An optimal control problem for the Oskolkov equation (Q1873485)

In this short communication the authors consider the Cauchy-Dirichlet problem for the equation \[ (1- \kappa\Delta) \Delta\partial\psi/\partial t= \nu\Delta^2\psi- \partial(\psi, \Delta\psi)/\partial(x, y)+ u \] on the cylinder \(\Omega\times \mathbb{R}\), where \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) with \(C^\infty\) boundary \(\partial\Omega\). The problem, which models the dynamics of certain viscoelastic incompressible fluids, is transformed into the Cauchy problem for a semilinear operator equation. The authors present conditions ensuring that the latter problem admits a unique solution \(v\in H^1((0,\tau);{\mathcal V})\), where \[ {\mathcal V}= \{v\in W^4_2(\Omega): v(x)=\Delta v(x)= 0, x\in\partial\Omega\}. \] An associated optimal control problem with respect to \(u\) is also considered.