scholarly journals Non-global solution for visco-elastic dynamical system with nonlinear source term in control problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xiaoqiang Dai ◽  
Wenke Li
Keyword(s):  
Dynamical System ◽  
Control System ◽  
Source Term ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlinear Source ◽  
Sharp Condition ◽  
Global Nonexistence ◽  
Existence And Nonexistence ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we study the initial boundary value problem of the visco-elastic dynamical system with the nonlinear source term in control system. By variational arguments and an improved convexity method, we prove the global nonexistence of solution, and we also give a sharp condition for global existence and nonexistence.</p>

scholarly journals Higher Order Wave Equation with Logarithmic Source Term

2017 ◽  
Author(s):  
Sh. Hajrulla ◽  
L. Bezati ◽  
F. Hoxha
Keyword(s):  
Wave Equation ◽  
Boundary Value Problem ◽  
Source Term ◽  
Boundary Value ◽  
Initial Boundary ◽  
Global Weak Solution ◽  
Initial Boundary Value Problem ◽  
Logarithmic Sobolev Inequality ◽  
Higher Order ◽  
Initial Boundary Value

In this paper we study the initial boundary value problem for logarithmic Higher Order Wave equation. Introducing the Logarithmic Sobolev inequality and using the combination of Galerkin method, we consider the theorem of existence of a global weak solution to problem for the initial boundary value problem of the logarithmic wave equation. By constructing an appropriate Lyapunov function, we obtain the decay estimates of energy for logarithmic Higher Order Wave equation. The proof of the main theorem is given.

scholarly journals Approximate Controllability of Infinite-Dimensional Degenerate Fractional Order Systems in the Sectorial Case

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 735 ◽  
Author(s):  
Dumitru Baleanu ◽  
Vladimir E. Fedorov ◽  
Dmitriy M. Gordievskikh ◽  
Kenan Taş
Keyword(s):  
Control System ◽  
Fractional Order ◽  
Approximate Controllability ◽  
Homogeneous Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Degenerate Equation ◽  
Fractional Order Control ◽  
Infinite Dimensional ◽  
Initial Boundary Value

We consider a class of linear inhomogeneous equations in a Banach space not solvable with respect to the fractional Caputo derivative. Such equations are called degenerate. We study the case of the existence of a resolving operators family for the respective homogeneous equation, which is an analytic in a sector. The existence of a unique solution of the Cauchy problem and of the Showalter—Sidorov problem to the inhomogeneous degenerate equation is proved. We also derive the form of the solution. The approximate controllability of infinite-dimensional control systems, described by the equations of the considered class, is researched. An approximate controllability criterion for the degenerate fractional order control system is obtained. The criterion is illustrated by the application to a system, which is described by an initial-boundary value problem for a partial differential equation, not solvable with respect to the time-fractional derivative. As a corollary of general results, an approximate controllability criterion is obtained for the degenerate fractional order control system with a finite-dimensional input.

scholarly journals A stiffly stable semi-discrete scheme for the characteristic linear hyperbolic relaxation with boundary

2020 ◽  
Vol 54 (5) ◽  
pp. 1569-1596
Author(s):  
Benjamin Boutin ◽  
Thi Hoai Thuong Nguyen ◽  
Nicolas Seguin
Keyword(s):  
Source Term ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Energy Estimates ◽  
Sufficient Condition ◽  
The Laplace Transform ◽  
Space Step ◽  
The Stability ◽  
Initial Boundary Value ◽  
Central Scheme

We study the stability of the semi-discrete central scheme for the linear damped wave equation with boundary. We exhibit a sufficient condition on the boundary to guarantee the uniform stability of the initial boundary value problem for the relaxation system independently of the stiffness of the source term and of the space step. The boundary is approximated using a summation-by-parts method and the stiff stability is proved using energy estimates and the Laplace transform. We also investigate if the condition is also necessary, following the continuous case studied by Xin and Xu (J. Differ. Equ. 167 (2000) 388–437).

Bifurcations and Chaos in an Experimental Based Quasi-Continuum Nonlinear Dynamical System for the ‘Clapper’ Nanoresonator

2007 ◽  
Author(s):  
O. Gottlieb ◽  
A. Gemintern ◽  
R. H. Blick
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical System ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Field Equations ◽  
Bifurcation Structure ◽  
Nonlinear Dynamical ◽  
Spatio Temporal ◽  
Initial Boundary Value ◽  
Temporal Field

In this paper we formulate and numerically investigate an experimentally based quasi-continuum nonlinear initial-boundary-value problem for the three-field ‘Clapper’ nanoresonator that consistently incorporates the system geometric nonlinearity with nonlinear contributions of both magnetomotive and electrodynamic excitation. The spatio-temporal field equations are then reduced via symmetry and a modal projection to an equivalent quasiperiodically excited, low order, nonlinear dynamical system. The governing parameters of the resulting system are matched with the experimentally measured resonance conditions for small amplitude response. Numerical analysis reveals a complex bifurcation structure of torus doubling culminating with a chaotic strange attractor that exhibits similar features to that previously measured in the ‘Clapper’ experiment.

Computational Methods for Solving Dynamic Ice Structure Interaction Problems

2011 ◽  
Author(s):  
Ibrahim Konuk
Keyword(s):  
Dynamical System ◽  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Element Model ◽  
Cohesive Element ◽  
Structure Interaction ◽  
Initial Boundary Value ◽  
Well Posed ◽  
Cohesive Element Model

A framework based on a complex dynamical system viewpoint for formulating and solving dynamic ice-structure interaction problems is introduced. Important constituents required for formulating a well posed initial-boundary value problem are discussed. Significance of these constituents is illustrated using a Cohesive Element model of several example problems.

Study of rigidity of a first-order algebro-differential system with perturbation in the right-hand side

2021 ◽  
pp. 172-181
Author(s):  
Vladimir I. Uskov
Keyword(s):  
Dynamical System ◽  
Boundary Layer ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Bifurcation Equation ◽  
First Order ◽  
Initial Boundary Value ◽  
The Right

The rigidity of a dynamical system described by a first-order differential equationwith an irreversible operator at the highest derivative is investigated. The system is perturbed by an operator addition of the order of the second power of a small parameter. Conditions under which the system is robust with respect to these disturbances are determined as well as conditions under which the influence of disturbances is significant. For this, the bifurcation equation is derived. It is used to set the type of boundary layer functions. As an example, we investigate the initial boundary value problem for a system of partial differential equations with a mixed second partial derivative which occurs in the study of the processes of sorption anddesorption of gases, drying processes, etc.

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