A General Minimum Principle for End-point Control Problems

Abstract

A very broad class of end -p oin t con trol problems is formulat ed in general t erms, and corresp ondingly gen eral procedures in the calcu lus of variations a re d ev eloped for their solut ion .Tho generality of tho treatment lies in a system a ti c application of the point of v iew of functional analysis. The state of the system being controlled is described by an abstract vector in a suitable linear sp ace; the evolution of th o sys tem state wi th t ime is d escribed in terms of an op erator on the st at e v ector; the measure of system p erformance t o be maximized or minimized is taken as a functional of the final state. The linearized system op erator and its adj oint w ith respect to an inner product in the st a t e space play central rol es in the va r iat iona l ar gu m ent s.The variational methods are in the spirit of those leading to t he familiar Pontrjagin maximum principle, and indeed this principle appears as the finite -dimen sional case of the present results. Theso results however apply equ ally to infinite-dimensional cases su ch as system s wi th distributed parameters whose mathematical description is given in tenus of partial rather than ordinary differential equations. Tho m ethods are illustrated here for a problem in h eat exch a nger co n t rol. Gener al numerical procedures aro discussed, but for distributed parameter systems tho calculations required to solve practical problems .will bo -ext.rcmely.formidable , _' , INTRODUCTIOK

Optimal control of tubular reactors. Part I. Computational considerations

“…More generally, however, some of the product stream may be recycled, after further proce::sing, with a time lag ( 5 ) F [ u ( t , r , ) , u ( t -t r , x f ) ] =o; t , , & t L t f ; F€C2…”

Optimization in a Class of Distributed-Parameter Systems

“…For convenience we consider only the case where the boundary conditions depend on the fluxes in the form The first-order variation in (P as a result of the variations in decisions may be written f1 dtF S<? = Jo dJ5*(V)^( 23) If we then complete the specification of the Green's vector by the additional conditions ~ 0 Vlwdr (26) That is, our use of the Green's functions has allowed us to obtain an explicit representation of the variation of (P in terms of the variations , , ?, and . It is clear that any other form for the boundary conditions could have been used in the place of Equations 7, 8, 19, and 20, leaving Equation 26essentially unchanged except for the coefficients of and ?.…”

Control of Plug-Flow Tubular Reactors by Variation of Flow Rate

“…We formulate the optimal control problem as follows: It is desired to determine t>(0) over the given time interval (0,0/) subject to * > ( ) > r*, the maximum and minimum allowable flow rates, such that the integral square error is minimized. By application of the necessary conditions for optimality for distributed parameter systems (Katz, 1964; Koppel et al, 1968; Seinfeld and Lapidus, 1968a), the optimal policy is found to be The two-point boundary value problem represented by Equations 1, 2, 11, 12, and 13 cannot be solved analytically. In cases when the optimal control is given by a bang-bang law, the switching times can be determined most easily by the method of direct search on the performance index (Seinfeld and Lapidus, 1968a).…”