On boundary control problems for the Klein-Gordon-Fock equation with an integrable coefficient. (English. Russian original) Zbl 1320.93051
Differ. Equ. 51, No. 5, 701-709 (2015); translation from Differ. Uravn. 51, No. 5, 688-696 (2015).
Summary: We consider the process described in the rectangle \(Q_T = [0 \leq x \leq l] \times [0 \leq t \leq T]\) by the equation \(u_{tt}-u_{xx}-q(x,t)u = 0\) with the condition \(u(l,t) = 0\), where the coefficient \(q(x,t)\) is only square integrable on \(Q_T\). We show that for \(T = 2l\) the problem of boundary control of this process by the condition \(u(0, t) = \mu(t)\) has exactly one solution in the class \(W_2^1(Q_T)\) under minimum requirements on the smoothness of the initial and terminal functions and under natural matching conditions at \(x=l\).
MSC:
| 93C20 | Control/observation systems governed by partial differential equations |
| 35L20 | Initial-boundary value problems for second-order hyperbolic equations |
| 35Q93 | PDEs in connection with control and optimization |
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