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Transactions of the American Mathematical Society

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Control problems governed by a pseudo-parabolic partial differential equation

Author: Luther W. White
Journal: Trans. Amer. Math. Soc. 250 (1979), 235-246
MSC: Primary 49B22; Secondary 49A22
MathSciNet review: 530053
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Abstract: Let G be a bounded domain in $ {R^n}$ and $ Q\, = \,G\, \times \,\left( {0,\,T} \right)$. We consider the solution $ y\left( u \right)$ of the pseudo-parabolic initial-value problem

\begin{multline}\left( {1\, + \,M\left( x \right)} \right)\,{y_t}\,\left( u \rig... ... \right)\, = \,0\,{\text{in}}\,{L^2}\,\left( G \right), \hfill \\ \end{multline}

, to be the state corresponding to the control u. Here $ M\left( x \right)$ and $ L\left( x \right)$ are symmetric uniformly strongly elliptic second-order partial differential operators. The control problem is to find a control $ {u_0}$ in a fixed ball in $ {L^2}\left( Q \right)$ such that (i) the endpoint of the corresponding state $ y\left( { \cdot ,\,T;\,{u_0}} \right)$ lies in a given neighborhood of a target Z in $ {L^2}\left( G \right)$ and (ii) $ {u_0}$ minimizes a certain energy functional. In this paper we establish results concerning the controllability of the states and the compatibility of the constraints, existence and uniqueness of the optimal control, existence and properties of Lagrange multipliers associated with the constraints, and regularity properties of the optimal control.

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Keywords: Pseudo-parabolic equation, optimal control, Lagrange multiplier, controllability, Lagrangian, regularity, target set
Article copyright: © Copyright 1979 American Mathematical Society

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