scholarly journals A Turnpike Property of Trajectories of Dynamical Systems with a Lyapunov Function

Games ◽  
2020 ◽  
Vol 11 (4) ◽  
pp. 63
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Economic Growth ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Weak Version ◽  
Turnpike Property ◽  
Set Valued Mappings

In this paper, we study the structure of trajectories of discrete disperse dynamical systems with a Lyapunov function which are generated by set-valued mappings. We establish a weak version of the turnpike property which holds for all trajectories of such dynamical systems which are of a sufficient length. This result is usually true for models of economic growth which are prototypes of our dynamical systems.

scholarly journals A Weak Turnpike Property for Perturbed Dynamical Systems with a Lyapunov Function

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 45
Author(s):  
Alexander J. Zaslavski
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Weak Version ◽  
Mathematical Economics ◽  
Turnpike Property

In this work, we obtain a weak version of the turnpike property of trajectories of perturbed discrete disperse dynamical systems, which have a prototype in mathematical economics.

scholarly journals Coevolving institutions and the paradox of informal constraints

2021 ◽  
pp. 1-20
Author(s):  
Daniel Seligson ◽  
Anne E. C. McCants
Keyword(s):  
Economic Growth ◽  
Game Theory ◽  
Dynamical Systems ◽  
Systems Theory ◽  
Institutional Economics ◽  
Dynamical Systems Theory ◽  
New Institutional Economics ◽  
Long Run

Abstract We can all agree that institutions matter, though as to which institutions matter most, and how much any of them matter, the matter is, paraphrasing Douglass North's words at the Nobel podium, unresolved after seven decades of immense effort. We suggest that the obstacle to progress is the paradigm of the New Institutional Economics itself. In this paper, we propose a new theory that is: grounded in institutions as coevolving sources of economic growth rather than as rules constraining growth; and deployed in dynamical systems theory rather than game theory. We show that with our approach some long-standing problems are resolved, in particular, the paradoxical and perplexingly pervasive influence of informal constraints on the long-run character of economies.

RAPID GROWTH PATHS IN CONVEX-VALUED RANDOM DYNAMICAL SYSTEMS

2001 ◽  
Vol 01 (04) ◽  
pp. 493-509 ◽  
Author(s):  
IGOR V. EVSTIGNEEV ◽  
MICHAEL I. TAKSAR
Keyword(s):  
Economic Growth ◽  
Asymptotic Behavior ◽  
Dynamical Systems ◽  
Growth Rates ◽  
Stochastic Models ◽  
Rapid Growth ◽  
Random Dynamical Systems ◽  
Random Vectors ◽  
Time Period

This paper examines set-valued random dynamical systems defined by convex homogeneous stochastic operators. The operators under consideration transform elements of a cone contained in a space of random vectors into subsets of the cone. We study rapid paths of such dynamical systems, i.e. those paths which maximize (appropriately defined) growth rates at every time period. Questions of existence, uniqueness and asymptotic behavior of infinite rapid trajectories are considered. The study is motivated by problems related to stochastic models of economic growth.

Control Vector Lyapunov Functions for Large-Scale Impulsive Systems

2011 ◽  
Author(s):  
Wassim M. Haddad ◽  
Sergey G. Nersesov
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Lyapunov Functions ◽  
Large Scale ◽  
Lyapunov Theory ◽  
System Stability ◽  
Dynamical System Theory ◽  
Control Vector ◽  
Vector Lyapunov Functions

This chapter extends the notion of control vector Lyapunov functions to impulsive dynamical systems. Vector Lyapunov theory has been developed to weaken the hypothesis of standard Lyapunov theory to enlarge the class of Lyapunov functions that can be used for analyzing system stability. In particular, the use of vector Lyapunov functions in dynamical system theory offers a very flexible framework since each component of the vector Lyapunov function can satisfy less rigid requirements as compared to a single scalar Lyapunov function. Using control vector Lyapunov functions, the chapter develops a universal hybrid decentralized feedback stabilizer for a decentralized affine in the control nonlinear impulsive dynamical system that possesses guaranteed gain and sector margins in each decentralized input channel. These results are used to develop hybrid decentralized controllers for large-scale impulsive dynamical systems with robustness guarantees against full modeling and input uncertainty.

ON THE STABILITY OF DYNAMIC SYSTEMS WITH CERTAIN SWITCHINGS, WHICH CONSISTS OF LINEAR SUBSYSTEMS WITHOUT DELAY

2021 ◽  
Vol 3 ◽  
pp. 5-17
Author(s):  
Denis Khusainov ◽  
◽  
Alexey Bychkov ◽  
Andrey Sirenko ◽  
Jamshid Buranov ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Lyapunov Function ◽  
Dynamic Systems ◽  
Lyapunov Functions ◽  
Lyapunov Method ◽  
Positive Definite Function ◽  
Definite Function ◽  
The Stability ◽  
Further Development ◽  
Zero Equilibrium

This work is devoted to the further development of the study of the stability of dynamic systems with switchings. There are many different classes of dynamical systems described by switched equations. The authors of the work divide systems with switches into two classes. Namely, on systems with definite and indefinite switchings. In this paper, the system with certain switching, namely a system composed of differential and difference sub-systems with the condition of decreasing Lyapunov function. One of the most versatile methods of studying the stability of the zero equilibrium state is the second Lyapunov method, or the method of Lyapunov functions. When using it, a positive definite function is selected that satisfies certain properties on the solutions of the system. If a system of differential equations is considered, then the condition of non-positiveness (negative definiteness) of the total derivative due to the system is imposed. If a difference system of equations is considered, then the first difference is considered by virtue of the system. For more general dynamical systems (in particular, for systems with switchings), the condition is imposed that the Lyapunov function does not increase (decrease) along the solutions of the system. Since the paper considers a system consisting of differential and difference subsystems, the condition of non-increase (decrease of the Lyapunov function) is used.For a specific type of subsystems (linear), the conditions for not increasing (decreasing) are specified. The basic idea of using the second Lyapunov method for systems of this type is to construct a sequence of Lyapunov functions, in which the level surfaces of the next Lyapunov function at the switching points are either «stitched» or «contain the level surface of the previous function».

GROUP ACTION PRESERVING THE HAAGERUP PROPERTY OF -ALGEBRAS

2015 ◽  
Vol 93 (2) ◽  
pp. 295-300 ◽  
Author(s):  
CHAO YOU
Keyword(s):  
Dynamical Systems ◽  
Group Action ◽  
Crossed Product ◽  
Weak Version ◽  
Haagerup Property ◽  
Tracial State

From the viewpoint of $C^{\ast }$-dynamical systems, we define a weak version of the Haagerup property for the group action on a $C^{\ast }$-algebra. We prove that this group action preserves the Haagerup property of $C^{\ast }$-algebras in the sense of Dong [‘Haagerup property for $C^{\ast }$-algebras’, J. Math. Anal. Appl.377 (2011), 631–644], that is, the reduced crossed product $C^{\ast }$-algebra $A\rtimes _{{\it\alpha},\text{r}}{\rm\Gamma}$ has the Haagerup property with respect to the induced faithful tracial state $\widetilde{{\it\tau}}$ if $A$ has the Haagerup property with respect to ${\it\tau}$.

Stability of Dynamical Systems With Multiple Nonlinearities

1987 ◽  
Vol 109 (4) ◽  
pp. 410-413 ◽  
Author(s):  
Norio Miyagi ◽  
Hayao Miyagi
Keyword(s):  
Dynamical System ◽  
Stability Analysis ◽  
Dynamical Systems ◽  
Lyapunov Function ◽  
Stability Region ◽  
Direct Method ◽  
Essential Feature ◽  
Point Of View ◽  
Stability Of Dynamical Systems ◽  
The Stability

This note applies the direct method of Lyapunov to stability analysis of a dynamical system with multiple nonlinearities. The essential feature of the Lyapunov function used in this note is a non-Lure´ type Lyapunov function which surpasses the Lure´-type Lyapunov function from the point of view of the stability region guaranteed. A modified version of the multivariable Popov criterion is used to construct non-Lure´ type Lyapunov function, which allow for the dynamical sytems with multiple nonlinearities.

Absolute Stability of a Class of Singular Nonlinear Systems with Set-Valued Mappings

2014 ◽  
Vol 24 (01) ◽  
pp. 1550015 ◽  
Author(s):  
Gaoming Feng ◽  
Xingguo Tan
Keyword(s):  
Time Delay ◽  
Nonlinear Systems ◽  
Lyapunov Function ◽  
Absolute Stability ◽  
Delay Systems ◽  
Stability Conditions ◽  
Time Delay Systems ◽  
Set Valued Mappings ◽  
The Absolute

A class of singular nonlinear systems with set-valued mappings are studied in this paper. Criteria are given based on the Lyapunov function to check the absolute stability of the systems, then the results are extended to the time delay systems and the time delay systems with uncertainty. Three examples are simulated to show the effectiveness of the proposed stability conditions.

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