Lyapunov Functionals Associated to Automata

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

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Lyapunov functionals for multistrain models with infinite delay

Discrete & Continuous Dynamical Systems - B ◽

10.3934/dcdsb.2017025 ◽

2017 ◽

Vol 22 (2) ◽

pp. 507-536

Author(s):

Yoji Otani ◽

◽

Tsuyoshi Kajiwara ◽

Toru Sasaki

Keyword(s):

Infinite Delay ◽

Lyapunov Functionals

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TIME-DEPENDENT RESCALINGS AND LYAPUNOV FUNCTIONALS FOR THE VLASOV–POISSON AND EULER–POISSON SYSTEMS, AND FOR RELATED MODELS OF KINETIC EQUATIONS, FLUID DYNAMICS AND QUANTUM PHYSICS

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s021820250100091x ◽

2001 ◽

Vol 11 (03) ◽

pp. 407-432 ◽

Cited By ~ 6

Author(s):

J. DOLBEAULT ◽

G. REIN

Keyword(s):

Fluid Dynamics ◽

Quantum Physics ◽

Kinetic Equations ◽

Euler System ◽

Lyapunov Functionals ◽

Time Behavior ◽

Long Time Behavior ◽

Poisson System ◽

Dispersion Effects ◽

Long Time

We investigate rescaling transformations for the Vlasov–Poisson and Euler–Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behavior of the solutions and can be applied to other models in kinetic theory (two-dimensional symmetric Vlasov–Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger–Poisson system, nonlinear Schrödinger equation).

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