Exact controllability of second order hyperbolic systems with control in the Dirichlet boundary conditions
For a given bounded domain \(\Omega\subset {\mathbb{R}}^ n \)with C 1 boundary and a point \(X_ 0\in {\mathbb{R}}^ n \)let \(S_ T=\partial \Omega \times (0,T)\) and \(S\) \(0_ T=\{(x,t)\in S_ T:\) \(<x-x_ 0,\nu (x)>>0\}\) where \(\nu\) (x) is the outward normal at \(x\in \partial \Omega\). The paper considers the initial-boundary problem \[ (1)\quad U_{tt}-\Delta U=0\quad in\quad \Omega \times (0,T),\quad U(\cdot,0)=U_ 0,\quad U_ t(\cdot,0)=U_ 1\quad in\quad \Omega,\quad U|_{S_ t}=v. \] It is shown that there exists \(T_ 0\), depending only on \(\Omega\) and \(X_ 0\), such that for \(T>T_ 0\) and every chosen pairs \((U_ 0,U_ 1)\) and \((U_ 0',U_ 1')\) from \(L_ 2(\Omega)\times H^{-1}(\Omega)\) there exists a control \(v\in L_ 2(S_ T)\) such that 1) \(v(x,t)=0\) on \(S_ T\setminus S\) \(0_ T\); 2) the solution U of (1) satisfies \(U(\cdot,T)=U_ 0'\), \(U_ t(\cdot,T)=U_ 1'\) in \(\Omega\).
scientific article; zbMATH DE number 4038615