The paper considers the transition of an economy from non-renewable to renewable energy. We set up a canonical growth model with damages in the household's welfare function and two energy sources – non-renewable and renewable energy. To produce renewable energy a capital stock must be built up. A socially optimal solution is considered that takes into account the negative externality from the non-renewable energy. We also study how the optimal solution can be mimicked in a market economy by policies using subsidies and tax rates. To solve the model numerically, we use Nonlinear Model Predictive Control. We study when a transition to renewable energy takes place and whether it occurs before the non-renewable resource is exhausted. In addition, we analyze the impact of the initial values of the non-renewable resource and of the capital stock on the time of paths of the variables.

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In this paper we present a spectral condition for the exponential stabilization of nonlinear control systems with constrained control range at singular points. The spectral approach in particular allows to formulate an equivalence result between exponential null controllability and exponential stabilization by means of a discrete feedback law. The key tool used is a discounted optimal control problem for the corresponding projected semilinear system, which also admits a numerical solution.

A time-varying nonlinear dynamical system is called a totally positive differential system (TPDS) if its Jacobian admits a special sign pattern: it is tri-diagonal with positive entries on the super-and sub-diagonals. If the vector field of a TPDS is T -periodic then every bounded trajectory converges to a T -periodic solution. In particular, when the vector field is time-invariant every bounded trajectory of a TPDS converges to an equlbrium. Here, we use the spectral theory of oscillatory matrices to analyze the behavior near a periodic solution of a TPDS. This yields information on the perturbation directions that lead to the fastest and slowest convergence to or divergence from the periodic solution. We demonstrate the theoretical results using a model from systems biology called the ribosome flow model.