IJPAM: Volume 76, No. 2 (2012)

WELL-POSEDNESS OF A QUASILINEAR PARABOLIC
OPTIMAL CONTROL PROBLEM

M.H. Farag$^1$, T.A. Talaat$^2$, E.M. Kamal$^3$
Department of Mathematics
Faculty of Science
Minia University
Minia, EGYPT


Abstract. In this paper we investigate the existence and uniqueness for the solution of the problem of determining the $v=(v_{0},v_{1},v_{2})$ in the quasilinear parabolic equation $\frac{\partial y}{\partial t} - \sum_{i=1}^{n} \frac{\partial }{\partial x_{i}... ... +\sum_{i=1}^{n} B_{i}(y,v_{1}) \frac{\partial y}{\partial x_{i}}=f(x,t,v_{2}) $. For the objective functional $ J_{\beta}(v)= \int_{S} [y(\zeta,t) - f_{0}(\zeta,t)]^{2} d\zeta dt + \beta \sum_{m=0}^{2} \Vert v_{m}-\omega_{m}\Vert^{2}_{l_{2}},$ it is proven that the problem has at least one solution for $\beta \ge 0$, and has a unique solution for $\beta > 0$.

Received: June 2, 2011

AMS Subject Classification: 49J20, 49K20, 49M29, 49M30

Key Words and Phrases: optimal control, quasilinear parabolic equation, existence and uniquness theorems

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 76
Issue: 2