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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

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