Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting

Summary The numerical solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known intergrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.     

Numerical treatment of delay differential equations by Hermite Interpolation

Summary A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically.Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.     

Facile synthesis and characterization of functionalized, monocrystalline rutile TiO2 nanorods

Functionalized, monocrystalline rutile TiO2 nanorods were prepared from TiCl4 in aqueous solution under acidic conditions in the presence of dopamine, followed by aging and hydrothermal treatment at 150 degrees C. The surface-bound organic ligand controls the morphology as well as the crystallinity and the phase selection of TiO2. The presence of monocrystalline rutile TiO2 was confirmed by X-ray powder diffraction and HRTEM investigations. The as-prepared nanorods are soluble in water at pH <3. The surface functionalization was analyzed by IR and 1H NMR, confirming the presence of dopamine on the surface. The surface amine groups can be tailored further with functional molecules such as dyes. Confocal laser scanning microscopy (CLSM) was used to characterize the binding of the fluorescent dye 4-chloro-7-nitrobenzofurazan (NBD) to the functionalized surface of the TiO2 nanorods.     

Numerical Computation of Optimal Trajectories for Coplanar, Aeroassisted Orbital Transfer

This paper is concerned with the problem of the optimal coplanaraeroassisted orbital transfer of a spacecraft from a high Earth orbitto a low Earth orbit. It is assumed that the initial and final orbits arecircular and that the gravitational field is central and is governed by theinverse square law. The whole trajectory is assumed to consist of twoimpulsive velocity changes at the begin and end of one interior atmosphericsubarc, where the vehicle is controlled via the lift coefficient.The problem is reduced to the atmospheric part of the trajectory, thusarriving at an optimal control problem with free final time and liftcoefficient as the only (bounded) control variable. For this problem,the necessary conditions of optimal control theory are derived. Applyingmultiple shooting techniques, two trajectories with different controlstructures are computed. The first trajectory is characterized by a liftcoefficient at its minimum value during the whole atmospheric pass. For thesecond trajectory, an optimal control history with a boundary subarcfollowed by a free subarc is chosen. It turns out, that this secondtrajectory satisfies the minimum principle, whereas the first one fails tosatisfy this necessary condition; nevertheless, the characteristicvelocities of the two trajectories differ only in the sixth significantdigit.In the second part of the paper, the assumption of impulsive velocitychanges is dropped. Instead, a more realistic modeling with twofinite-thrust subarcs in the nonatmospheric part of the trajectory isconsidered. The resulting optimal control problem now describes the wholemaneuver including the nonatmospheric parts. It contains as controlvariables the thrust, thrust angle, and lift coefficient. Further,the mass of the vehicle is treated as an additional state variable. For thisoptimal control problem, numerical solutions are presented. They are comparedwith the solutions of the impulsive model.     

The influence of pair potentials on the structure of polyvalent liquid metals

The structure factors of some polyvalent liquid metals show a shoulder on their first peak. We explain these shoulders on the basis of ionic pair potentials : a shoulder is expected near the wave vector 2k_F, the wave length of the oscillations in the metallic pair potentials. To show this quantitatively, we used the “optimized random phase approximation” of Andersen et al.     

Nonnegative splines, in particular of degree five

Summary. We investigate splines from a variational point of view, which have the following properties: (a) they interpolate given data, (b) they stay nonnegative, when the data are positive, (c) for a given integer they minimize the functional for all nonnegative, interpolating . We extend known results for to larger , in particular to and we find general necessary conditions for solutions of this restricted minimization problem. These conditions imply that solutions are splines in an augmented grid. In addition, we find that the solutions are in and consist of piecewise polynomials in with respect to the augmented grid. We find that for general, odd there will be no boundary arcs which means (nontrivial) subintervals in which the spline is identically zero. We show also that the occurrence of a boundary arc in an interval between two neighboring knots prohibits the existence of any further knot in that interval. For we show that between given neighboring interpolation knots, the augmented grid has at most two additional grid points. In the case of two interpolation knots (the local problem) we develop polynomial equations for the additional grid points which can be used directly for numerical computation. For the general (global) problem we propose an algorithm which is based on a Newton iteration for the additional grid points and which uses the local spline data as an initial guess. There are extensions to other types of constraints such as two-sided restrictions, also ones which vary from interval to interval. As an illustration several numerical examples including graphs of splines manufactured by MATLAB- and FORTRAN-programs are given. Received November 16, 1995 / Revised version received February 24, 1997     

Numerical Computation of Optimal Feed Rates for a Fed-Batch Fermentation Model

In this paper, we consider a model for a fed-batch fermentation process which describes the biosynthesis of penicillin. First, we solve the problem numerically by using a direct shooting method. By discretization of the control variable, we transform the basic optimal control problem to a finite-dimensional nonlinear programming problem, which is solved numerically by a standard SQP method. Contrary to earlier investigations (Luus, 1993), we consider the problem as a free final time problem, thus obtaining an improved value of the penicillin output. The results indicate that the assumption of a continuous control which underlies the discretization scheme seems not to be valid. In a second step, we apply classical optimal control theory to the fed-batch fermentation problem. We derive a boundary-value problem (BVP) with switching conditions, which can be solved numerically by multiple shooting techniques. It turns out that this BVP is sensitive, which is due to the rigid behavior of the specific growth rate functions. By relaxation of the characteristic parameters, we obtain a simpler BVP, which can be solved by using the predicted control structure (Lim et al., 1986). Now, by path continuation methods, the parameters are changed up to the original values. Thus, we obtain a solution which satisfies all first-order and second-order necessary conditions of optimal control theory. The solution is similar to the one obtained by direct methods, but in addition it contains certain very small bang-bang subarcs of the control. Earlier results on the maximal output of penicillin are improved.     

On the numerical computation of minimum-fuel,Earth-Mars transfer

The minimum-fuel space travel problem for the planar Earth-Mars transfer by a low-thrust ion rocket is considered. The investigation shows that the use of the classical switching function and the natural boundary condition of the Hamiltonian leads to numerical difficulties, caused by the nondifferentiability of the Hamiltonian. It is shown, however, that these difficulties can be overcome by substitution of the switching function and the Hamiltonian condition. The numerical solution by multiple shooting techniques is presented.     

The derivation of cubic splines with obstacles by methods of optimization and optimal control

Summary Interpolating splines which are restricted in their movement by the presence of obstacles are investigated. For simplicity we mainly treat cubic splines which are required to be non-negative. The extension to splines of higher order and to certain other forms of obstacles is straightforward. Methods of optimization and of optimal control are used to obtain necessary optimality criteria. These criteria are applied to derive an algorithm to compute splines which are restricted to constant lower or upper bounds. There is a numerical example which illustrates the method presented.Dedicated to Günter Meinardus on the occasion of his 60th birthday     

Electrical resistivity of ternary glassy and liquid Pd-alloys

The electrical resistivity of liquid (Pd₁₀₀Cu_100?x)₈₀Ge₂₀ alloys has been measured as a function of temperature. For Cu-rich negative temperature coefficients and for Pd-rich alloys positive temperature coefficients of the electrical resistivity have been observed. This behavior is very similar to recent observed resistivity temperature curves of glassy (Pd₁₀₀Cu_100?x)₈₀P₂₀ alloys. An explanation of the resistivity behavior in terms of liquid metals theory is suggested.