639 research outputs found
Backstepping PDE Design: A Convex Optimization Approach
Abstract\u2014Backstepping design for boundary linear PDE is
formulated as a convex optimization problem. Some classes of
parabolic PDEs and a first-order hyperbolic PDE are studied,
with particular attention to non-strict feedback structures. Based
on the compactness of the Volterra and Fredholm-type operators
involved, their Kernels are approximated via polynomial
functions. The resulting Kernel-PDEs are optimized using Sumof-
Squares (SOS) decomposition and solved via semidefinite
programming, with sufficient precision to guarantee the stability
of the system in the L2-norm. This formulation allows optimizing
extra degrees of freedom where the Kernel-PDEs are included
as constraints. Uniqueness and invertibility of the Fredholm-type
transformation are proved for polynomial Kernels in the space
of continuous functions. The effectiveness and limitations of the
approach proposed are illustrated by numerical solutions of some
Kernel-PDEs
First passage times of two-correlated processes: analytical results for the Wiener process and a numerical method for diffusion processes
Given a two-dimensional correlated diffusion process, we determine the joint
density of the first passage times of the process to some constant boundaries.
This quantity depends on the joint density of the first passage time of the
first crossing component and of the position of the second crossing component
before its crossing time. First we show that these densities are solutions of a
system of Volterra-Fredholm first kind integral equations. Then we propose a
numerical algorithm to solve it and we describe how to use the algorithm to
approximate the joint density of the first passage times. The convergence of
the method is theoretically proved for bivariate diffusion processes. We derive
explicit expressions for these and other quantities of interest in the case of
a bivariate Wiener process, correcting previous misprints appearing in the
literature. Finally we illustrate the application of the method through a set
of examples.Comment: 18 pages, 3 figure
Solving Fuzzy Nonlinear Volterra-Fredholm Integral Equations by Using Homotopy Analysis and Adomian Decomposition Methods
In this paper, Adomian decomposition method (ADM) and homotopy analysis method (HAM) are proposed to solving the fuzzy nonlinear Volterra-Fredholm integral equation of the second kind. we convert a fuzzy nonlinear Volterra-Fredholm integral equation to a nonlinear system of Volterra-Fredholm integral equation in crisp case. we use ADM , HAM and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy Volterra-Fredholm integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed methods are proved. Examples is given and the results reveal that homotopy analysis method is very effective and simple compared with the Adomian decomposition method
Approximate Optimal Control of Volterra-Fredholm Integral Equations Based on Parametrization and Variational Iteration Method
This article presents appropriate hybrid methods to solve optimal control problems ruled by Volterra-Fredholm integral equations. The techniques are grounded on variational iteration together with a shooting method like procedure and parametrization methods to resolve optimal control problems ruled by Volterra - Fredholm integral equations. The resulting value shows that the proposed method is trustworthy and is able to provide analytic treatment that clarifies such equations and is usable for a large class of nonlinear optimal control problems governed by integral equations
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