Modelling and Regularity of Nonlinear Impulsive Switching Dynamical System in Fed-Batch Culture

and Applied Analysis 3 According to the factual experiments, we make the following assumptions. H1 The concentrations of reactants are uniform in reactor, while time delay and nonuniform space distribution are ignored. H2 The feeding media includes only fixed concentrations of glycerol and alkali. In the laboratory, the effect of alkali feeding on the pH value is much bigger than the effect of glycerol feeding on the order of magnitude, and the time of alkali feeding is relatively shorter than the whole experiment time. The impulsive switching system is fitter for describing the actual process than the continuous system. According to the above description, in this paper, we let FN 0 in 13 , that is, the flow rate of alkali is zero. Glycerol feeding strategy is determined by a preassigned times sequence. Let t0 t0 < t1 < t2 < · · · < t2N T be a partition of I : t0, T . t0, t1 is a period of time of batch fermentation, ti, i ∈ {1, 3, . . . , 2N − 1} is the moment of starting the inlet flow of glycerol, and ti 1 is the moment of ending this input. So, for convenience, we let j t ⎧ ⎪⎨ ⎪⎩ 0, t ∈ [ ti, ti 1 ) ⇐⇒ FG v, 1, t ∈ [ ti 1, ti 2 ) ⇐⇒ FG 0, i ∈ {1, 3, . . . , 2N − 1}. 2.1 Under the assumptions H1 and H2 , the fed-batch process can be formulated by dx1 t dt ( μ − d)x1 t − FG x7 t x1 t : f j t 1 ( t, x, p ) , dx2 t dt −q2x1 t FG x7 t Cs0 − x2 t : f t 2 ( t, x, p ) , dx3 t dt q3x1 t − FG x7 t x3 t : f j t 3 ( t, x, p ) , dx4 t dt q4x1 t − FG x7 t x4 t : f j t 4 ( t, x, p ) , dx5 t dt q5x1 t − FG x7 t x5 t : f j t 5 ( t, x, p ) , dx6 t dt − FG x7 t x6 t : f j t 6 ( t, x, p ) , dx7 t dt FG : f j t 7 ( t, x, p ) , 2.2 where x : x1, x2, x3, x4, x5, x6, x7 is the continuous state vector; p : d, μm, ks, m2, m3, m4, m5, Y2, Y3, Y4, Y5, Δ2, Δ3, Δ4, Δ5, K∗ 2, K ∗ 3, K ∗ 4, K ∗ 5, γ is the parameter vector. The specific growth rate of cells μ, specific consumption rate of substrate q2, and specific formation rate 4 Abstract and Applied Analysis of products qi, i 3, 4, 5, are expressed by the following equations based on previous works 5, 13, 15 :