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Some results on controllability for linear and nonlinear heat equations in unbounded domains. (English) Zbl 1170.93007

The authors consider the null controllability of heat equation in unbounded domain for two problems. First one is the following linear parabolic problem
\[ \begin{aligned} y_t-\Delta y+B\cdot\nabla y+ay=vl_\omega&\quad\text{in }Q=\Omega\times(0,T),\\ y=0&\quad \text{on }\Sigma=\partial\Omega\times(0,T),\\ y(x,0)=y_0(x) &\quad\text{in }\Omega, \end{aligned}\tag{1} \]
where \(B\in L^\infty(Q)^N\), \(a\in L^\infty(Q)\), \(v\in L^2(\Omega)\) and \(y_0\in L^2(\Omega)\), \(\Omega\subset\mathbb R^N\) is an unbounded connected open set with boundary \(\partial\Omega\) of class \(C^{0,1}\), \(\omega\subset\Omega\) a nonempty open subset, time \(T>0\), \(l_\omega\) denotes the characteristic function of subset \(\omega\), \(y=y(x,t)\) is the state and \(v=v(x,t)\) is the control function.
Second one is the nonlinear problem
\[ \begin{aligned} y_t-\Delta y+f(y,\nabla y)+ay=vl_\omega &\quad\text{in }Q=\Omega\times(0,T),\\ y=0&\quad \text{on }\Sigma=\partial\Omega\times(0,T),\\ y(x,0)=y_0(x) &\quad\text{in }\Omega, \end{aligned}\tag{2} \]
where \(f:\mathbb R\times\mathbb R^N\to\mathbb R\) is a locally Lipschitz-continuous function, \(y_0\) is given in Banach space, and \(v\) is control which action to the system through the open set \(\omega\). The systems (1) and (2) are null controllable at time \(T\) if for every \(y_0\in L^2(\Omega)\) for (1), respectively for (2), there exists a solution \(y\) that satisfies \(y(x,T)=0\) in \(\Omega\). The authors prove null controllability for system (1) and (2) under formulated assumptions.

MSC:

93B05 Controllability
93B07 Observability
35M20 PDE of composite type (MSC2000)
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