Optimal boundary control of distributed systems involving dynamic boundary conditions (Q1380943)

An example of the type of control system studied here is that described by the equation \[ {\partial \over \partial t}T(t, x)= \nabla \cdot(k(x) \nabla T(t, x)) + \nu(t, x) \cdot \nabla T(t, x),\qquad T(0, x)= T_0(x), \] in a bounded domain \(\Omega \subseteq\mathbb{R}^3\). The boundary \(\Gamma\) of \(\Omega\) consists of two isolated pieces \(\Gamma_1\) and \(\Gamma_2.\) In the second, we have the usual type of boundary condition, \[ T(t,x)=h(t,x) \quad (x\in\Gamma) \] while in the first there is a ``dynamic'' boundary condition: the restriction \(T(t, x)|_{\Gamma_1}\) satisfies \[ {\partial \over \partial t} T(t, x) \big| _{\Gamma_1}= \Delta_{\Gamma_1}T(t, x)\big| _{\Gamma_1} - \beta D_\nu T(t, x)\big| _{\Gamma_1} + g(x, T(t, x)\big| _{\Gamma_1}, u(t, x)), \quad T(0, x)= T_1(x), \] where \(D_\nu\) is the outer normal derivative at the boundary and \(u(t, x)\) is the control. The authors deal with this kind of system through the abstract model \[ {d \over dt} Bx(t) = A(t)x(t) + f(Bx(t), u(t)) , \] with \(A(t), B\) unbounded operators.