Optimal spatial patterns in feeding, fishing, and pollution

<p style='text-indent:20px;'>Infinite time horizon spatially distributed optimal control problems may show so–called optimal diffusion induced instabilities, which may lead to patterned optimal steady states, although the problem itself is completely homogeneous. Here we show that this can be considered as a generic phenomenon, in problems with scalar distributed states, by computing optimal spatial patterns and their canonical paths in three examples: optimal feeding, optimal fishing, and optimal pollution. The (numerical) analysis uses the continuation and bifurcation package <inline-formula><tex-math id="M1">\begin{document}$\mathtt{pde2path} $\end{document}</tex-math></inline-formula> to first compute bifurcation diagrams of canonical steady states, and then time–dependent optimal controls to control the systems from some initial states to a target steady state as <inline-formula><tex-math id="M2">\begin{document}$ t\to\infty $\end{document}</tex-math></inline-formula>. We consider two setups: The case of discrete patches in space, which allows to gain intuition and to compute domains of attraction of canonical steady states, and the spatially continuous (PDE) case.</p>

Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

<p style='text-indent:20px;'>We consider one-dimensional distributed optimal control problems with the state equation being given by the viscous Burgers equation. We discretize using a space-time discontinuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.</p>

Solution stability to parametric distributed optimal control problems with finite unilateral constraints

<p style='text-indent:20px;'>This paper deals with stability of solution map to a parametric control problem governed by semilinear elliptic equations with finite unilateral constraints, where the objective functional is not convex. By using the first-order necessary optimality conditions, we derive some sufficient conditions under which the solution map is upper semicontinuous with respect to parameters.</p>

A Nitsche-eXtended Finite Element Method for Distributed Optimal Control Problems of Elliptic Interface Equations

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the {L^{2}} norm are derived. Numerical results are provided to verify the theoretical results.