## International Journal of Differential Equations

- Int. J. Differ. Equ.
- Volume 2018, Special Issue (2018), Article ID 7417590, 6 pages.

### On the Control of Coefficient Function in a Hyperbolic Problem with Dirichlet Conditions

Seda İğret Araz and Murat Subaşi

#### Abstract

This paper presents theoretical results about control of the coefficient function in a hyperbolic problem with Dirichlet conditions. The existence and uniqueness of the optimal solution for optimal control problem are proved and adjoint problem is used to obtain gradient of the functional. However, a second adjoint problem is given to calculate the gradient on the space ${W}_{\mathrm{2}}^{\mathrm{1}}\left(\mathrm{0},l\right).$ After calculating gradient of the cost functional and proving the Lipschitz continuity of the gradient, necessary condition for optimal solution is constructed.

#### Article information

**Source**

Int. J. Differ. Equ., Volume 2018, Special Issue (2018), Article ID 7417590, 6 pages.

**Dates**

Received: 6 February 2018

Revised: 2 April 2018

Accepted: 4 April 2018

First available in Project Euclid: 10 October 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ijde/1539136980

**Digital Object Identifier**

doi:10.1155/2018/7417590

**Mathematical Reviews number (MathSciNet)**

MR3861847

#### Citation

İğret Araz, Seda; Subaşi, Murat. On the Control of Coefficient Function in a Hyperbolic Problem with Dirichlet Conditions. Int. J. Differ. Equ. 2018, Special Issue (2018), Article ID 7417590, 6 pages. doi:10.1155/2018/7417590. https://projecteuclid.org/euclid.ijde/1539136980