scholarly journals Solutions to Lyapunov stability problems of sets: nonlinear systems with differentiable motions

1994 ◽  
Vol 17 (1) ◽  
pp. 103-112 ◽  
Author(s):  
Ljubomir T. Grujic
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Invariant Set ◽  
Necessary And Sufficient ◽  
Time Invariant ◽  
Selection Of

Time-invariant nonlinear systems with differentiable motions are considered. The algorithmic necessary and sufficient conditions are established in various forms for one-shot construction of a Lyapunov function, for asymptotic stability of a compact invariant set and for the exact determination of the asymptotic stability domain of the invariant set.The classical conditions are expressed in terms of existence of a system Lyapunov functions. The conditions of theorems presented herein are expressed via properties of the solution?to??=-p, or of the solution?to??=-(1-?)p, for arbitrarily selectedp?P(S;f)orp?P1(S;f), where familiesP(S;f)andP1(S;f)are well defined. The equation??=-p, or its equivalent??=-(1-?)p, should be solved only for one selection of the functionp.

scholarly journals New approach to asymptotic stability: time-varying nonlinear systems

1997 ◽  
Vol 20 (2) ◽  
pp. 347-366 ◽  
Author(s):  
L. T. Grujić
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Time Varying ◽  
Uniform Asymptotic Stability ◽  
Stability Time ◽  
Necessary And Sufficient

The results of the paper concern a broad family of time-varying nonlinear systems with differentiable motions. The solutions are established in a form of the necessary and sufficient conditions for: 1) uniform asymptotic stability of the zero state, 2) for an exact single construction of a system Lyapunov function and 3) for an accurate single determination of the (uniform) asymptotic stability domain. They permit arbitrary selection of a functionp(⋅)from a defined functional family to determine a Lyapunov functionv(⋅),[v(⋅)], by solvingv′(⋅)=−p(⋅){or equivalently,v′(⋅)=−p(⋅)[1−v(⋅)]}, respectively. Illstrative examples are worked out.

scholarly journals Analysis of positivity and stability of fractional discrete-time nonlinear systems

2016 ◽  
Vol 64 (3) ◽  
pp. 491-494 ◽  
Author(s):  
T. Kaczorek
Keyword(s):  
Nonlinear Systems ◽  
Asymptotic Stability ◽  
Discrete Time ◽  
Lyapunov Method ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stability Tests ◽  
Necessary And Sufficient

Abstract The positivity and asymptotic stability of the fractional discrete-time nonlinear systems are addressed. Necessary and sufficient conditions for the positivity and sufficient conditions for the asymptotic stability of the fractional nonlinear systems are established. The proposed stability tests are based on an extension of the Lyapunov method to the positive fractional nonlinear systems. The effectiveness of tests is demonstrated on examples.

scholarly journals Solutions to Lyapunov stability problems: nonlinear systems with continuous motions

1994 ◽  
Vol 17 (3) ◽  
pp. 587-596 ◽  
Author(s):  
Ljubomir T. Grujic
Keyword(s):  
Lyapunov Function ◽  
Nonlinear Dynamical System ◽  
Sufficient Conditions ◽  
Stability Domain ◽  
Necessary And Sufficient Conditions ◽  
Nonlinear Dynamical ◽  
Bounded Set ◽  
Necessary And Sufficient ◽  
Bounded Sets ◽  
Selection Of

The necessary and sufficient conditions for accurate construction of a Lyapunov function and the necessary and sufficient conditions for a set to be the asymptotic stability domain are algorithmically solved for a nonlinear dynamical system with continuous motions. The conditions are established by utilizing properties of o-uniquely bounded sets, which are explained in the paper. They allow arbitrary selection of an o-uniquely bounded set to generate a Lyapunov function.Simple examples illustrate the theory and its applications.

Positive stable realizations of discrete-time linear systems

2012 ◽  
Vol 60 (3) ◽  
pp. 605-616
Author(s):  
T. Kaczorek
Keyword(s):  
Transfer Function ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Asymptotically Stable ◽  
Numerical Examples ◽  
Discrete Time Systems ◽  
Necessary And Sufficient ◽  
Time Systems ◽  
Time Linear

Abstract The problem of existence and determination of the set of positive asymptotically stable realizations of a proper transfer function of linear discrete-time systems is formulated and solved. Necessary and sufficient conditions for existence of the set of the realizations are established. A procedure for computation of the set of realizations are proposed and illustrated by numerical examples.

Stabilizing Feedback Controls for Nonlinear Hamiltonian Systems and Nonconservative Bilinear Systems in Elasticity

1982 ◽  
Vol 104 (1) ◽  
pp. 27-32 ◽  
Author(s):  
S. N. Singh
Keyword(s):  
Asymptotic Stability ◽  
Hamiltonian Systems ◽  
Attitude Control ◽  
Sufficient Conditions ◽  
Bilinear Systems ◽  
Feedback Controls ◽  
Control Laws ◽  
Nonlinear Maps ◽  
The Stability ◽  
Necessary And Sufficient

Using the invariance principle of LaSalle [1], sufficient conditions for the existence of linear and nonlinear control laws for local and global asymptotic stability of nonlinear Hamiltonian systems are derived. An instability theorem is also presented which identifies the control laws from the given class which cannot achieve asymptotic stability. Some of the stability results are based on certain results for the univalence of nonlinear maps. A similar approach for the stabilization of bilinear systems which include nonconservative systems in elasticity is used and a necessary and sufficient condition for stabilization is obtained. An application to attitude control of a gyrostat Satellite is presented.

Asymptotic stability of heteroclinic cycles in systems with symmetry. II

2004 ◽  
Vol 134 (6) ◽  
pp. 1177-1197 ◽  
Author(s):  
Martin Krupa ◽  
Ian Melbourne
Keyword(s):  
Asymptotic Stability ◽  
Transition Matrix ◽  
Sufficient Conditions ◽  
Systematic Investigation ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Heteroclinic Cycles ◽  
Higher Dimensional ◽  
Necessary And Sufficient

Systems possessing symmetries often admit robust heteroclinic cycles that persist under perturbations that respect the symmetry. In previous work, we began a systematic investigation into the asymptotic stability of such cycles. In particular, we found a sufficient condition for asymptotic stability, and we gave algebraic criteria for deciding when this condition is also necessary. These criteria are satisfied for cycles in R3.Field and Swift, and Hofbauer, considered examples in R4 for which our sufficient condition for stability is not optimal. They obtained necessary and sufficient conditions for asymptotic stability using a transition-matrix technique.In this paper, we combine our previous methods with the transition-matrix technique and obtain necessary and sufficient conditions for asymptotic stability for a larger class of heteroclinic cycles. In particular, we obtain a complete theory for ‘simple’ heteroclinic cycles in R4 (thereby proving and extending results for homoclinic cycles that were stated without proof by Chossat, Krupa, Melbourne and Scheel). A partial classification of simple heteroclinic cycles in R4 is also given. Finally, our stability results generalize naturally to higher dimensions and many of the higher-dimensional examples in the literature are covered by this theory.

scholarly journals Positive 2D Discrete-Time Linear Lyapunov Systems

2009 ◽  
Vol 19 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Przemysław Przyborowski ◽  
Tadeusz Kaczorek
Keyword(s):  
Asymptotic Stability ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Numerical Examples ◽  
Necessary And Sufficient ◽  
Time Linear

Positive 2D Discrete-Time Linear Lyapunov SystemsTwo models of positive 2D discrete-time linear Lyapunov systems are introduced. For both the models necessary and sufficient conditions for positivity, asymptotic stability, reachability and observability are established. The discussion is illustrated with numerical examples.

Arithmetical Semigroup Rings

1980 ◽  
Vol 32 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
Bonnie R. Hardy ◽  
Thomas S. Shores
Keyword(s):  
Principal Ideal ◽  
Sufficient Conditions ◽  
Forthcoming Paper ◽  
Semigroup Ring ◽  
Necessary And Sufficient Conditions ◽  
Semigroup Rings ◽  
Principal Ideal Ring ◽  
Arithmetical Semigroup ◽  
Necessary And Sufficient

Throughout this paper the ring R and the semigroup S are commutative with identity; moreover, it is assumed that S is cancellative, i.e., that S can be embedded in a group. The aim of this note is to determine necessary and sufficient conditions on R and S that the semigroup ring R[S] should be one of the following types of rings: principal ideal ring (PIR), ZPI-ring, Bezout, semihereditary or arithmetical. These results shed some light on the structure of semigroup rings and provide a source of examples of the rings listed above. They also play a key role in the determination of all commutative reduced arithmetical semigroup rings (without the cancellative hypothesis on S) which will appear in a forthcoming paper by Leo Chouinard and the authors [4].

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