We present an optimal control model to simultaneously determine the optimal planting density, thinning schedules, harvest age, and revenues of an even-aged longleaf pine (Pinus palustris Mill) stand, an iconic species in the Southeastern United States. We assume that the forest stand is managed for timber production and carbon sequestration under different site indexes—a measurement of potential forest productivity. Our simulation results show that the optimal planting density tends to increase when longleaf pine is managed in medium and high site indexes. Furthermore, the optimal harvest age tends to be extended with payments for carbon sequestration.

A conceptual optimal control theory model which considers farm level decision making with respect to soil management is developed. A simplified version of the theoretical model is applied to the Piedmont area of Virginia. The model includes the productivity impacts of both soil erosion and technological progress. Both the theoretical model and its empirical application are improvements over previous efforts. Results suggest that farmers in the study area can achieve substantial reductions in soil erosion by adopting alternative farming practices.

A finite-element multigrid scheme for elliptic Nash-equilibrium multiobjective optimal control problems with control constraints is investigated. The multigrid computational framework implements a nonlinear multigrid strategy with collective smoothing for solving the multiobjective optimality system discretized with finite elements. Error estimates for the optimal solution and two-grid local Fourier analysis of the multigrid scheme are presented. Results of numerical experiments are presented to demonstrate the effectiveness of the proposed framework.

A mesh-independent, robust, and accurate multigrid scheme to solve a linear state-constrained parabolic optimal control problem is presented. We first consider a Lavrentiev regularization of the state-constrained optimization problem. Then, a multi-grid scheme is designed for the numerical solution of the regularized optimality system. Central to this scheme is the construction of an iterative pointwise smoother which can be formulated as a local semismooth Newton iteration. Results of numerical experiments and theoretical twogrid local Fourier analysis estimates demonstrate that the proposed scheme is able to solve parabolic state-constrained optimality systems with textbook multigrid efficiency.

The inadequacy of the traditional sliding mode variable structure (SMVS) control method for cruise missiles is addressed. An improved SMVS control method is developed, in which the reaching mode segment of the SMVS control is decomposed into an acceleration accessing segment, a speed keeping segment, and a deceleration buffer segment. A time-fuel optimal control problem is formulated as an optimal control problem involving a switched system with unknown switching times and subject to a continuous state inequality constraint. The new design method is developed based on a control parametrization, a time scaling transform and the constraint transcription method. A sequence of approximate optimal parameter selection problems is obtained with fixed switching time points and a canonical state inequality constraint. Each approximate optimal parameter selection problem can be solved effectively by using existing gradient-based optimization techniques. The convergence of these approximate optimal solutions to the true optimal solution is assured. Simulation results show that the proposed method is highly effective. The response speed of the missile under the control law obtained by the proposed method is improved significantly, while the elevator of the missile is constrained to operate within its permitted range.

A method to obtain linear selection indexes that maximize objectives that involve average profit in one or more generations within a planning horizon based on non-linear profit functions, was derived through application of optimal control theory. The method involves simultaneous optimization of indexes for each generation in the planning horizon. Optimum linear indexes were found to be conform indexes derived from selection index theory, with economic values equal to a weighted sum of partial derivatives of the profit function at the trait means which result in each generation of the planning horizon. Numerical procedures to derive optimum indexes are presented. Methods and properties of alternative strategies for selection witli non-linear profit functions are illustrated for selection on egg weight and rate of lay in poultry. In the example, the additional benefit of selection indexes that maximize cumulative net present value of profit over a planning horizon of10 years was small relative to use of traditional selection procedures. Optimal indexes were also derived with a derivative-free non-linear programming optimizer, with identical results. The latter method also allows incorporation of additional constraints.

Possible extensions of the optimal control methodology to address other problems related to optimization of selection over multiple generations are discussed.

The Pareto-optimal design for profit-sharing is derived under general assumptions as to the utility function of both the insured and the insurer. This generalizes the result of Jones and Gerber and explains commonly used dividend formulas in terms of risk aversion.