Stabilization and simultaneous boundary controllability for a class of evolution systems
The author considers the following initial boundary value problem: \[ \begin{aligned} &\begin{cases} i\frac{\partial u}{\partial t}- \sum_{p,q=1}^n \frac{\partial }{\partial x_p} \Bigl( A_{pq} \frac{\partial u}{\partial x_q}\Bigr)=0,\\ i\frac{\partial v}{\partial t}- \sum_{p,q=1}^n \frac{\partial }{\partial x_p} \Bigl( A_{pq} \frac{\partial v}{\partial x_q}\Bigr)=0, \end{cases} \qquad (x,t)\in \Omega\times (0,T),\tag{1}\\ &u\bigl|_{t=0}= f(x), \quad v\bigl|_{t=0}= g(x), \quad x\in\Omega,\tag{2}\\ &\begin{cases} \sum_{p,q=1}^n A_{pq} \frac{\partial u}{\partial x_q} \nu_p+au+b \frac{\partial u}{\partial t}+ \xi\frac{\partial v}{\partial t} \Bigl|_{S_0\times (0,T)}= h(x,t), \quad u\Bigl|_{S_1\times (0,t)} =0,\\ \sum_{p,q=1}^n A_{pq} \frac{\partial v}{\partial x_q} \nu_p+ cv-\xi \frac{\partial u}{\partial t} \Bigl|_{S_0\times (0,T)}=0, \quad v\Bigl|_{S_1\times (0,T)} =0, \end{cases} \tag{3} \end{aligned} \] where \(\Omega\) is a bounded domain in \(R^n\) with smooth boundary \(S\) which contains two disjoint closed surfaces \(S_0\) and \(S_1\), \(u= (u^1(x,t),\dots, u^m(x,t))\), \(v= (v^1(x,t),\dots, v^m(x,t))\), \(A_{pq}= A_{qp}^*\) are \(m\times m\) square matrices, \(\nu= (\nu_1,\dots, \nu_n)\) is the unit outer normal and \(a=a(x)\geq 0\), \(c=c(x)\geq 0\), \(b(x)\geq 0\), \(\xi(x)\geq 0\) are scalar functions on \(S_0\). The results on uniform stabilization of the solutions are established using the classical semigroup theory. The uniform exponential decay of the energy is obtained by a certain estimate of \({\mathcal U}(t)\), combined with the semigroup property, where \({\mathcal U}(t)\), \(t\geq 0\), is the strongly continuous semigroup of contractions generated by the operator \(B\) defined on an appropriate Hilbert space as follows: \[ B(u,v)= \{-iAu, -iAv\}, \quad Au= \sum_{p,q=1}^n \frac{\partial }{\partial x_p} \biggl(A_{pq} \frac{\partial u}{\partial x_q} \biggr). \] Exact boundary controllability is obtained through Russell's ``controllability via stabilizability'' principle. Moreover, simultaneous exact controllability is proved for the problems: \[ \begin{aligned} &\begin{cases} i\frac{\partial\varphi}{\partial t}- \sum_{p,q=1}^n \frac{\partial}{\partial x_p} \Bigl( A_{p,q} \frac{\partial\varphi}{\partial x_q} \Bigr)=0, \quad (x,t)\in \Omega\times (0,T),\\ \varphi\bigl|_{t=0}= f_1(x), \quad x\in \Omega,\\ \varphi\bigl|_{S_0\times (0,T)}=P(x,t), \quad \varphi\bigl|_{S_1\times (0,T)}=0,\end{cases}\\ &\begin{cases} i\frac{\partial\psi}{\partial t}- \sum_{p,q=1}^n \frac{\partial}{\partial x_p} \Bigl( A_{pq} \frac{\partial\psi} {\partial x_q}\Bigr)=0, \quad (x,t)\in \Omega\times (0,T),\\ \psi\bigl|_{t=0}= f_2(x), \quad x\in \Omega,\\ \sum_{p,q=1}^n A_{pq} \frac{\partial\psi}{\partial x_q} \nu_p+ c\psi\bigl|_{S_0\times (0,T)}= Q(x,t), \quad \psi\bigl|_{S_1\times (0,T)}=0. \end{cases} \end{aligned} \] {}.
scientific article; zbMATH DE number 1229867