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On the exact controllability of a Petrowsky system. (English) Zbl 0711.34077
Qualitative theory of differential equations, 3rd Colloq., Szeged/Hung. 1988, Colloq. Math. Soc. János Bolyai 53, 357-363 (1990).
[For the entire collection see Zbl 0695.00015.]
Given a bounded open set \(\Omega\) of \({\mathbb{R}}^ n\) having a \({\mathbb{C}}^ 2\) boundary \(\Gamma\) with \(\Gamma ={\bar \Gamma}_+\cup {\bar \Gamma}_-\) (where \(\Gamma_+\) and \(\Gamma_-\) are disjoint open subsets of \(\Gamma\)) and \(q\in L^{p+\epsilon}({\mathbb{R}}^ n)\), consider the control system \(\ddot y+\Delta^ 2y+qy=0\), with \(y(0)=y^ 0\), \(\dot y(0)=y^ 1\), \(y=\partial_ vy=0\) on \(\Gamma_ -\times (0,T)\) and \(y=0\), \(\partial_ vy=v\) on \(\Gamma_+\times (0,T)\). Here \(\partial_ v\) is the normal derivative and v is a control function. It is known that for every initial condition \((y^ 0,y^ 1)\in L^ 2(\Omega)\times H^{-2}(\Omega)\), the above system has a solution \((y,\dot y)\in C(0,T,L^ 2(\Omega)\times H^{-2}(\Omega))\) \((T>0\) is arbitrary). The author proves the following
Theorem 1. The above system is exactly controllable, i.e. for any \((y^ 0,y^ 1)\in L^ 2(\Omega)\times H^{-2}(\Omega)\), there exists a \(v\in L^ 2(\Gamma_+\times (0,T))\) such that \(y(T)=\dot y(T)=0\) on \(\Omega\) (i.e. any initial state can be driven to 0). This result was known in the case \(q=0\). The proof developes a crucial a priori estimate for the solution of the corresponding homogeneous problem and then uses the Hilbert uniqueness method.
Theorem 2. Let I be a compact interval and u be a solution to the corresponding homogeneous problem with initial data \((u^ 0,u^ 1)\), then there is a constant \(C>0\) such that \[ \int_{I}\int_{\Gamma_+}(\Delta u)^ 2 d\Gamma dt\geq C(\| u^ 0\|^ 2_{H^ 2_ 0(\Omega)}+\| u^ 1\|^ 2_{L^ 2(\Omega)}). \]
Reviewer: M.G.Nerurkar

34H05 Control problems involving ordinary differential equations
93B05 Controllability
Zbl 0695.00015