Bilinear optimal control for a wave equation. (English) Zbl 0939.49016

Liang, Min

Math. Models Methods Appl. Sci. 9, No. 1, 45-68 (1999).

The paper considers the optimal control problem \[ \int_Q [(w- w_d)^2+\alpha h^2] dx dt\to \min,\tag{1} \]
\[ w_{tt}=\Delta w+ hw+ f\quad\text{in }\Omega\times(0, T),\quad w=0\quad\text{on }\partial\Omega\times (0,T),\tag{2} \]
\[ w(0,x)= w_0(x),\quad w_t(0, x)= w_1(x), \] where \(\Omega\subset \mathbb{R}^n\) is a bounded domain and \(h\in U= \{h\in L_\infty(Q)\mid-M\leq h(x,t)\leq M\}\) is the control. The author shows the existence of an optimal control and that the mapping \(h\to w_h\) (\(w_h\) being the solution of (2) corresponding to a chosen \(h\in U\)) is Gâteaux differentiable in appropriate spaces. On this basis the necessary optimality conditions are derived and it is shown that for \(T>0\) small enough there exists only one solution of the optimality system, provided that the state and the solution of the conjugate equation are bounded in \(Q\).

Reviewer: U.Raitums (Riga)

MSC:

49K20	Optimality conditions for problems involving partial differential equations
35L05	Wave equation

Keywords:

wave equation; optimal control; necessary optimality conditions

References:

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