Solvability of stationary boundary control problems for heat convection equations
The following optimal control problem is studied for the motion of a viscous heat conducting fluid in \(\Omega\): \[ \begin{aligned} -&\nu\Delta u + (u\cdot\text{grad})u + \text{grad} p + \beta GT =f,\\ -&\lambda\Delta T + u\cdot\text{grad} T = f,\qquad \text{div} u=0,\\ u=g \;&\text{on} \;\Gamma_1,\qquad u=u_0 \;\text{on} \;\Gamma\setminus \Gamma_1,\qquad T=g\;\text{on} \;\Gamma_2,\\ &\lambda\left(\frac{\partial T}{\partial n} + \alpha T\right)=\psi\;\text{on} \;\Gamma\setminus\Gamma_2, \end{aligned} \] where \(\Omega\) is a bounded domain in \(\mathbb R^d\), \(d=2,3\), with Lipschitz boundary \(\Gamma\), which satisfies the condition \(\Gamma\in C^{0,1}\) and \(\Gamma\) has finitely many connected components \(\Gamma^{(1)},\dots,\Gamma^{(n)}\). The functions \(u\), \(\rho\), and \(T\) are the velocity, pressure, and temperature; the function \(g\) describes the Dirichlet boundary control for the velocity \(u\) on the part \(\Gamma_1\) of \(\Gamma\). The parts \(\Gamma_1\) and \(\Gamma_2\) may coincide or differ; moreover, any of them may coincide with \(\Gamma\). In the article under consideration, the optimal control problem is studied for the case \(\beta\neq 0\). This enables us to state some new extremal problems of hydrodynamics; in particular, the problem of controlling the temperature distribution in the domain by choosing the boundary regime for velocity and, conversely, the problem of controlling the flow of a viscous heat conducting fluid by the distribution of the boundary temperature. The aim of the article under review is to analyze optimal control problems. This requires proving existence for weak solutions to the problem, solvability of the optimal boundary control problem for a weakly lower semicontinuous quality functional, and justifying applicability of the Lagrange multiplier principle by using the apparatus of the theory of extremal problems. Using the Lagrange principle, the author derives an optimality system for the sought and adjoint states.
scientific article; zbMATH DE number 1285995
scientific article