On the Construction of Optimal Monotone Cubic Spline Interpolations

In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call thelocalcase, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs.     

Numerical solution of minimax optimal control problems by multiple shooting technique

This paper presents the application of the multiple shooting technique to minimax optimal control problems (optimal control problems with Chebyshev performance index). A standard transformation is used to convert the minimax problem into an equivalent optimal control problem with state variable inequality constraints. Using this technique, the highly developed theory on the necessary conditions for state-restricted optimal control problems can be applied advantageously. It is shown that, in general, these necessary conditions lead to a boundary-value problem with switching conditions, which can be treated numerically by a special version of the multiple shooting algorithm. The method is tested on the problem of the optimal heating and cooling of a house. This application shows some typical difficulties arising with minimax optimal control problems, i.e., the estimation of the switching structure which is dependent on the parameters of the problem. This difficulty can be overcome by a careful application of a continuity method. Numerical solutions for the example are presented which demonstrate the efficiency of the method proposed.     

Cross-linked poly(4-vinylpyridine-N-oxide) as a polymer-supported oxygen atom transfer reagent

Oxygen atom transfer (OAT) reagents are common in biological and industrial oxidation reactions. While many heterogeneous catalysts have been utilized in OAT reactions, heterogeneous OAT reagents have not been explored. Here, cross-linked poly(4-vinylpyridine-N-oxide), called x-PVP-N-oxide, was tested as a heterogeneous OAT reagent and its oxidation chemistry compared to its molecular counterpart, pyridine-N-oxide. The insoluble oxidant x-PVP-N-oxide demonstrated comparable reactivity to pyridine-N-oxide in direct oxidation reactions of phosphines and phosphites in acetonitrile, but x-PVP-N-oxide did not react in other solvents. The polymer backbone of x-PVP-N-oxide, however, allowed for easy filtering and recycling in sequential oxidation reactions. In addition, x-PVP-N-oxide was tested as the stoichiometric oxidant in a copper-catalyzed OAT reaction to α-diazo-benzeneacetic acid methyl ester. The heterogeneous oxidant was much less reactive than pyridine-N-oxide, indicating that interaction with the metal catalyst was challenging. These results demonstrated a proof-of-concept that recyclable, polymer-supported OAT reagents could be a viable OAT reagents in direct oxidation reactions without metal catalysts.     

Existence and Multiple Solutions of the Minimum-Fuel Orbit Transfer Problem

In this paper, the well-known problem of piloting a rocket with a low thrust propulsion system in an inverse square law field (say from Earth orbit to Mars orbit or from Earth orbit to Mars) is considered. By direct methods, it is shown that the existence of a fuel-optimal solution of this problem can be guaranteed, if one restricts the admissible transfer times by an arbitrarily prescribed upper bound. Numerical solutions of the problem with different numbers of thrust subarcs are presented which are obtained by multiple shooting techniques. Further, a general principle for the construction of such solutions with increasing numbers of thrust subarcs is given. The numerical results indicate that there might not exist an overall optimal solution of the Earth-orbit problem with unbounded free transfer time.     

Numerical computation of singular control problems with application to optimal heating and cooling by solar energy

The method presented here is an extension of the multiple shooting algorithm in order to handle multipoint boundary-value problems and problems of optimal control in the special situation of singular controls or constraints on the state variables. This generalization allows a direct treatment of (nonlinear) conditions at switching points. As an example a model of optimal heating and cooling by solar energy is considered. The model is given in the form of an optimal control problem with three control functions appearing linearly and a first order constraint on the state variables. Numerical solutions of this problem by multiple shooting techniques are presented.