Optimal periodic control problem: a modified trigonometric approximation and quasi-stationarity conditions

The constrained optimal periodic control problem for a system described by differential equations and endowed with inertial controllers is considered, A sequence of discretized problems using trigonometric polynomials is proposed to approximate the problem. Instantaneous constraints for the state and control are handled by a new and more precise approach that imposes only a small number of non-linear but easily computable constraints. The convergence conditions for a sequence of optimal solutions of discretized problems are derived. The inclusion in the approximating scheme of various quasi-stationarity conditions for the control and state variables is analysed. Extension of a new approximating approach for inertialess and smooth problems is also discussed.     

ENZYMATIC HYDROLYSIS OF SOYBEAN FOR SOLVENT AND MECHANICAL OIL EXTRACTION

Due to inefficient extractability of its low oil content, soybeans are often bypassed in village‐scale processing. Soygrits, flakes, and expanded collets were hydrolyzed by proteases, cellulases, and pectinases before oil extraction by solvent and static mechanical pressure. Driselase with multi‐enzyme activity and two proteases improved solvent extraction rates but only Driselase enhanced mechanical pressing. Up to 58% of seed oil was pressed from enzyme‐hydrolyzed flakes but 88% was pressed from Driselase‐treated collets. Either pretreatment is a feasible adjunct to mechanical pressing in small batch operations.     

SEARCH FOH OPTIMUM CONDITIONS OF EXTRACTION OF METAL COMPLEXES WITH ALKYLIMIDAZOLES. III. STRUCTURE - EXTRACTABILITY RELATIONSHIPS FOR 1,4-DIMETHYLIMIDAZOLE COMPLEXES OF Co(II), Ni(II), Cu(II), Zn(II), AND Cd(II)

Stability constants of the 1,4-dimethylimidazole complexes of Co(II),Ni{II),Cu(II),Zn(II), and Cd(II) have been determined in aqueous solution and their structure has been elucidated. Conformational equilibria of the type octahedron-tetrahedron have been found in the case of the Co(II) and Sn(II) complexes. The extraction of all complexes has been shown to be dependent on their'molar contributions, d?n? expressed in per cent. The distribution constants have determined for each complex and the composition of the species extracted with organic solvent has been suggested.The measurements were run at 298?K, at the ionic strength of the aqueous phase of 0.5 maintained by means of KNO?3?.     

Approximate discrete maximum principle for the difference approximation of optimal periodic control problems

Distinctive features of the difference approximation for optimal periodic control (OPC) problems are considered. It is shown that the convergence conditions on approximate solutions imply consistency conditions on the time grid step and the accuracy of the periodicity constraint approximation such that the approximate discrete maximum principle holds for a sequence of discretized problems. A specific form of this principle when applied to the difference approximation method for OPC problems is analysed and computational methods are discussed. Application of this approach to a class of systems non-linearly dependent on control variables, and occurring in the chemical industry, is suggested.     

Difference approximation of optimal periodic control problems

The constrained optimal periodic control problem is approximated by a sequence of discretized problems in which the system of differential equations of the basic continuous problem is replaced by a system of one–step difference equations. Two kinds of approximate optimal controls are derived from the optimal solutions of discretized problems: the first in the form of a step function and the second in the form of a special trigonometric polynomial generated by a positive kernel. Sufficient conditions for approximate solutions to be weakly convergent to the optimal solution of the basic problem are given. Certain improvements in the difference approximation considered are discussed and potential applications given.