Numerical solution of minimax problems of optimal control,part 2

In a previous paper (Part 1), we presented general transformation techniques useful to convert minimax problems of optimal control into the Mayer-Bolza problem of the calculus of variations [Problem (P)]. We considered two types of minimax problems: minimax problems of Type (Q), in which the minimax function depends on the state and does not depend on the control; and minimax problems of Type (R), in which the minimax function depends on both the state and the control. Both Problem (Q) and Problem (R) can be reduced to Problem (P).In this paper, the transformation techniques presented in Part 1 are employed in conjunction with the sequential gradient-restoration algorithm for solving optimal control problems on a digital computer. Both the single-subarc approach and the multiple-subarc approach are employed. Three test problems characterized by known analytical solutions are solved numerically.It is found that the combination of transformation techniques and sequential gradient-restoration algorithm yields numerical solutions which are quite close to the analytical solutions from the point of view of the minimax performance index. The relative differences between the numerical values and the analytical values of the minimax performance index are of order 10^–3 if the single-subarc approach is employed. These relative differences are of order 10^–4 or better if the multiple-subarc approach is employed.This research was supported by the National Science Foundation, Grant No. ENG-79-18667, and by Wright-Patterson Air Force Base, Contract No. F33615-80-C3000. This paper is a condensation of the investigations reported in Refs. 1–7. The authors are indebted to E. M. Coker and E. M. Sims for analytical and computational assistance.     

Propagation of electromagnetic waves in a nonlinear dielectric slab

The early phases of propagation of a large amplitude electromagnetic disturbance in a nonlinear dielectric slab embedded between two linear media are investigated by the method of characteristics. This disturbance is triggered by the arrival of an electromagnetic shock wave at one of the interfaces. Reflection and transmission of an arbitrary signal when it arrives at one of the slab boundaries is characterized in terms of nonlinear reflection and transmission coefficients for the interface. No restrictions are placed on the form of the constitutive laws of the material comprising the slab. By introducing, for the nonlinear dielectric, a class of model equations, an exact solution to the characteristic equations which describes the interaction of a centered wave with anarbitrary oncoming signal is obtained. Solutions for the electromagnetic fields are derived for the special case in which the incident disturbance interacts with the reflected signal from the slab interface. A particular case of the disturbance propagating across a nonmagnetic slab is also examined.     

Polymer–polymer interaction parameters of polystyrene and polybutadiene from studies in solutions of toluene or tetrahydrofuran

Equations used for the determination of polymer–polymer interaction parameters for narrow molecular weight distribution polymers which have undergone phase separation in solutions are discussed. Binodals with tie lines and plait points are presented for ternary systems polystyrene-polybutadiene-toluene and polystyrene-polybutadiene-tetrahydrofuran. Typical interaction parameters determined from the experimental data for the polystyrene-polybutadiene systems are presented.     

Studies on chemical stability in CO2 and H2O and electrical conductivity of perovskite-type Ba3In2Zr1?xCexO8 (x?= 0, 0.5, 1)

We report the synthesis, structure, microstructure, chemical stability in H₂O and CO₂, and electrical transport properties of an oxide ion-conducting perovskite-related structure Ba₃In₂MO₈ (M = Zr, Ce, Zr_0.5Ce_0.5). Powder X-ray diffraction confirmed the formation of a simple cubic perovskite-like structure for Ba₃In₂ZrO₈ (a = 4.205(9) ?), Ba₃In₂CeO₈ (a = 4.234(1) ?), and Ba₃In₂Zr_0.5Ce_0.5O₈ (a = 4.285(8) ?). The increase in lattice constant is consistent with the Shannon’s ionic radius trend. Among the three samples investigated, Ba₃In₂ZrO₈ was found to be stable against reaction with pure CO₂ at elevated temperature, while the Ce and 1:1 Zr and Ce compounds were unstable at 600 °C. Ba₃In₂ZrO₈, Ba₃In₂CeO₈, and Ba₃In₂Zr_0.5Ce_0.5O₈ were found to be chemically unstable in H₂O at about 50 °C. The bulk electrical conductivity of the samples prepared at different temperatures was found to be nearly the same; the total conductivity (bulk + grain–boundary + electrode) seems to change with sintering temperature. Both Ba₃In₂ZrO₈ and Ba₃In₂CeO₈, prepared at 1,400 °C, exhibited comparable electrical conductivity of about 6 × 10⁻³ S cm⁻¹ at 800 °C, which is comparable to that of conventional Y₂O₃-doped ZrO₂ electrolyte. These compounds are very promising electroltes, provided that their chemical and mechnical stabitities are improved without losing any ionic conductivity.