Max
SIR:We accept the correction of Bailey and Horn concerning the generality of the description of the relaxed steady state in our preliminary discussion (Chang and Bankoff, 1968). Apart from the fact that this technique was not used in our work, it should be pointed out that our description was, in fact, correct for the class of problems considered therein. Essentially, this implies that one can get the same results as in relaxed steady-state operation by a suitable combination of steady-state reactors. This circumstance is by no means unusual; however, there is no present set of clear guideposts as to when this will occur. Although the work of Warga (1962) is sometimes cited for relaxed control of distributedparameter systems (Horn and Bailey, 1969), it is, in fact, limited to systems of ordinary differential equations. I n particular, a theorem extending Warga’s Thm. 2.3, giving sufficient conditions for an original minimizing curve to be also a relaxed minimizing curve, to partial differential equation systems is not yet available. The extensions are not trivial, as the following examples will show.
Referring to Figure 13, the curved arc (denoted by I’) represents the locus of solutions of P2, where = { v(0)Ivl(0) x E [0.65, 0.901 gram mole/liter) . Because of the dependence of the optimal control along any characteristic purely upon the initial condition for that characteristic, the optimal yield of P1 maps along r with exactly the period 7 ;and the solution of P1 is a convex combination of solutions of P2. This result is indpendent of 7 . Hence, the upper chord in Figure 13 represents the locus of time-average optimal yields in relaxed operation between the bounding concentrations, 0.65 and 0.90 gram mole per liter. The lower chord represents the zero-frequency limit obtained when the switching time is allowed to become large, rather than the time on either bound. Incidentally, one can determine the sign of the zerofrequency effect from the curvature of in the neighborhood of the steady-state inlet concentration, vlS = 0.775, if the neighborhood N = (vl(0) (vl(0)e[0.65,0.90]).
Example 1
Example 3
Let
Lv
=
g(v,u)
a + at ax’
where L is a linear partial differential operator, p-
b
Y
with (t,x)eT x R; where T is the closed interval [to,tj]and RcE’, s = 1, 2, or 3, is a closed and bounded set. The vector function v(t,x) maps: T x R 4A , where A is compact in E3. Y > 0 is a constant while p = 1; g(o,u) : A x U + B , where U is a convex compact set of controls, and B is compact in Ea. Further, let
+ (1 - a)g(v,uz)= g[v,aul+ ( 1 -
ag(u,ul)
4
~
2
1
=
$(t)
(34
But Equation 2a implies that
B
co(c1B)
Hence, every relaxed steady state is, indeed, a linear combination of steady states corresponding to the selected controls.
I
*
In Example 1, let Y = 0. With obvious modifications, the system then reduces to the ordinary differential equation set considered by Warga (1962). An obvious, but important, consequence of Equation 2a is that the identity of Equation 3b holds here also, so that relaxed control does not enlarge the set of attainable states, if the dynamic equations are linear in control. Example 4
ev = g ( v )
Let where
(2a)
( g linear in control) vu1, uzeU, 0 5 a 5 1. This is the set of Equations 1, 2 considered by Chang and Bankoff (1968). The initial and boundary conditions are
v(0,x)= @(z);v(t,O)
UEU’{J’ = v 2(L)
b L =I bt
+ axa VI‘-
I is the identity matrix, and I’ is a diagonal matrix with a t least one zero diagonal entry. The initial conditions are of form 3a and 5a. This problem has been considered by Bailey and Horn in connection with relaxed operation of a packed tubular reactor, with interchange between the gas phase and adsorbed phase (Horn and Bailey, 1969). -4s indicated b y them, the attainable set in relaxed operation is greater than in any combination of steady states. However, if we modify the operator to take into account axial diffusion
Example 2
Referring to the system of Example 1, consider the Problem P1
where R
[O,L] is a segment of the real line, and +‘t
$(t
+
7) =
$(t)&, a compact set in E 3
>0 (5a)
Since Y is a constant scalar, it is clear that the characteristics of System l a merge into a single set of nonintersecting lines. Along these lines the P D E system decomposes into a set of ordinary differential equations in the Lagrangian coordinate S. Problem P1 then becomes identical to Problem P2: I n Example 1, let p = 0 ; v = ~ ( 2 )R: + A ’ C A
u = u(z)aU’CU g = (v,u):A’ X Ut-+ B‘CB
v(O)e!V. We wish to find subject t o Equation l a :
where E > 0, the amplitude of the gas-phase concentration oscillation goes to zero as 7 + 0 everywhere except a t the inlet, so that relaxed operation in this event gives no theoretical improvement over steady-state operation, for every E > 0. I n practice, of course, 7 > 0, and the degradation of the results, which may be very serious, depends upon both and E . We recognize t,as a singular perturbation operator, since the solution does not depend smoothly on E as E + 0 and 7’0. literature Cited
Chang, K. S., Bankoff, S. G., IND.ENG.CHEM.FUNDAMENTALS 7,633 (1968).
Horn, F. J. M., Bailey, J. E., Cleveland Meeting of AIChE, May 1a
m
W&l,’J., J. Math. Anal. A p p l . 4, 111, 129 (1962).
Northwestern University Evanston, Ill. 60201 University of Waterloo Waterloo, Ont., Canada
S . G. Bankoff
K . S . Chang
Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
301
