The necessary conditions for optimization of a system governed by a nonlinear vector firstorder partial differential equation with two (space and time) independent variables, such as governs the unsteady behavior of tubular flow reactors, ore derived. Rather general objective functionals and boundary conditions, such as the recycle of unconverted reactant with an appropriate time delay for separation and a free choice of final time, are allowed. A gradient techniaue in control smce is formulated. and it is shown that distinct computational advantages can accrue from the use of the method of characteristics.Despite the formidable computational problems, quite a bit of attention has been devoted to the optimal coiltrol of distributed parameter processes in recent years. Work on the problem was initiated by Butkovskii and Lerner( 1 ) , and necessary conditions were developed by Butkovskii ( 2 , 3 ) , Lurk ( 4 ) , Katz ( 5 ) , and Wang and Tung (6, 7). The latter workers, in particular, discussed the observability, controllability, and stability of these systems. Schmaedeke (8) has established some existence theorems for optimal control. Butkovskii (9) suggested as computational methods spatial discretization of the partial differential equations, and the use of the method of moments ( l o ) , which requires a knowledge of the system eigenfunctions. A function space method has been applied by Chaudhuri ( 1 1 ) to a linear system. In the chemical field, Volin and Ostrovskii ( 1 2 ) and Jackson ( 1 3 ) have dealt with optimal control of tubular reactors with slowly decaying catalyst, Denn, et al.( 1 4 ) with a reactor with radial temperature gradients, Jackson (15) with processes, such as absorption columns, which are approximately describable by partial differential equations, and Denn (:16), as well as others, with temperature regulation of dabs. The present work may be looked upon as an extens8ion of Sirazetdinov's ( 1 7) formulation of the necessary conditions for a process governed by a first-order partial differential equation to a system of these equations, such as arise in the description of a tubular reactor. Very general boundary conditions are permitted, such as the recycle of unconverted reactant, with an appropriate time delay, back to the inlet, perhaps after some separation has occurred. In addition, the final time of the optimal control period may be left free, and hence be chosen optimally.A steepest descent scheme in control space is evolved, and the method of characteristics examined to determine its applicability. In the case where the adjoint and state equation characteristics are identical, it offers considerable computational advantages.
STATEMENT OF PROBLEMWe begin by defining a very general optimal control problem for Aow processes governed by a system of firstorder partial differential equations (PDE) in space, x, and time, t . The system is where w 3 [ul (t, x)
Page 410 AlChEwith avi (in the class of all twice piece-hy-piece continuously differentiable functions). The vector v = w ( t , x) d...