admit one or more inflection points (d2P/dS2 = 0 at these points). The locus of the inflection points is provided by

The intersections of the singular arc and the locus of inflection points have special significance with respect to singular control. These intersections are discrete points (S — Si, P — Pi) where

The feeding policy defined in Eqs. 7.58 and 7.72 reduces to

at each such intersection.

It follows from Eqs. 7.74 and 7.75 that Si and Pi are equilibrium (time-invariant) solutions of Eqs. 7.30 and 7.31, respectively. Singular control operation at such points is characterized by time invariance of S and P and a gradual approach of X to the equilibrium value Xt (Xl = (¿(Si, Pi)/p(Si, Pi)) unless already at it. Upon reaching such a quasi-steady state ((X, S, P) — (Xi, Si, Pi), V varies with t) in a cycle, the bioreactor will be operated in an exponential fed-batch mode until a transition to the boundary control (F = 0, batch operation) occurs upon saturation of bioreactor volume.

In view of the nonlinear dependence of the three specific rates on S and P, multiplicity of limit points for a fixed feed composition cannot be ruled out. In a repeated fed-batch operation, the bioreactor trajectories during singular control interval of each cycle must terminate at a locally asymptotically stable (accessible) limit point. The necessary and sufficient conditions for local accessibility of a limit point can be obtained via linearized stability analysis of Eqs. 7.29-7.31 [447],

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