Second order sufficient optimality conditions for a nonlinear elliptic boundary control problem. (English) Zbl 0879.49020

Casas, E.; Tröltzsch, F.; Unger, A.

Z. Anal. Anwend. 15, No. 3, 687-707 (1996).

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 \[ \int_{\partial\Omega} (\partial_2\psi(w(u_0),u_0)+ \partial_2b(w(u_0),u_0)v_0)(u- u_0)d\sigma\geq 0\quad\text{for all }u\in U, \]
\[ \begin{split}\int_\Omega \Biggl(\sum^n_{i=1}\partial_i v_0\partial_iv+ v_0v- \varphi'\circ w(u_0)v\Biggr)dx=\\ \int_{\partial\Omega}(\partial_1b(w(u_0),u_0)v_0+ \partial_1\psi(w(u_0),u_0))vd\sigma\text{ for all }v\in W^{1,2}(\Omega);\end{split} \] 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 \[ \begin{split} \int_\Omega \Biggl(\sum^n_{i=1} \partial_ih\partial_iv+ hv\Biggr)dx=\\ \int_{\partial\Omega}(\partial_1b(w(u_0),u_0)h+ \partial_2b(w(u_0),u_0)(u- u_0))vd\sigma\text{ for all }v\in W^{1,2}(\Omega).\end{split} \] 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.

Reviewer: G.Bottaro (Genova)

MSC:

49K20	Optimality conditions for problems involving partial differential equations
35J99	Elliptic equations and elliptic systems
49K27	Optimality conditions for problems in abstract spaces

Keywords:

sufficient optimality conditions; nonlinear elliptic boundary control

References:

[1]	Adams, R. A.: Sobolev Spaces (Pure and Applied Mathematics: Vol. 65). New York: Academic Press Inc. 1978.
[2]	Casas, E.: Pontryagin’s Principle for Optimal Control Problems Governed by Semilinear Elliptic Equations (International Series of Numerical Mathematics (ISNM): Vol. 118). Basel: Birkhãuser Verlag 1994, pp. 97 - 114. · Zbl 0810.49025
[3]	Casas, E. and J. Yong: Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations. Duff. mt. Equ. 8 (1995), 1 - 18. · Zbl 0817.49025
[4]	Collatz, L. and W. Wetterling: Optimiernngsaufgaben (2. Aufi.). Berlin-Heidelberg-New York: Springer-Verlag 1971.
[5]	Dontchev, A. L., Hager, W. W., Poore, A. B. and B. Yang: Optimality, stability, and convergence in nonlinear control. Appi. Math. Optim. 31(1995), 297 - 326. · Zbl 0821.49022 · doi:10.1007/BF01215994
[6]	Eppler, K. and A. Unger: Boundary control of semilinear elliptic equations - existence of optimal solutions. Control and Cybernetics 1995 (to appear). (7] Goldberg, H. and F. Tröltzsch: Second order optimality conditions for a class of control problems governed by non-linear integral equations with applications to parabolic boundary control. Optimization 20 (1989), 687 - 698.
[7]	Goldberg, H. and F. Tróltzsch: Second-order sufficient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J. Control Optim. 31 (1993), 1007 - 1025. · Zbl 0787.49011 · doi:10.1137/0331045
[8]	[10] loffe, A. D.: Necessary and sufficient conditions for a local minimum. Part 3: Second order conditions and augmented duality. SIAM J. Control Optim. 17 (1979), 266 - 288. · Zbl 0417.49029 · doi:10.1137/0317021
[9]	Kinderlehrer, D. and G. Stampacchia: An Introduction to Variational Inequalities and their Applications. New York: Academic Press 1980. · Zbl 0457.35001
[10]	Ladyzhenskaya, 0. A. and N. N. Ural’tseva: Linear and Quasilinear Elliptic Equations. New York - London: Academic Press 1968. · Zbl 0164.13002
[11]	Malanowski, K.: Sufficient optimality, conditions in optimal control. Technical report. Warsaw: Systems Res. Inst. Polish Acad. Sci. 1994. · Zbl 0810.49009
[12]	Maurer, H.: First and second order sufficient optimality conditions in mathematical pro- gramming and optimal control. Math. Progr. Study 14 (1981), 163 - 177. · Zbl 0448.90069 · doi:10.1007/BFb0120927
[13]	[16] Tiba, D.: Lectures on the Optimal Control of Elliptic Equations (Lecture Notes: Vol. 32). Jyvàskylä: Univ./ Dep. Math. 1995.