Observability and Controllability of the 1-D Wave Equation in Domains with Moving Boundary

By mean of generalized Fourier series and Parseval’s equality in weighted \(L^{2}\)-spaces, we derive a sharp energy estimate for the wave equation in a bounded interval with a moving endpoint. Then, we show the observability, in a sharp time, at each of the endpoints of the interval. The observability constants are explicitly given. Using the Hilbert Uniqueness Method we deduce the exact boundary controllability of the wave equation.     

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 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.     

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.     

Complete observability of differential-algebraic systems with delays

We study the statement and solvability of complete observability problems for linear stationary differential-algebraic dynamical systems with delays (DAD systems. Since in the general case, the state space of such systems is infinite-dimensional and is not necessarily “minimal,” we consider various statements of problems depending on what states are observed. Our attention is focused on the simplest DAD system in symmetric form. We obtain efficient parametric criteria and analyze relationships between various notions of complete observability for DAD systems. In the case of DAD systems with scalar coefficients, we obtain a complete classification of notions of complete observability in the class of continuous initial functions with the continuous matching condition. We analyze the problem of computing the minimum number of outputs of a spectrally observable DAD system.     

Quasidifferentiability and observability of linear nonstationary systems

We suggest a method for studying the observability of linear nonstationary ordinary differential systems on the basis of the quasidifferentiability of the output variables with respect to some lower-triangular matrix. This approach permits one to weaken the smoothness requirement on the coefficients in the statement of observability criteria.     

Carleman estimates for the one-dimensional heat equation with a discontinuous coefficient and applications to controllability and an inverse problem

We study the observability and some of its consequences (controllability, identification of diffusion coefficients) for one-dimensional heat equations with discontinuous coefficients (piecewise C¹). The observability, for a linear equation, is obtained by a Carleman-type estimate. This kind of observability inequality yields controllability results for a semi-linear equation as well as a stability result for the identification of the diffusion coefficient.     

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.     

Controllability and influence of the spatial discretization of the beam equation

We study the problem of the control of a finite difference semi-discrete scheme for the 1D beam equation modelling the transverse vibrations of a beam with fixed ends. We show that for most numerical schemes, due to high frequency spurious solutions, the observability may be lost under numerical discretizations when the mesh size tends to zero. We then prove that the uniform observability property is obtained by truncating the high frequencies. We also show how spatial discretization can be performed by using our nonstandard finite difference scheme.     

q-Difference linear control systems

We propose q-difference versions of some basic results of complete controllability and observability of linear systems such as criterions of controllability and observability, controllability and observability canonical forms, the q-duality theorem, the interconnection between these concepts and that of polynomials without common zeros. As an appendix, is given a simple but computationally meaningful Maple Procedure for the controllability criterion for time constant systems.     

Observability and nonlinear filtering

This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability is explored. Finally, the results are partially extended to non-compact signal state spaces.     

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.     

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.     

Stochastic uniform observability of linear differential equations with multiplicative noise

The aim of this paper is to give a deterministic characterization of the uniform observability property of linear differential equations with multiplicative white noise in infinite dimensions. We also investigate the properties of a class of perturbed evolution operators and we used these properties to give a new representation of the covariance operators associated to the mild solutions of the investigated stochastic differential equations. The obtained results play an important role in obtaining necessary and sufficient conditions for the stochastic uniform observability property.     

Reachability and Observability Concepts for Stochastic Systems

We define observability and reachability and introduce the corresponding Gramians for a Levy driven linear system like Benner, Damm in [3] which focused on the case of Wiener noise. We additionally show that the sets of observable and reachable states are characterized by these Gramians. This is analogous to deterministic systems, where observability and reachability concepts are described in Sections 4.2.1 and 4.2.2 in Antoulas [1]. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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.     

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.