Controllability of switched time-delay systems under constrained switching

Different from the existing mathematical models for switched systems, where the switching from one subsystem to another subsystem is finished instantly, in this paper it is assumed that the switching is a transfer process. Moreover, there exists a basic transfer subsystem such that in the transfer process, the transfer subsystem is active. Based on the model of switched systems under constrained switching, this paper studies the controllability of such systems with time delay in the control function. A necessary and sufficient condition for controllability of such systems is established. Finally, an example is given to illustrate the utility of our results.     

On controllability of systems of the form\dot x=A(t)x+g(t,u)

In an earlier paper, the author established a sufficient condition for controllability of systems of the form =A(t)x+g(t, u). This condition is a growth condition which generalizes the concept of an asymptotically proper system introduced by LaSalle for linear systems. The purpose of this paper is examine and apply this growth condition. We first show that the condition is also necessary for controllability. Then, we use these results to consider the controllability of perturbations of the above system. The main result of the paper is a class of systems which in many applications can be assumed to be controllable.During the writing of this paper, the author held a Junior Faculty Summer Fellowship from the Research Council of the University of Nebraska.     

On the controllability of some singularly perturbed nonlinear systems

The controllability of a large scale dynamic system which depends singularly upon a small parameter λ is considered. When λ = 0, the large scale system degenerates into a reduced order subsystem representing its slow dynamics while neglecting the fast phenomena. Another subsystem, often called a boundary layer system, represents the fast dynamics. In this paper sufficient conditions are established under which the controllability of the overall large scale system is inferred from the same property of the two subsystems.     

An efficiency measurement framework for multi-stage production systems

We develop an efficiency measurement framework for systems composed of two subsystems arranged in series that simultaneously computes the efficiency of the aggregate system and each subsystem. Our approach expands the technology sets of each subsystem by allowing each to acquire resources from the other in exchange for delivery of the appropriate (intermediate or final) product, and to form composites from both subsystems. Managers of each subsystem will not agree to ‘`vertical integration’' initiatives unless each subsystem will be more efficient than what each can achieve by separately applying conventional efficiency analysis. A Pareto Efficient frontier characterizes the acceptable set of efficiencies of each subsystem from which the managers will negotiate to select the final outcome. Three proposals for the choice for the Pareto efficient point are discussed: the one that achieves the largest equiproportionate reduction in the classical efficiencies; the one that achieves the largest equal reduction in efficiency; and the one that maximizes the radial contraction in the aggregate consumption of resources originally employed before integration. We show how each choice for the Pareto efficient point determines a derived measure of aggregate efficiency. An extensive numerical example is used to illustrate exactly how the 2 subsystems can significantly improve their operational efficiencies via integration beyond what would be predicted by conventional analysis.     

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 of nonlinear systems with restrained controls

In this paper, we examine the controllability of nonlinear differential systems which are perturbations of systems of the formx=A(t)x+g(t, u). Under appropriate assumptions, we show that integrably bounded perturbations preserve the controllability properties of the base system in the case when the control values are restrained. Our approach is based on the necessary and sufficient growth condition for controllability developed earlier by the author.     

On the controllability of infinite-dimensional linear systems

Two types of controllability, namely weak controllability and strong controllability, are defined for general linear systems. Both types of controllability are useful in control theory. If the system is finite-dimensional, the two types of controllability are equivalent to each other and to the standard concept of complete state-controllability.     

Approximate controllability of semilinear functional equations in Hilbert spaces

In this paper approximate and complete controllability for semilinear functional differential systems is studied in Hilbert spaces. Sufficient conditions are established for each of these types of controllability. The results address the limitation that linear systems in infinite-dimensional spaces with compact semigroup cannot be completely controllable. The conditions are obtained by using the Schauder fixed point theorem when the semigroup is compact and the Banach fixed point theorem when the semigroup is not compact.     

Barycentric systems and stretchability

Using a general resolution of barycentric systems we give a generalization of Tutte's theorem on convex drawing of planar graphs. We deduce a characterization of the edge coverings into pairwise non-crossing paths which are stretchable: such a system is stretchable if and only if each subsystem of at least two paths has at least three free vertices (vertices of the outer face of the induced subgraph which are internal to none of the paths of the subsystem). We also deduce that a contact system of pseudo-segments is stretchable if and only if it is extendible.     

Exact controllability of a cascade system of conservative equations

We consider a cascade system of two conservative equations and prove the controllability of the full system when each equation is controllable, provided that the unitary group corresponding to the free evolution is time-periodic. Applications to systems of Schrödinger (resp. wave) equations are given. With the aid of Kannai transform we infer that a one-dimensional system of heat equations is null controllable even if the supports of the control function and of the coupling term do not intersect.     

Controllability questions for nonlinear systems in abstract spaces

In abstract spaces, we consider certain constrained controllability and approximate controllability properties of a nonlinear system that can be deduced from various controllability properties of its associated linear system. Several examples involving partial differential operators and functional delay operators are given to illustrate the theory.The research of the second author was supported in part by NSF Grant DMS-85-088651 and by the University of Tennessee Science Alliance Award.     

Some remarks on Boolean control systems: Controllability domains and realization theory

We construct a theory of realizations and controllability domains for linear stationary systems in the category of finitely generated free semimodules over a Boolean semiring. We show that the classical realization theorems cannot be generalized to this case, and we prove some incomplete analogs of these theorems. We analyze the structure of controllability domains and the reachability and observability characteristics. In particular, we define a geometric object representing the reachability properties of a system, namely, the generalized reachability topology on the state space.     

Novel Controllability Conditions for a Class of Singularly-Perturbed Systems with Small State Delays

A singularly-perturbed linear time-dependent controlled system with small pointwise and distributed delays in the state variable is considered. Two simpler parameter-free systems, the slow and fast ones, can be associated with the original system. It was established in the literature that the Euclidean space controllability of the original system, valid for all sufficiently small values of the parameter of singular perturbations, follows from the controllability properties of the slow and fast systems. It also was established that such a connection between the controllability properties of the original system and the slow and fast systems is correct, in general, only in one direction. Namely, the controllability of the slow and fast systems provides the controllability of the original system, while the controllability of the original system not always yields the controllability of both the slow and fast systems. In this paper, we consider the original system such that the respective fast system is uncontrollable, meaning that the previously established controllability conditions are not applicable to this original system. In this case, novel parameter-free sufficient conditions for the Euclidean space controllability of the original system, robust with respect to the small parameter of singular perturbations, are derived. Illustrative examples are presented.     

Irreducibly inconsistent systems of linear inequalities

Irreducibly inconsistent systems of linear inequalities are considered from the point of view of applying simplex-like methods for investigation.We give necessary and sufficient conditions, for a system to be irreducibly inconsistent, which can be found by applying a simplex algorithm. Next we consider general inconsistent systems. The simplex method can be used to identify irreducibly inconsistent subsystem. Such subsystems can give structural insight in the inconsistency and are used in trying to make the overall system consistent. This can be done in an easy way e.g. by right-hand side manipulations. Finally the problem of finding a ‘cheapest’ way of doing this can be formulated and solved as an extended linear program.     

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.     

A controllability minimum principle

A controllability minimum principle and two associated transversality conditions are presented, dealing with the controllability of nonlinear systems. The theorems represent necessary conditions for a control function to generate a system path which lies in the boundary of the set of points that are controllable to a target. The theorems presented here are controllability counterparts to Pontryagin's maximum principle, and undoubtedly these results will seem familiar or may have occurred to other researchers in the area of optimal control. The purpose of this paper is to make the distinction explicit and to establish the validity of these controllability theorems on their own merits. The theorems are demonstrated using a simple example and the principal result (a controllability minimum principle) is shown to be equivalent to the Kalman controllability criterion for linear systems.     

Canonical forms for positive discrete-time linear control systems

In this paper, the properties of reachability, controllability and essential reachability of positive discrete-time linear control systems are studied. These properties are characterized in terms of the directed graph of the state matrix. From these characterizations canonical forms of those properties are deduced.     

Controllability of Some Nonlinear Systems in Hilbert Spaces

In this paper, several abstract results concerning the controllability of semilinear evolution systems are obtained. First, approximate controllability conditions for semilinear systems are obtained by means of a fixed-point theorem of the Rothe type; in this case, the compactness of the linear operator is assumed. Next, the exact controllability of semilinear systems with nonlinearities having small Lipschitz constants is derived by means of the Banach fixed-point theorem; in this case, the compactness of the operators is not assumed. In both cases, it is proven that the controllability of the linear system implies the controllability of the associated semilinear system. Finally, these abstract results are applied to the controllability of the semilinear wave and heat equations.     

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.