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Zbl 0938.93008
Aniţa, Sebastian; Barbu, Viorel
Null controllability of nonlinear convective heat equations. (English)
[J] ESAIM, Control Optim. Calc. Var. 5, 157-173 (2000). ISSN 1292-8119; ISSN 1262-3377/e

The authors study the internal and boundary exact null controllability of the nonlinear system $$\cases y_t-\Delta y+ \text{div}(by(x,t))= m(x)u(x,t),\quad & (x,t)\in Q,\\ y(x,t)= 0,\quad & (x,t)\in\Sigma,\\ y(x,0)= y_0(x),\quad & x\in\Omega,\endcases\tag 1$$ in the following sense: for every $T>0$ and for all $y_0$ in a suitable space, there are $(y,u)\in H^{2,1}(Q)\times L^2(\Omega)$ which satisfy (1) and such that $y(x,T)= 0$ a.e. $x\in\Omega$. Here $m$ denotes the characteristic function of some nonempty open subset $\omega\subset\Omega$ and $H^{2,1}(Q)= W^{2,1}_2(Q)\cap L^2(0,T; H^1_0(\Omega))$.\par A first result states that, for the case $n=1$, if $$\align |b'(r)|&\le (1+\ln^{1/2}(|r|+ 1))\varphi(r),\quad\forall r\in\bbfR,\\ |b''(r)|&\le(1+ \ln^{1/2}(|r|+ 1))(|r|+ 1)^{-1} \varphi_0(r),\quad \text{a.e. }r\in\bbfR,\endalign$$ where $\varphi,\varphi_0: \bbfR\to\bbfR$ are continuous functions such that $\lim_{|r|\to\infty} \varphi(r)= \lim_{|r|\to\infty} \varphi_0(r)= 0$, then the system (1) is exactly null controllable for all $y_0\in H^1_0(\Omega)$, and for the cases $n= 2,3$, if $$\align |b'(r)|&\le (1+ \ln^{1/2}(|r|+ 1))\varphi(r),\quad \forall r\in\bbfR,\\ |b''(r)|&\le (1+ \ln^{1-2\delta/2}(|r|+ 1))(|r|+ 1)^{-1}\varphi_0(r),\quad\text{a.e. }r\in\bbfR,\endalign$$ where $\delta\in \left(0,{1\over 2}\right)$ and $\varphi$, $\varphi_0$ are as above, then (1) is exactly null controllable for all $y_0\in H^1_0(\Omega)\cap H^2(\Omega)$.\par Also, according to the above assumptions, there are $v\in L^2(\Sigma_0)$ and $y\in W^{2,1}_2(Q)$ such that $$\cases y_t-\Delta y+\text{div}(b(y))= 0,\quad (x,t)\in Q,\\ y=\cases V,\ (x,t)\in\Sigma_0,\\ 0, (x,t)\in \Sigma\setminus\Sigma_0,\endcases\\ y(x,0)= y_0(x),\quad y(x,T)= 0,\quad x\in\Omega,\endcases$$ where $\Sigma_0= \Sigma\cap B_0(x_0; \varepsilon)$, $x_0\in\partial\Omega$ and $\varepsilon> 0$ is a constant. For the following more general equation $$\cases y_t-\Delta y+ \text{div}(b(y(x,t))+ f(x,t,y(x,t))= m(x)u(x,t),\quad (x,t)\in Q,\\ y(x,t)= 0,\quad (x,t)\in\Sigma,\\ y(x,0)= y_0(x),\quad x\in\Omega,\endcases\tag 2$$ if, in addition, the function $f: Q\times\bbfR\to \bbfR$ is continuous in the third variable, measurable in $(x,t)$ and satisfies $$\align f(x,t,r) &\ge -\mu_0r^2,\quad \forall r\in\bbfR,\quad (x,t)\in Q,\\ |f(x,t,r)|&\le|r|\xi(r)(1+ (\ln(1+|r|))^{3/2}),\quad\forall(x,t,r)\in Q\times\bbfR,\endalign$$ where $\mu_0\ge 0$ and $\xi:\bbfR\to\bbfR$ is continuous function, such that $\lim_{|r|\to\infty} \xi(r)=0$, then the system (2) is exactly null controllable for all $y_0\in H^1_0(\Omega)\cap H^2(\Omega)$; if $n=1$ and $y_0\in H^1_0(\Omega)\cap W^2_{5/2}(\Omega)$, if $n= 2,3$.\par These results are proved by using Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system, interpolation inequalities and some estimates in the theory of parabolic boundary value problems in $L^k$.
[M.Ivanovici (Craiova)]

MSC 2000:: *93B05 Controllability
93C20 Control systems governed by PDE
35K05 Heat equation
46B70 Interpolation between normed linear spaces
46E35 Sobolev spaces and generalizations

Keywords: Carleman estimates; interpolation inequality; Kakutani fixed point theorem; nonlinear partial differential equation; exact null controllability

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