Optimal control of the gradient systems

In this paper we consider the optimal control problems governed by the gradient systems for the Ginzburg-Landau free energy where denotes a potential function and ε is the diffusivity. One example of gradient systems are the Schlögl equation arising in chemical waves with a quartic potential function F(y). Gradient systems are characterized by energy decreasing property . Numerical integrators that preserve the energy decreasing property in the discrete setting are called energy or gradient stable. It is known that the implicit Euler method is first order unconditionally energy stable method. The only second order unconditionally energy stable method is the average vector field (AVF) method. We discretize the gradients systems by discontinuous Galerkin method in space and by AVF integrator in time. We solve optimal control problems for the Schlögl equation with traveling and spiraling waves using sparse and regularized controls .