Continuous dependence of the solution of an infinite-dimensional system of integrodifferential equations (which are unsolved with respect to the derivative) on a parameter, and the averaging method

1974 ◽  
Vol 25 (6) ◽  
pp. 685-688
Author(s):  
T. V. Melikidze
Keyword(s):  
Continuous Dependence ◽  
Averaging Method ◽  
Integrodifferential Equations ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
System Of Integrodifferential Equations ◽  
Infinite Dimensional System

An adaptive multimodal approach to nonlinear sloshing in a rectangular tank

2001 ◽  
Vol 432 ◽  
pp. 167-200 ◽  
Author(s):  
ODD M. FALTINSEN ◽  
ALEXANDER N. TIMOKHA
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Fluid Motion ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Modal Theory ◽  
Nonlinear Sloshing ◽  
Rectangular Tank ◽  
Wide Range ◽  
Infinite Dimensional System

Two-dimensional nonlinear sloshing of an incompressible fluid with irrotational flow in a rectangular tank is analysed by a modal theory. Infinite tank roof height and no overturning waves are assumed. The modal theory is based on an infinite-dimensional system of nonlinear ordinary differential equations coupling generalized coordinates of the free surface and fluid motion associated with the amplitude response of natural modes. This modal system is asymptotically reduced to an infinite-dimensional system of ordinary differential equations with fifth-order polynomial nonlinearity by assuming sufficiently small fluid motion relative to fluid depth and tank breadth. When introducing inter-modal ordering, the system can be detuned and truncated to describe resonant sloshing in different domains of the excitation period. Resonant sloshing due to surge and pitch sinusoidal excitation of the primary mode is considered. By assuming that each mode has only one main harmonic an adaptive procedure is proposed to describe direct and secondary resonant responses when Moiseyev-like relations do not agree with experiments, i.e. when the excitation amplitude is not very small, and the fluid depth is close to the critical depth or small. Adaptive procedures have been established for a wide range of excitation periods as long as the mean fluid depth h is larger than 0.24 times the tank breadth l. Steady-state results for wave elevation, horizontal force and pitch moment are experimentally validated except when heavy roof impact occurs. The analysis of small depth requires that many modes have primary order and that each mode may have more than one main harmonic. This is illustrated by an example for h/l = 0.173, where the previous model by Faltinsen et al. (2000) failed. The new model agrees well with experiments.

A Monte Carlo method for backward stochastic differential equations with Hermite martingales

2019 ◽  
Vol 25 (1) ◽  
pp. 37-60
Author(s):  
Antoon Pelsser ◽  
Kossi Gnameho
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Approximation ◽  
Mathematical Finance ◽  
Backward Stochastic Differential Equations ◽  
Terminal Condition ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
The Family ◽  
Infinite Dimensional System

Abstract Backward stochastic differential equations (BSDEs) appear in many problems in stochastic optimal control theory, mathematical finance, insurance and economics. This work deals with the numerical approximation of the class of Markovian BSDEs where the terminal condition is a functional of a Brownian motion. Using Hermite martingales, we show that the problem of solving a BSDE is identical to solving a countable infinite-dimensional system of ordinary differential equations (ODEs). The family of ODEs belongs to the class of stiff ODEs, where the associated functional is one-sided Lipschitz. On this basis, we derive a numerical scheme and provide numerical applications.

scholarly journals Recovering the initial state of an infinite-dimensional system using observers

Automatica ◽  
2010 ◽  
Vol 46 (10) ◽  
pp. 1616-1625 ◽  
Author(s):  
Karim Ramdani ◽  
Marius Tucsnak ◽  
George Weiss
Keyword(s):  
Dimensional System ◽  
Initial State ◽  
Infinite Dimensional ◽  
Infinite Dimensional System

scholarly journals Controllability of a Family of Nonlinear Population Dynamics Models

2021 ◽  
Vol 2021 ◽  
pp. 1-17
Author(s):  
Yacouba Simporé
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical System ◽  
Null Controllability ◽  
Dimensional System ◽  
Observability Inequality ◽  
Nonlinear Dynamical ◽  
Infinite Dimensional ◽  
Population Dynamics Models ◽  
Dynamics Models ◽  
Infinite Dimensional System

Considering a nonlinear dynamical system, we study the nonlinear infinite-dimensional system obtained by grafting an operator A and an age structure. This system is such that the nonlinearity is at the level of births. We show that there is a time T dependent on the constraints on the age and the observability minimal time T 0 of the pair A , B ( B is the control operator), from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply Schauder’s fixed point in the proof of the null controllability of the nonlinear system..

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