2 edition of **cone associated with the Lyapunov mapping** found in the catalog.

cone associated with the Lyapunov mapping

Charles Gordon Lindberg

- 318 Want to read
- 22 Currently reading

Published
**1968**
.

Written in English

- Matrices.

**Edition Notes**

Other titles | Lyapunov mapping, a cone associated with. |

Statement | by Charles Gordon Lindberg. |

The Physical Object | |
---|---|

Pagination | [5], 43 leaves, bound ; |

Number of Pages | 43 |

ID Numbers | |

Open Library | OL15078210M |

In this paper, we study the existence and the stability in the sense of Lyapunov of differential inclusions governed by the normal cone to a given prox-regular set, which is subject to a. Lyapunov functions and stability problems Gunnar S oderbacka, Workshop Ghana, , 1 Introduction In these notes we explain the power of Lyapunov functions in determining stability of equilibria and estimating basins of attraction. We concentrate on two dimensional g: cone.

In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. Also, again under some regularity conditions, the criticality of the Lyapunov exponent function implies some rigidity of the map, in the sense that the volume decomposes as a product along the two.

Controller design based on Lyapunov stability theory. Lyapunov stability theory provides a means of stabilizing unstable nonlinear systems using feedback control. The idea is that if one can select a suitable Lyapunov function and force it to decrease along the trajectories of the system, the resulting system will converge to its g: cone. where or, is called Lyapunov stable (asymptotically, exponentially stable) if it becomes such after equipping the space (or) with a property of the solution does not depend on the choice of the norm. 2) Let a mapping be given, where is a metric space. The point is called Lyapunov stable relative to the mapping if for every there exists a such that for any satisfying the inequality.

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In this paper we investigate the Lyapunov mapping P --> AP + PA * where A is a positive stable matrix and P is a hermitian matrix. In particular, for special positive stable A we characterize the image of the cone of positive definite matrices under this : Charles Gordon Lindberg.

It was observed that this difficulty is because of the choice of the cone relative to the comparison system, namely R n +, the cone of nonnegative elements of R n and that a possible approach to overcome this limitation is to choose an appropriate cone other than R n +.

The chapter presents this idea and by developing the theory of differential inequalities through cones, the method of cone-valued Lyapunov functions Cited by: 1. We describe systematically the relation between Lyapunov functions and nonvanishing Lyapunov exponents, both for maps and flows.

This includes a brief survey of the existing results in the area. In particular, we consider separately the cases of nonpositive and arbitrary Lyapunov functions, thus yielding optimal criteria for negativity and positivity of the Lyapunov exponents of linear Cited by: 6.

2. Gowda, M.S., Sznajder, R.: Some global uniqueness and solvability results for linear complementarity problems over symmetric cones. SIAM J. by: 8. LYAPUNOV EQUATION We can now state in the stable case the first fundamental connection between Lyapunov the equation and the novel notion of cic.

LEMMA The set sf_ p, where P Eis a maximal open stable cic in CnXn. Proof. Using the remark after Lemmait suffices to show that sV_, is a maximal open stable by: For the discrete-time linear Lyapunov cone-systems, the necessary and sufficient conditions for being the cone-system, the asymptotic stability, reachability, observability and controllability to.

Lyapunov Functions and Cone Families with the explicit construction of eventually strict Lyapunov functions for any map or flow with nonzero Lyapunov exponents. of cones are associated.

L. Barreira, D. Dragičević, C. Valls, Lyapunov functions and cone families. Stat. Phys.– () MathSciNet CrossRef zbMATH Google ScholarAuthor: Luís Barreira.

Lyapunov Exponent. Standard map orbits rendered with Std Map. Descriptions of the sort given at the end of the prevous page are unnatural and clumsy. It would be nice to have a simple measure that could discriminate among the types of orbits in the same. Define for (t, x) e R + S(p), D+V(t,x) = limsup-j [V(t + h,x + hf(t,x)) - V(t,x)].

t h~O+ h The following comparison theorem plays a prominent role whenever we. The equilibrium state 0 of (1) is exponentially stable, if it is stable in the sense of Lyapunov and there exists a δ′>0 and constants M 0 such that xt e Mxtt o ()≤−−α()o (L.3) for all xt ()o Lyapunov Stability Theorems For Autonomous (or Time-Invariant) SystemsFile Size: 78KB.

introduce the concept of bilinearity rank of a cone. Gowda and Tao [3] showed that this rank could also be described by means of the so-called Lyapunov-like matrices on the cone, and renamed the rank as the Lyapunov rank.

We say that a matrix A 2R n is Lyapunov-like [2] on K if hAx;si= 0 for all (x;s) 2C(K). The associated Lyapunov function is given by V (x) = x T P x. Since the SOS decomposition technique introduced about 10 years ago [50], the system analysis for polynomial systems can be performed more efficiently because it helps to answer many difficult questions on system analysis that were hard to.

Some stability deﬁnitions we consider nonlinear time-invariant system x˙ = f(x), where f: Rn → Rn a point xe ∈ R n is an equilibrium point of the system if f(xe) = 0 xe is an equilibrium point ⇐⇒ x(t) = xe is a trajectory suppose xe is an equilibrium point • system is globally asymptotically stable.

The direct method of Lyapunov. Lyapunov’s direct method (also called the second method of Lyapunov) allows us to determine the stability of a system without explicitly inte-grating the diﬀerential equation (). The method is a generalization of the idea that File Size: KB.

Cone and Lyapunov Function Techniques 95 Lyapunov Functions 96 A Criterion for Nonvanishing Lyapunov Exponents 98 Invariant Cone Families Cocycles with Values in the Symplectic Group Monotone Operators and Lyapunov Exponents The Algebra of Potapov Lyapunov Exponents for J-Separated File Size: KB.

FIG. Signs of Lyapunov exponents for various attractors. and their signs characterize the attractor. This idea is illustrated for three-dimensional state space in Fig. 3, taken from Ruelle (). Lyapunov's first method Lyapunov's first method provides theoretical validity File Size: 2MB.

Lyapunov functions have been used in various contexts (stability, convergence analysis, design of model reference adaptive systems, etc.).

The Lyapunov approach is based on the physical idea that the energy of an isolated system decreases. A Lyapunov function maps scalar or vector variables to real numbers (ℜ N → ℜ +) and decreases with time.

The main attribute of the Lyapunov approach that Missing: cone. Journals & Books; Register Sign in. Vol Issue 9, MayPages Higher derivatives of Lyapunov functions and cone-valued Lyapunov functions. Higher derivatives of Lyapunov functions and cone-valued Lyapunov functions.

Author links. July 8, Optimization Methods & Software the˙lyapunov˙rank˙of˙an˙improper˙cone notation (1) does not allow this: the expression ‘K∗’ is ambiguous when we may think of Kas living in more than one ambient space, since ‘K ∗in V’ and ‘K in W’ are two diﬀerent sets.

Associated with a matrix A E ~, let us define the Lyapunov map LA by a linear map on Sn: LA(X):=AX + XAT. It is well known that a matrix representation of LA is obtained from the Sn-invariant linear map LA(X):=vec-l((I + A I)vec(X)), X E [~nxn, by restricting it to the subspace S, where indicates the Kronecker product and vec(X):= (xT.

x v.,x,) E IR"~ for X = (xI. x,) 6 E,x.Cited by: The Lyapunov first and second methods are investigated and the stability analysis of fractional differential systems is highlighted. A new Lemma for the Caputo fractional derivative is reviewed and a class of fractional-order gene regulatory networks is : Ronak Saeed.Moreover, we describe converse results of these criteria with the explicit construction of eventually strict Lyapunov functions for any map or flow with nonzero Lyapunov exponents.

We also construct examples showing that in general the existence of an eventually strict invariant cone family does not imply the existence of an eventually strict.