用

本文的主要目的是介绍一个统一的处理分布参数系统能控能观性问题的方法,并给出该方法在分布参数控制理论中的应用。为此,本文将从一类“类抛物” 偏微分算子（即没有椭圆性条件）的带权恒等式出发,给出所有已知的关于抛物型方程、双曲型方程、Schrödinger 方程和板方程的基于整体Carleman 估计的能控能观性结果。同时,基于该带权的恒等式,本文还给出它在双曲型系统的稳定性问题和在拟线性复Ginzburg-Landau 方程能控能观性等问题中的应用。     

Exact boundary observability for a kind of second-order quasilinear hyperbolic system

In this paper, by means of a constructive method based on the existence and uniqueness of the semi-global classical solution, we establish the local exact boundary observability for a kind of second-order quasilinear hyperbolic system in which the number of positive eigenvalues and the number of negative ones are not equal. As an application, we obtain the one-sided local exact boundary observability and two-sided local exact boundary observability with fewer observed values for first-order quasilinear hyperbolic systems of diagonal form with boundary conditions in which the diagonal variables corresponding to the positive eigenvalues and those corresponding to the negative ones are decoupled.     

Observability of the 1-D Wave Equation with Mixed Boundary Conditions in a Non-cylindrical Domain

We study a wave equation with mixed boundary conditions in a bounded interval with a moving endpoint. We derive precise estimates for the energy of the solution and show observability results at each endpoint. The observability constants are explicitly given and the obtained time of observability is sharp.     

Controllability and Observability of Linear Quaternion-valued Systems

The aim of this paper is to define an extension of the controllability and observability for linear quaternion-valued systems (QVS). Some criteria for controllability and observability are derived, and the minimum norm control and duality theorem are also investigated. Compared with real-valued or complex-valued linear systems, it is shown that the classical Caylay-Hamilton Theorem as well as Popov-Belevitch-Hautus (PBH) type controllability and observability test do not hold for linear QVS. Hence, a modified PBH type necessary condition is studied for the controllability and observability, respectively. Finally, some examples are given to illustrate the effectiveness of the obtained results.     

Simultaneous observability and stabilization of some uncoupled wave equations

We consider uncoupled wave equations with different speed of propagation in a bounded domain. Using a combination of the Bardos–Lebeau–Rauch observability result for a single wave equation and a new unique continuation result for uncoupled wave equations, we prove an observability estimate for that system. Applying Lions? Hilbert uniqueness method (HUM), one may derive simultaneous exact controllability results for the uncoupled system; the controls being locally distributed, with their supports satisfying the geometric control condition of Bardos, Lebeau and Rauch. Afterwards, we discuss the related simultaneous stabilization problem; this latter problem is solved by a combination of the new observability inequality, and a result of Haraux establishing an equivalence between observability and stabilization for second order evolution equations with bounded damping operators. Our observability and stabilization results generalize to higher space dimensions some earlier results of Haraux established in the one-dimensional setting.     

Production capacity buildup and double marginalization mitigation in a dynamic supply chain

The paper investigates the extent to which capacity investment considerations interact with the double marginalization effect in a simple supply chain governed by a wholesale price contract. To do so, a non-cooperative differential game model is formulated to study the pricing and capacity investment decisions in a supply chain, which consists of a supplier and a manufacturer. In such a game, there are different decision rules—open-loop, closed-loop, feedback—that are available to the supply chain participants, depending on the observability of the current state of the supply chain. While closed-loop and feedback equilibrium strategies involve the observability of other chain member’s production capacity, open-loop equilibrium strategies do not have such requirement. We examine how the supplier and the manufacturer determine, with the different decision rules, their production capacities and pricing policies to maximize their profits over an infinite planning horizon, and determine how the observability of other supply chain’s members’ production capacity affects the magnitude of the double marginalization effect. Our study suggests that the observability of other chain member’s current production capacity entails a lower production efficiency that results in a greater double marginalization effect. This allows us to conclude that observability of other chain member’s current production capacity is associated with a greater double marginalization effect.     

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.     

Reduced order dead-beat observers for the chemostat

This paper studies the strong observability property and the reduced-order dead-beat observer design problem for a continuous bioreactor. New relationships between coexistence and strong observability, and checkable sufficient conditions for strong observability, are established for a chemostat with two competing microbial species. Furthermore, the dynamic output feedback stabilization problem is solved for the case of one species.     

Generalized Gram Matrix and Its Application to the Observability Problem for Linear Nonsteady Systems

Sufficient conditions for the nondegeneracy of a generalized Gram matrix are obtained. In particular, it is shown that the generalized Gram matrix is nondegenerate for the Chebyshev systems of functions. An application of the results to the observability problems for linear nonsteady systems of ordinary differential equations are given. In terms of the observability matrix, necessary and sufficient conditions of the complete and total observability by means of finite-parameter solving operations are established.     

Observability of coupled systems

By applying the theory of semigroups, we generalize an earlier result of Komornik and Loreti [5] on the observability of compactly perturbed systems. As an application, we answer a question of the same authors concerning the observability of weakly coupled linear distributed systems. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exact controllability of thermoelastic plate equation with memory

In this paper, we consider exact control problem for a coupled system of plate with Gurtin‐Pipkin equation. Using duality arguments, the problem is reduced to the obtention of suitable observability estimates for the dual system. Firstly, we obtain the observability inequality of the dual system by means of multiplier method. Then, we prove that the system is exactly controllable based on the Hilbert Unique Method.     

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.     

Cauchy Matrices in the Observation of Diffusion Equations

Observability Gramians of diffusion equations have been recently connected to infinite Pick and Cauchy matrices. In fact, inverse or observability inequalities can be obtained after estimating the extreme eigenvalues of these structured matrices,with respect to the diffusion semi-group matrix. The purpose is hence to conduct a spectral study of a subclass of symmetric Cauchy matrices and present an algebraic way to show the desired observability results. We revisit observability inequalities for three different observation problems of the diffusion equation and show how they can be (re)stated through simple proofs.     

Criteria for controllability and observability in complex domain models of stationary systems

We propose criteria for controllability and observability of stationary systems specified by complex domain models. Conclusions concerning controllability and observability are made on the basis of coefficients in the corresponding transfer functions. Translated fromDinamicheskie Sistemy, Vol. 12, pp. 120–126, 1993.