THE THEOREM OF BOHL–PERRON ON THE ASIMPTOTIC STABILITY OF HYBRID SYSTEMS AND INVERSE THEOREM

2018 ◽  
pp. 726-737
Author(s):  
Pyotr Mikhailovich Simonov
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Difference Equation ◽  
Hybrid Systems ◽  
Vector Function ◽  
Hybrid System ◽  
Finite Interval ◽  
Converse Theorem ◽  
Absolutely Continuous ◽  
Functional Differential

We consider an abstract hybrid system of two equations with two unknowns: a vector function x that is absolutely continuous on each finite interval [0,T], T>0, and a sequence of numerical vectors y. The study uses the W-method N.V. Azbelev. As a model, a system containing a functional differential equation with respect to x is used, and a difference equation with respect to y. Solutions spaces are studied. For a hybrid system, the Bohl–Perron theorem on asymptotic stability and the converse theorem are obtained.

Conical approximation and asymptotic stability

2012 ◽  
Author(s):  
Rafal Goebel ◽  
Ricardo G. Sanfelice ◽  
Andrew R. Teel
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Hybrid System ◽  
Original System ◽  
Conical Flow ◽  
Linear Flow ◽  
System A

This chapter presents a technique of approximating a hybrid system with a conical hybrid system: a system with conical flow and jump sets and with constant or linear flow and jump maps. The main result here deduces pre-asymptotic stability for the original system from pre-asymptotic stability for the conical approximation. This result generalizes, to a hybrid system, the result that asymptotic stability for the linearization of a differential equation implies asymptotic stability for the differential equation. In many cases, the analysis of the conical approximation is simpler than of the original hybrid system; this is illustrated in several examples later in the chapter.

scholarly journals Asymptotic behaviour of solutions to neutral functional differential equations

1989 ◽  
Vol 40 (3) ◽  
pp. 345-355
Author(s):  
Shaozhu Chen ◽  
Qingguang Huang
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Differential Equations ◽  
Asymptotic Behaviour ◽  
Functional Differential Equation ◽  
Necessary Conditions ◽  
Functional Differential Equations ◽  
Neutral Functional Differential Equation ◽  
Neutral Functional Differential Equations ◽  
Functional Differential

Sufficient or necessary conditions are established so that the neutral functional differential equation [x(t) − G(t, xt)]″ + F(t, xt) = 0 has a solution which is asymptotic to a given solution of the related difference equation x(t) = G(t, xt) + a + bt, where a and b are constants.

scholarly journals 19.—Asymptotic Stability for Some Functional Differential Equations

1976 ◽  
Vol 74 ◽  
pp. 253-255 ◽  
Author(s):  
Hans-Otto Walther ◽  
W. N. Everitt
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Differential Equations ◽  
Functional Differential Equation ◽  
Characteristic Equation ◽  
Population Biology ◽  
Linear Functional ◽  
Functional Differential Equations ◽  
Linear Functional Differential Equation ◽  
Functional Differential

SynopsisFor a non-linear functional differential equation from population biology, a result on asymptotic stability is obtained by investigating the zeros of the characteristic equation of the linearised functional differential equation.

scholarly journals ON ASYMPTOTIC PROPERTIES OF THE CAUCHY FUNCTION FOR AUTONOMOUS FUNCTIONAL DIFFERENTIAL EQUATION OF NEUTRAL TYPE

2020 ◽  
pp. 7-26
Author(s):  
Vera Malygina ◽  
◽  
Kirill Chudinov ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Initial Data ◽  
Exponential Stability ◽  
Functional Differential Equation ◽  
Initial Function ◽  
Neutral Type ◽  
Cauchy Function ◽  
Functional Differential ◽  
Equation Of Neutral Type

We investigate stability of a linear autonomous functional differential equation of neutral type. The basis of the study is the well-known explicit solution representation formula including an integral operator, the kernel of which is called the Cauchy function of the equation under study. It is shown that the definitions of Lyapunov, asymptotic and exponential stabilities can be formulated without loss of generality in terms of the corresponding properties of the Cauchy function. The conclusion is drawn that stability with respect to initial data depends on the functional space which the initial function belongs to, and, as a consequence, that there is the need to indicate this space in the definition of stability. It is shown that, along with the concept of asymptotic stability, a certain stronger property should be introduced, which we call strong asymptotic stability. The main study is devoted to stability with respect to initial function from spaces of integrable functions. Special attention is paid to the study of asymptotic and exponential stability. We use the following known properties of the Cauchy function of an equation of neutral type: this function is piecewise continuous, and its jumps are determined by a Cauchy problem for a linear difference equation. We obtain that the strong asymptotic stability of the equation under consideration for initial data from the space L1 is equivalent to an exponential estimate of the Cauchy function and; moreover, we show that these properties are equivalent to the exponential stability with respect to initial data from the spaces Lp for all p from 1 to infinity inclusive. However, we show that strong asymptotic stability with respect to the initial data from the space Lp for p greater than one may not coincide with exponential stability.

scholarly journals Asymptotic Stability of Neutral Set-Valued Functional Differential Equation by Fixed Point Method

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Junyan Bao ◽  
Peiguang Wang
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Asymptotic Stability ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Fixed Point Method ◽  
Stability Theorem ◽  
Point Method ◽  
Functional Differential ◽  
Necessary And Sufficient

This paper studies a class of nonlinear neutral set-valued functional differential equations. The globally asymptotic stability theorem with necessary and sufficient conditions is obtained via the fixed point method. Meanwhile, we give an example to illustrate the obtained result.

Asymptotic Stability of an Abstract Delay Functional-Differential Equation

2006 ◽  
Vol 11 (1) ◽  
pp. 79-93 ◽  
Author(s):  
J. M. Tchuenche
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Functional Differential Equation ◽  
Chaotic Dynamics ◽  
Integral Form ◽  
Transform Method ◽  
Deviating Arguments ◽  
Interesting Connection ◽  
Exponential Asymptotic Stability ◽  
Functional Differential

We study the exponential asymptotic stability of an abstract functional-differential equation with a mixed type of deviating arguments, namely: c which might represent the gestation period of the population and r(u(t)), a suitably defined function. The equation is reduced to its equivalent integral form and solved via Laplace transform method. An interesting connection of our study is with generalizations of populations with potentially complex (chaotic) dynamics.

Autonomy and Responsibility in Hybrid Systems

2017 ◽  
Author(s):  
Wulf Loh ◽  
Janina Loh
Keyword(s):  
Hybrid Systems ◽  
Hybrid System ◽  
Moral Dilemma ◽  
Autonomous Driving ◽  
Shared Responsibility ◽  
Driving System ◽  
Autonomous Cars ◽  
Self Driving Cars ◽  
Trolley Cases ◽  
Autonomous Driving System

In this chapter, we give a brief overview of the traditional notion of responsibility and introduce a concept of distributed responsibility within a responsibility network of engineers, driver, and autonomous driving system. In order to evaluate this concept, we explore the notion of man–machine hybrid systems with regard to self-driving cars and conclude that the unit comprising the car and the operator/driver consists of such a hybrid system that can assume a shared responsibility different from the responsibility of other actors in the responsibility network. Discussing certain moral dilemma situations that are structured much like trolley cases, we deduce that as long as there is something like a driver in autonomous cars as part of the hybrid system, she will have to bear the responsibility for making the morally relevant decisions that are not covered by traffic rules.

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