# On boundary control problems for the Klein–Gordon–Fock equation with an integrable coefficient

Control Theory

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## Abstract

We consider the process described in the rectangle *Q* _{ T } = [0 ≤ *x* ≤ *l*] × [0 ≤ *t* ≤ *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* = 2*l* the problem of boundary control of this process by the condition *u*(0, *t*) = µ(*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*.

## Keywords

Integration Domain Boundary Control Mixed Problem Volterra Equation Exact Controllability
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