Elsevier

Automatica

Volume 37, Issue 10, October 2001, Pages 1609-1617
Automatica

Brief Paper
Sufficient conditions for the existence of an unbounded solution

https://doi.org/10.1016/S0005-1098(01)00114-5Get rights and content

Abstract

Readily verifiable conditions under which a dynamical system of the form ẋ=f(x) possesses an unbounded solution are presented. The results are illustrated by showing they can be used to infer results about lack of global stabilizability for nonlinear control systems. The key observation in the paper is that behaviour at infinity can be studied using local methods applied to an auxiliary system.

Introduction

Let R denote the real numbers. Given a dynamical system of the formẋ=f(x)defined on Rn, it is often of interest to know if all possible solutions of the system are bounded or if the system possesses an unbounded solution. Determining this for systems without explicit solutions can be a highly nontrivial task. Presented in this paper are sufficient conditions for a system of the form (1) to possess an unbounded solution.

When considering whether a system possesses an unbounded solution, one is asking how the system behaves arbitrarily far away from the origin, that is, how it behaves near “infinity”. The key observation in the paper is that behaviour at infinity can be studied using local methods. It will be shown that the existence of appropriate auxiliary functions and variables allows us to construct a new system of dimension one greater than the original from which the existence of an unbounded solution of the original system can be inferred using local methods.

As mentioned above, the result given in the paper relies on finding appropriate auxiliary functions and variables. In this manner it is similar to Lyapunov's stability theorem which requires one to find an appropriate auxiliary function, namely a Lyapunov function, in order to give a positive result about system stability.

The paper is structured as follows. The main ideas of the paper are further introduced in Section 2. The material in Section 2 is quite concrete and is presented to motivate the slightly more abstract material in the remainder of the paper. The main result of the paper is presented in Section 3. Section 4 contains some examples. Included are examples that show how the results of the paper can be used to infer results about lack of global stabilizability for nonlinear control systems. A partial converse theorem to the main result is presented in Section 5. Section 6 contains some additional comments and the paper ends with some concluding remarks in Section 7. (Two technical lemmas have been placed at the end of the paper in an appendix.)

Section snippets

Main ideas

Presented in this section, in a rather nonrigorous manner, are sufficient conditions for a system to possess an unbounded solution. The material presented here is intended only as an introduction to the main ideas of the paper and rigorous proofs of results based on these ideas are given in the main body of the paper.

Consider an arbitrary system of the form (1) defined on Rn. Let λ∈R be a fixed positive number and let z∈R be a new variable governed by the dynamicsż=−λz.Further let α1,…,αn be

Main result

This section contains the main result of the paper giving sufficient conditions for a system to possess an unbounded solution. The material in this section is directly motivated by the ideas in Section 2 and the results of that section are presented in rigorous form in Corollary 3.5.

Let Z denote the integers, R>0 the set {z∈R|z>0} and C1 the class of continuously differentiable functions. If ϕ(z,x) is a map from R×Rn to Rn which is differentiable at (z,x)=(a,b), let D1ϕ(a,b) denote the partial

Examples

In this section the results of the previous section are illustrated with some examples. In particular, some of the examples in this section show how the results of Section 3 can be used to infer results about lack of global stabilizability for nonlinear control systems.

Example 4.1

For arbitrary (but fixed) real numbers a,b,c and d, with c and d nonzero, the systemẋ1=ax2+x1x3+bx32,ẋ2=cx1,ẋ3=dx2x3has an unbounded solution. This can be verified by checking thatF(x)=1+x32,λ=0.5,α1=2,α2=1,α3=1,ȳ=(c/d,|c|/d,

A partial converse theorem

In this section a partial converse theorem to Theorem 3.2 is presented.

Theorem 5.1

Supposef:RnRnisC1and that the systemẋ=f(x)has an unbounded solution. Then there exists aC1functionφ:R>0×RnRnand a pointȳRnthat together satisfy properties (P1)–(P3), and

(P4′) limsupz→0+|φ(1/z,ȳ)|=∞.

Furthermore, there exists a continuous functionF:RnR>0and a scalarλ∈R>0that together withφandȳsatisfy conditions (1) and (2) of Theorem 3.2.

Proof

Define F=1+|f|2 and g=f/F. For each x∈Rn, let ξ(·,x) denote the solution ofẇ=g(w),

Additional comments

Given a system ẋ=f(x) and a continuous function F:RnR>0, let g=f/F and consider the normalized system ẋ=g(x). If the requirements of Theorem 3.2 are met, it follows from the proof of Theorem 3.2 that there exists a solution to (6) passing through a point (z0,y0),z0>0, and converging to (0,ȳ). If (z(·),y(·)) denotes such a solution then it was shown that x(t)=φ(1/z(t),y(t)) is a solution of ẋ=g(x). Suppose now that φ(z,x)=(zα1x1,…,zαnxn) and that for some j∈{1,…,n}, ȳj≠0 and αj>0 as in

Concluding remarks

In this paper a start was made at exploring the use of local methods to analyse behaviour at infinity. Presented were sufficient conditions for a dynamical system to possess an unbounded solution and it was shown that these results can be used to infer results about lack of global stabilizability for nonlinear control systems.

Robert Orsi was born in Canberra, Australia, in April 1972. He received his B.Sc. degree in mathematics and his B.E. and M.E. degrees in systems engineering from the Australian National University in 1992, 1994 and 1996 respectively. He then went on to complete a Ph.D. in electrical engineering at the University of Melbourne, Australia, in the area of nonlinear dynamical systems. From February to September 2000 he worked part time as a researcher at the University of Melbourne and part time as

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Robert Orsi was born in Canberra, Australia, in April 1972. He received his B.Sc. degree in mathematics and his B.E. and M.E. degrees in systems engineering from the Australian National University in 1992, 1994 and 1996 respectively. He then went on to complete a Ph.D. in electrical engineering at the University of Melbourne, Australia, in the area of nonlinear dynamical systems. From February to September 2000 he worked part time as a researcher at the University of Melbourne and part time as a consultant to Voyan Technology, a company specializing in noise management for DSL and wireless communication networks. In October 2000 he moved to the US and took up a full time position with Voyan. His current work involves helping to develop methods of increasing loop reach and bandwidth in DSL networks. His interests include nonlinear control and many parts of signal processing and digital communications.

Dr. Laurent Praly graduated from École Nationale Supérieure des Mines de Paris in 1976. After working in industry for three years, in 1980 he joined the Centre Automatique et Systèmes at École des Mines de Paris. From July 1984 to June 1985, he spent a sabbatical year as a visiting assistant professor in the Department of Electrical and Computer Engineering at the University of Illinois at Urbana-Champaign. Since 1985 he has continued at the Centre Automatique et Systèmes where he served as director for two years. In 1993, he spent a quarter at the Institute for Mathematics and its Applications at the University of Minnesota where he was an invited researcher. His main interest is in feedback stabilization of controlled dynamical systems under various aspects – linear and nonlinear, dynamic, output, under constraints, with parametric or dynamic uncertainty. On these topics he is contributing both on the theoretical aspect with many academic publications and the practical aspect with applications in power systems, mechanical systems, aerodynamical and space vehicles.

Iven Mareels was born in Aalst Belgium 11 August 1959. He obtained the Bachelor of Electromechanical Engineering from Gent University, Belgium in 1982 and the PhD in Systems Engineering from the Australian National University, Canberra, Australia in 1987. He is presently Professor and Head of Department, Department of Electrical and Electronic Engineering, the University of Melbourne, a position he took up in 1996. Previously he was a Reader at the Australian National University (1990–1996), a lecturer at the University of Newcastle (1988–1990) and the University of Gent (1986–1988). In 1994 he was a recipient of the Vice-Chancellor's Award for Excellence in Teaching. He is a co-editor in chief, together with Prof. A. Antoulas for the journal Systems & Control Letters. He is a senior member of the Institute of Electrical and Electronics Engineers, a member of the Society for Industrial and Applied Mathematics, a fellow of the Institute of Engineers Australia, a member of the Asian Control Professors Association, and Chairman of the Education Committee of the latter, member of the Steering Committee for the Asian Control Conference. He is registered with the IEAust as a professional engineer with extensive experience in consulting for both industry and government. His research interests are in adaptive and learning systems, non-linear control and modelling. At present he has a particular interest in modelling and controlling of environmental systems with applications in natural resource management. He has published widely, about 60 journal papers and over 100 conference publications. He received several awards for his publications.

This paper is an extended version of the NOLCOS ’98 paper “Sufficient conditions for a dynamical system to possess an unbounded solution”. This paper was recommended for publication in revised form by Associate Editor Eugene P. Ryan under the direction of Editor Roberto Tempo.

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