Second order sufficient optimality conditions for a nonlinear elliptic boundary control problem
Let \(\varphi:\mathbb{R}\to\mathbb{R}\), \(\psi,b:\mathbb{R}\times\mathbb{R}\to\mathbb{R}\) and all derivatives up to second-order be globally Lipschitz continuous, \(b(.,y)\) decreasing for all \(y\in\mathbb{R}\); \(\Omega\) a bounded domain of \(\mathbb{R}^n\) with a sufficiently smooth boundary such that \(W^{1,p}(\Omega)\) is continuously embedded in \(C(\overline\Omega)\) and \(w(u)\in W^{1,p}(\Omega)\) for some \(p>n\), where \(u\in U\equiv\{u\in L^\infty(\partial\Omega): m\leq u\leq M\}\) and \(m,M\in\mathbb{R}\), \[ \int_\Omega \Biggl(\sum^n_{i=1} \partial_iw(u)\partial_iv+ w(u)v\Biggr)dx= \int_{\partial\Omega} b(w(u),u)vd\sigma\quad\text{for all }v\in W^{1,2}(\Omega), \] \[ |w(u_1)- w(u_2)|_{W^{1,p}(\Omega)}\leq c_p|u_1- u_2|_{L^p(\partial\Omega)}\quad\text{for }c_p>0; \] \[ L:(C(\overline\Omega)\cap W^{1,2}(\Omega))\times U\times(C(\overline\Omega)\cap W^{1,2}(\Omega))\to \mathbb{R}, \] \[ L(w,u,v)= \int_\Omega\varphi\circ wdx+ \int_{\partial\Omega} \psi(w,u)d\sigma-\int_\Omega \Biggl(\sum^n_{i=1} \partial_iw\partial_iv+ wv\Biggr)dx+ \int_{\partial\Omega}b(w,u)vd\sigma. \] Theorem. Let \(u_0\in U\), \(v_0\in C(\overline\Omega)\cap W^{1,2}(\Omega)\) such that \[ \int_{\partial\Omega} (\partial_2\psi(w(u_0),u_0)+ \partial_2b(w(u_0),u_0)v_0)(u- u_0)d\sigma\geq 0\quad\text{for all }u\in U, \] \[ \begin{multlined}\int_\Omega \Biggl(\sum^n_{i=1}\partial_i v_0\partial_iv+ v_0v- \varphi'\circ w(u_0)v\Biggr)dx=\\ \int_{\partial\Omega}(\partial_1b(w(u_0),u_0)v_0+ \partial_1\psi(w(u_0),u_0))vd\sigma\text{ for all }v\in W^{1,2}(\Omega);\end{multlined} \] let \(\varepsilon>0\) and \[ \Gamma_\varepsilon= \{x\in\partial\Omega:|\partial_2 \psi(w(u_0)(x), u_0(x))+ \partial_2b(w(u_0)(x), u_0(x))v_0(x)|\geq \varepsilon\}; \] suppose that there exists \(\alpha>0\) such that \[ L(.,.,v_0)''(w(u_0),u_0)((h,u- u_0),(h,u- u_0))\geq\alpha|(h,u- u_0)|^2_{W^{1,2}(\Omega)\times L^2(\partial\Omega)}\text{ for }u\in U \] such that \(u(x)= u_0(x)\) for a.e. \(x\in\Gamma_\varepsilon\) and where \(h\) is such that \[ \begin{multlined} \int_\Omega \Biggl(\sum^n_{i=1} \partial_ih\partial_iv+ hv\Biggr)dx=\\ \int_{\partial\Omega}(\partial_1b(w(u_0),u_0)h+ \partial_2b(w(u_0),u_0)(u- u_0))vd\sigma\text{ for all }v\in W^{1,2}(\Omega).\end{multlined} \] Then there exist \(\beta>0\) and \(\rho>0\) such that \[ L(w(u),u,v_0)\geq L(w(u_0),u_0,v_0)+ \beta|(w(u),u)- (w(u_0),u_0)|^2_{W^{1,2}(\Omega)\times L^2(\partial\Omega)}\text{ for all }u\in U \] such that \[ |(w(u),u)- (w(u_0),u_0)|^2_{W^{1,p}(\Omega)\times L^q(\partial\Omega)}\leq\rho\text{ for }q=\infty; \] for \(q=p\) if furthermore \(b(w,u)= b_1(w)+ b_2(w)u\) and \(\psi(w,u)= \psi_1(w)+ \psi_2(w)u+\gamma u^2\), where \(\gamma\geq 0\) and \(\psi_1,\psi_2,b_1,b_2:\mathbb{R}\to \mathbb{R}\) and all derivatives up to second-order are globally Lipschitz continuous.
scientific article; zbMATH DE number 935179
scientific article