Method of Penalty Functions in One Problem of Optimal Control of a Process In a Chemical Reactor

Abstract

The paper considered the problem of optimal control of a process in a nonadiabatic tubular reactor used in chemical technology. Gas feeds the reactor that undergoes a first order exothermic reaction. The reactor has an outer jacket - casing. Through the casing flows reactor coolant liquid. In turn, the reactor changes the temperature in the casing.

As a control function, the coolant supply rate to the casing is taken. The liquid supplied to the casing has a constant temperature. So the control function depends only on time. The values of the reactor temperature, the concentration of the reacting mixture vary along the length of the reactor and the reaction time.

The mathematical model of the reactor consists of differential equations in partial derivatives and the corresponding boundary, initial conditions. In this case, appropriate restrictions are imposed on the temperature in the reactor and on the control function. The restriction on the temperature in the reactor is taken into account by introducing in the objective functional (as a term) a penalty function of the type of a linear cutoff. This type of penalty function in mathematical programming is usually driven to meet the constraints exactly. Constraints on the control function are given in the form of inequalities. As the target functional, the total amount of unreacted substance at the reactor outlet for a fixed period of time is taken. As noted above, a penalty function is added to the objective functional. The aim of the control is to minimize this functional.

The paper proves a theorem on the existence of an optimal control in such a problem. In the course of the proof, the boundedness of solutions of the system of partial differential equations in the H?lder norms is used. This allows us to use Arzel's criterion on the compactness of a set of continuous functions. The proof also uses the weak compactness sets of control functions in the space $L_2 (0,T)$. With an unlimited increase in the penalty coefficient, the convergence to zero of the penalty function is proved, i.e. the feasibility, in the limit, of the phase limitation on the temperature in the reactor is shown. We also obtain some properties of the terms of the objective functional, which will be necessary in what follows.