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.     

Duality in controllability and observability problems for nonstationary hybrid impulsive differential-difference systems

We consider linear nonstationary hybrid differential-difference dynamical observable systems under the action of impulses, which generates jumps in the corresponding solutions of the systems. For such systems, we construct dual controllability problems and prove a general duality relation, which permits one to state a general duality principle in controllability and observability problems for hybrid differential-difference systems. The results are illustrated by an example.     

On constraint controllability of linear systems in Banach spaces

In this paper, we give an addendum to a result of Dolecki and Russell (Ref. 1) related to the duality relationship between observation and control for linear systems in Banach spaces. Our results relate the controllability of a system to the constraint controllability of that system and to the observability of an adjoint system. The main tool used here is an extension of the classical open mapping theorem.     

How to define the dual of a higher-dimensional linear system

The classical duality of linear systems is generalized to higher dimensions. Consequently, a new insight is given to the concepts of controllability and observability, and autonomy.     

Controllability and observability of coordinated linear systems

This paper develops concepts of controllability and observability for the class of coordinated linear systems, a special class of linear hierarchical systems with a top-to-bottom information structure. The usual concepts of reachability and indistinguishability for linear systems are refined, taking into account the decentralized nature of coordinated linear systems. The corresponding controllability and observability decompositions are derived, and several concepts of controllability and observability for coordinated linear systems are introduced and discussed. These results are then combined to a definition and characterization of stabilizability via dynamic measurement feedback for this class of systems.     

Exact controllability for stochastic Schrödinger equations

This paper is addressed to studying the exact controllability of stochastic Schrödinger equations by two controls. One is a boundary control and the other is an internal control in the diffusion term. By means of the duality argument, the control problem is converted into an observability problem for backward stochastic Schrödinger equations, and the desired observability estimate is obtained by a global Carleman estimate. At last, we give a result about the lack of exact controllability, which shows that the action of two controls is necessary.     

Exact Boundary Controllability of Weak Solutions for a Kind of First Order Hyperbolic System — the HUM Method*

The exact boundary controllability and the exact boundary observability for the 1-D first order linear hyperbolic system were studied by the constructive method in the framework of weak solutions in the work [Lu, X. and Li, T. T., Exact boundary controllability of weak solutions for a kind of first order hyperbolic system — the constructive method, Chin. Ann. Math. Ser. B, 42(5), 2021, 643–676]. In this paper, in order to study these problems from the viewpoint of duality, the authors establish ...     

Sharp Observability Estimates for a System of Two Coupled Nonconservative Hyperbolic Equations

We consider a system of two coupled nonconservative wave equations. For this system, we prove several observability estimates. Those observability estimates are sharp in the sense that they lead by duality to the controllability (exact or approximate) of the coupled system with a single control acting through one of the equations only while keeping the same controllability time as for a single equation. Existing results in the literature either require two controls, or in the case of a single control, they have a controllability time that blows up as the coupling parameter goes to zero. Our proofs rely on: (i) Carleman estimates, (ii) energy estimates and (iii) localizing arguments. The results obtained complement and improve, in some sense, earlier results while at the same time providing new uniqueness results.     

On a Class of Population Dynamics Problem with Mixed Boundary Conditions

In this paper, we consider a linear population dynamics model with age dependence and spatial structure in which birth process is described by a nonlocal term. Carleman estimate for the backward nonlocal adjoint problem providing the observability results is fundamental to the analysis and by duality it provides the controllability result. More precisely, we observe that one can expect to compute the initial distribution using observations on a small part of the boundary.     

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.     

Controllability and observability for a class of time-varying impulsive systems

This paper is concerned with the controllability and observability for a class of piecewise linear time-varying impulsive systems. Several sufficient and necessary conditions for state controllability and observability of such systems are established. Meanwhile, corresponding criteria for time-invariant impulsive systems are also obtained and the criteria are compared with the existing results.     

Recursiveness of zero structures for system matrices of interconnected linear systems via module theoretic tools

The importance of the zeros of multivariable linear systems is well-known in terms of measure obstructions to the controllability and the observability. In this paper, a recursive decomposition of interconnected systems is outlined by taking into account the sequential structure of the connections. The paper extends the coordinate, module-theoretic studies from the elementary algebraic systems theory to include the case of such linear interconnected systems which need not to be controllable or observable. Also, the properties of controllability and observability, the decoupling zeros and the signal bloking issues are characterized.     

On duality between observation and control for implicit linear discrete-time systems

Some results concerning the problem of duality between observationand control for implicit linear discrete-time systems are presented.It is not assumed that the considered systems are regular. Inthis case, some new concepts of observability and controllabilityturn out to be necessary for studying the problem of duality.     

广义区间系统可控和可观测的充分条件

分别给出了单输入多输出广义区间系统可控和多输入单输出广义区间系统可观测的充分条件     

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 of Unsteady Supercritical Flows in a Tree-Like Network of Open Canals

The author establishes the exact boundary observability of unsteady supercritical flows in a tree-like network of open canals with general topology. An implicit duality between the exact boundary controllability and the exact boundary observability is also given for unsteady supercritical flows.     

Exact boundary observability for 1‐D quasilinear wave equations

By means of a direct and constructive method based on the theory of semi‐global C² solution, the local exact boundary observability and an implicit duality between the exact boundary controllability and the exact boundary observability are shown for 1‐D quasilinear wave equations with various boundary conditions. Copyright © 2006 John Wiley & Sons, Ltd.     

An approach to non-linear control problems using fixed-point methods,degree theory and pseudo-inverses.

We indicate how techniques from fixed point and degree theory may be combined with the linear theory of control and observation, to create controllability and observability results for a certain class of non-linear systems.     

Controlled and conditioned invariant subspaces in linear system theory

The concept of invariance of a subspace under a linear transformation is strongly connected with controllability and observability of linear dynamical systems. In this paper, we definecontrolled andconditioned invariant subspaces as a generalization of the simple invariants, for the purpose of investigating some further structural properties of linear systems. Moreover, we prove some fundamental theorems on which the computation of the above-mentioned subspaces is based. Then, we give two examples of practical application of the previous concepts concerning the determination of the constant output and perfect output controllability subspaces.     

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.