2013 Relaxation Problems Involving Second-Order Differential Inclusions
Abstr. Appl. Anal. 2013: 1-9 (2013). DOI: 10.1155/2013/792431

## Abstract

We present relaxation problems in control theory for the second-order differential inclusions, with four boundary conditions, $\stackrel{¨}{u}\left(t\right)\in F\left(t,u\left(t\right),\stackrel{˙}{u}\left(t\right)\right)$ a.e. on $\left[\mathrm{0,1}\right]$; $u\left(\mathrm{0}\right)=\mathrm{0}, u\left(\eta \right)=u\left(\theta \right)=u\left(\mathrm{1}\right)$ and, with $m\ge \mathrm{3}$ boundary conditions, $\stackrel{¨}{u}\left(t\right)\in F\left(t,u\left(t\right),\stackrel{˙}{u}\left(t\right)\right)$ a.e. on $\left[\mathrm{0,1}\right]; \stackrel{˙}{u}\left(\mathrm{0}\right)=\mathrm{0}, u\left(\mathrm{1}\right)={\sum }_{i=\mathrm{1}}^{m-\mathrm{2}}\mathrm{‍}{a}_{i}u\left({\xi }_{i}\right)$, where $\mathrm{0}<\eta <\theta <\mathrm{1}$, $\mathrm{0}<{\xi }_{\mathrm{1}}<{\xi }_{\mathrm{2}}<\cdots <{\xi }_{m-\mathrm{2}}<\mathrm{1}$ and $F$ is a multifunction from $\left[\mathrm{0,1}\right]×{ℝ}^{n}×{ℝ}^{n}$ to the nonempty compact convex subsets of ${ℝ}^{n}$. We have results that improve earlier theorems.

## Citation

Adel Mahmoud Gomaa. "Relaxation Problems Involving Second-Order Differential Inclusions." Abstr. Appl. Anal. 2013 1 - 9, 2013. https://doi.org/10.1155/2013/792431

## Information

Published: 2013
First available in Project Euclid: 27 February 2014

zbMATH: 1272.49025
MathSciNet: MR3044990
Digital Object Identifier: 10.1155/2013/792431 