The performance of optimal steady state continuous chemical processes can be improved in some cases by forced periodic operation of these processes. Significant experimental and theoretical effort has been undertaken to identify operations that lead to improved productivity, yield and/or selectivity of chemical reactors by periodic variations in one or more reactor inputs [30, 34, 102, 103, 104, 115, 124, 136, 141, 237, 240, 256, 318, 319, 385, 442, 443, 444, 445, 509, 550, 561, 571, 572, 573, 632, 633, 634, 653]. Optimization of batch and fed-batch operations of bioprocesses has received much more attention compared to optimization of continuous operations. With the increasing significance of continuous bioreactor operations, it is important to know how these can be operated more effectively. The effect of cycling of feed conditions on behavior of continuous cultures has been examined experimentally and theoretically in few prior studies [4, 5, 74, 322, 448, 449, 450, 451, 455, 460, 468, 469, 525, 549, 568, 569, 570, 610, 636, 678, 688].
A matter of primary concern in the periodic control problem is whether and when forced periodic control is superior to steady state control. The three major approaches taken for analysis of forced periodic operations [202, 571] are: (1) the Hamilton-Jacobi approach based on the maximum principle [34, 202, 358, 431], (2) a frequency-domain approach using second-variations methods [66, 634], and (3) numerical approach based on, among other things, vibrational control [102, 103, 104, 442, 443, 444, 445, 509]. Sufficient conditions for optimality of periodic control have been proposed using the Hamilton-Jacobi approach, relaxed steady state analysis, and second-variations methods. For periodic operations employing high frequencies, the sufficient condition is either based on the maximum principle or relaxed steady state analysis [30, 34]. In very low frequency periodic operations, description of process dynamics is based on the quasi-steady state assumption. For the intermediate frequency range, the sufficient condition, based on second-variations methods, is provided by the 7r-criterion [66, 202]. While the relaxed steady state analysis allows for strong variations in control variables, both the 7r-criterion and the maximum principle are applicable only for weak variations in control variables.
A generalized 7r-criterion based on perturbations around arbitrary steady states which are locally, asymptotically stable has been proposed . The development of the generalization, which is based on a line of reasoning similar to that outlined by Bryson and Ho  and the averaging result of Tikhonov et al. , allows application of the 7r-criterion to a broader range of problems. Continuous processes may not always operate at an optimal steady state and in some situations, optimal steady states may not be admissible [448, 449, 573]. The generalized 7r-critcrion is useful in these cases to explore the possibility of improving the process performance via forced periodic operation.
7.3.1 Preliminaries on the 7r-Criterion
Consider the steady r-periodic operation of a continuous process described
The optimal periodic control problem is to maximize a scalar objective function subject to the integral constraints in Eq. 7.5 with tf = r. Let there be a steady-state solution x* of Eq. 7.76 corresponding to u = u* at which the performance index in Eq. 7.77 is maximized, subject of course by
to satisfaction of constraints in Eq. 7.5. The forced periodic control is said to be proper if the objective function for a periodic operation exceeds that for the steady-state operation. A sufficient (but not necessary) condition for this is provided by the 7r-criterion. This criterion, originally developed for forced periodic operations around an optimal steady state [66, 202], has been generalized to be applicable to forced periodic operation around an arbitrary steady state [571, 572, 573]. Assuming that f, h, <fi and ^ are continuously differentiate in x and u, the Hermitian matrix Ilia;) is defined as nH = Gc(joj)PG(ju) + QTG(ju>) + Gc(jw)Q + R (7.78)
G(s) = (si - A)-XB, A - fx(x, u), P - tfxx(x, u, A, p, 77), B = fu(x, u), Q = iixu(x, u, A, p, rj), R = £Tuu(x, u, A, p, T7)(7.79)
and the Hamiltonian H is defined as
£T(x, u, A, p, rj) ~ h + ATf + pT</>. (7.80)
In Eqs. 7.78-7.80, x and u assume their respective values at a steady state, viz., xo and uo, respectively. The superscript c in Eqs. 7.78, 7.83 and 7.84 denotes the complex conjugate. The adjoint variable vectors A, p, and r] satisfy the conditions
Zx(x, u, A, p, V) = 0T, Z = H + rf*, r) < 0, J7T^(x, U) = 0, (7.81)
at an arbitrary steady state. At the optimal steady state, the following additional condition must be satisfied
For superior performance of forced periodic operation of continuous bioreac-tors vis-a-vis operation of the same at a steady-state, it is sufficient (but not necessary) to have positive-definite II for some values of u> (0 < u> < 00). For small variations in the control variables relative to their values at a steady state, the differential between the magnitudes of the objective function in a forced periodic operation and the corresponding steady-state operation can be expressed as 
5 J = —- / (duc)TH(u>)6udt, <5*u = u-uo, SJ = J-Jo, Jo = Mxo, uo), 2T Jo
w being the overall frequency of the periodically perturbed system.
Periodic variation in only two inputs is considered here since the case study to be discussed later pertains to two inputs. The results obtained here can be extended to higher dimensions, with tedious algebraic manipulations [449, 450]. Since the domain of II is complex vectors, Su is assigned the following form:
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