Analysis of an optimal control problem connected to bioprocesses involving a saturated singular arc

We study a minimal time control problem under the presence of a saturation point on the singular locus. The system describes a fed-batch reactor with one species and one substrate. Our aim is to find an optimal feedback control steering the system to a given target in minimal time. The growth function is of Haldane type implying the existence of a singular arc which is non-necessary admissible everywhere (i.e. the singular control can take values outside the admissible control set). Thanks to Pontrygin’s Principle, we provide an optimal synthesis of the problem that exhibits a frame point at the intersection of the singular arc and a switching curve. Numerical simulations allow to compute this curve and the frame point.


Introduction.
Minimal time control problems for affine systems with one input such as:ẋ = f (x) + ug(x), x ∈ R n , |u| ≤ 1, (1) have been investigated a lot in the literature, see e.g. [9,10,20,21,28,29,30] and [8] for n = 2 and references herein. One often encounters singular trajectories which appear when the switching function of the system is vanishing on a time interval. In order to find an issue to a minimal time control problem governed by (1), one usually requires that the singular control u s is admissible, which means that |u s | ≤ 1. ( This allows the trajectory to stay on the singular arc. However, one cannot in general show that this assumption holds. In fact, the expression of the singular control in terms of the state and adjoint state does not always guarantee that (2) is satisfied. One can argue that it is enough to consider a larger admissible upper bound for the controls, but this seems rather artificial, and not necessarily feasible from a practical point of view. The objective of this work is to study a minimal time control problem in the plane where the singular control satisfies (2) only on a sub-domain of the state space (the part of the singular arc where (2) does not hold is usually called barrier [8,9,10,20,21]). This may happen in many engineering problems in particular when the singular control can take arbitrary large values in the state space.
The system that we consider is a fed-batch bioreactor with one species and one substrate [19]. Our aim is to find an optimal feedback control that steers the system in minimal time to a given target where the substrate concentration is less than a prescribed value, see [19]. Finding an optimal feeding strategy can significantly increase the performance of the system and has several advantages from a practical point of view (see e.g. [1,2,6,7,12,13,14,16,19]). The design of adequate feedback control laws is used for instance in wastewater treatment industries [13].
Whenever the growth function is of Monod type [18,26], then one can prove that the optimal feeding strategy is bang-bang [19]. This means that the reactor is filled until its maximum volume with the maximum input flow rate. Then, microorganisms consume the substrate until reaching a reference value. In the case where the growth function is of Haldane type (in case of substrate inhibition [18,26]), this strategy is no longer optimal. In fact, one can prove that the optimal strategy is singular provided that the singular arc is always admissible (see [19]). It consists in reaching in minimal time a substrate concentrations corresponding to the maximum of the growth rate function, and which coincides with the singular set. Then, the substrate concentration is kept constant at this value until reaching the maximal volume (a more detailed description of singular strategies can be found in [1,2,12,19]).
In the present work, we are interested in studying the optimal synthesis for Haldane-type growth function whenever the singular arc is no longer admissible from a certain volume value. This can happen when the singular control becomes larger than the maximal input flow rate which is allowed in the system. It follows that there exists a volume value above which singular extremal trajectories are no longer admissible. Such a point is usually called saturation point [5,15,23,24]. Whereas in [1,19], the maximal volume can be reached by a singular trajectory, this is no longer possible as the singular control saturates before reaching this value.
In this setting, the optimal synthesis of the problem is more intricate to obtain as one cannot apply the clock form (see [4]). Thanks to the Pontryagin's Principle, we provide an optimal synthesis of the problem. In particular, we show that singular optimal trajectories leave the singular arc at a frame point defined as the intersection between the singular arc and a switching curve (see [8] for a description of frame points). As a consequence, a singular extremal trajectory ceases to be optimal before reaching the saturation point. This phenomena is studied in [8,20,23,24] where local results are given. In [23], a local regular synthesis is given in low dimension for the minimal time problem with the saturation phenomena. It appears in several systems such as a model of tumor-cancer therapy [15,24], in the problem of NMR [5], and in chemical engineering when optimizing a batch reactor (i.e. without input substrate) w.r.t. the temperature [7,6].
The paper is organized as follows. The second section states the optimal control problem. We also recall the optimality result of [19] and we apply Pontryagin's Principle. The third section is devoted to the optimal synthesis of the problem whenever the singular arc is not admissible. In Proposition 4, we show that optimal controls are of type BSBB or BBSB with at most three switching points (here B denotes a bang arc and S a singular arc). Proposition 2 studies the case where the singular arc is never admissible in the state space. This allows to obtain an optimal OPTIMAL CONTROL OF A FED-BATCH REACTOR 3 feedback control of the problem (see Theorems 3.2 and 3.3) in line with [15,23,24]. In section 4, we provide numerical simulations of optimal trajectories and computations of the switching curve and the frame point. Several questions on the optimal synthesis are discussed in the last section (implicit equation of the switching curve, regularity of the value function, and uniqueness of optimal trajectories).
2. General results. In this section, we state the optimal control problem and we recall the optimal synthesis as in [19] that will allow us to introduce the optimal control problem when the singular arc is partially admissible. We also apply the Pontryagin Maximum Principle (PMP) [22] that will be used in the next section.
2.1. Statement of the problem. We consider a system describing a fed-batch bioreactor with one species and one substrate [14,26]: Here x represents the concentration of micro-organisms (biomass), s the concentration of substrate, and v is the volume of the tank. The input substrate concentration is denoted by s in > 0, and u is the input flow rate in the system. For convenience, we have taken a yield coefficient equal to one (by rescaling the equation). The function s −→ µ(s) is the growth function of Monod or Haldane type (see [18,26]). In the following, we consider that u takes values within the set: Here u max denotes the maximum input flow rate in the system. By time scaling, we can take u max = 1. The target we consider is defined by: The optimal control problem becomes: inf where ξ(·) denotes the unique solution of (3) for the control u that starts at ξ 0 . One essential feature in the system (3) is that the quantity is conserved along any trajectory of (3), hence M is constant and equal to v 0 (x 0 + s 0 − s in ). From (5), we obtain: and system (3) can be put into a two-dimensional system: One can easily show that the set [0, s in ] × R * + is invariant by (7). Notice that if we define x by (6), the micro-organisms concentration may not be positive. This can happen when M ≤ 0 which means that initial conditions of micro-organisms and substrate are low. Therefore, we consider initial conditions for (7) in the domain D defined by: We denote by ∂D the boundary of D. In the rest of the paper, we also write u(·) a control in open loop and u[·] a feedback control depending on the state (s, v). Also, we define two vector fields f, g : D → R 2 associated to (7) by:

2.2.
Optimal synthesis in presence of an admissible singular arc. In this part, we review a result of [19] on optimal trajectories for problem (4) in the case where the singular arc is always admissible. First, we consider the case where the growth function µ is of Monod type i.e. µ(s) =μ s k+s withμ > 0 and k > 0 [18]. Theorem 2.1. Assume that µ is of Monod type. Then, the optimal feedback control u M steering any initial condition in D to the target T is: In other words, the optimal strategy is fill and wait, and it consists in filling the tank with maximum input flow rate until v = v m , and then we let u = 0 until s reaches the value s ref (if necessary). In the rest of the paper, we only consider the case where the growth function µ is of Haldane type i.e. µ(s) =μ s gs 2 + s + k , withμ > 0, g > 0, and k > 0. In this case, µ has exactly one maximum over R + , that we denotes, and we suppose thats > s ref (which means that the reference concentration to achieve is sufficiently small). The optimal synthesis in this case is rather different than for Monod growth function (see an illustration Fig. 4).
Theorem 2.2. Assume that µ is of Haldane type and that the following assumption holds: Then, the optimal feedback control u H to reach the target is given by This can be proved by using either the PMP or the clock form [4,17]. Here we have emphasized that (8) is necessary in order to state the optimality result (see e.g. [1,12] The objective of this paper is to provide an optimal synthesis of the problem whenever (8) is not satisfied. From (9), one can see that the singular control can take arbitrary large values. Note that in practice, one should start the fed-batch with a high biomass concentration (i.e. high M ) in order to speed up the process, so that condition (8) can no longer be satisfied.
2.3. Pontryagin maximum principle. In this part we apply the PMP on (4). The existence of an optimal control is straightforward from standard arguments. First, notice that any trajectory of the system takes values within the compact set [0, s in ] × [0, v m ]. Moreover, a trajectory corresponding to the feedback u M steers any initial condition to the target. As the system is affine w.r.t. the control which takes values within a compact set, the result follows. Let H := H(s, v, λ s , λ v , λ 0 , u) the Hamiltonian of the system defined by: If u is an optimal control and z := (s, v) the corresponding solution of (7), there exists t f > 0, λ 0 ≤ 0, and an absolutely continuous and we have the maximality condition: for almost every t ∈ [0, t f ]. We call extremal trajectory a sextuplet (s(·), v(·), λ s (·), λ v (·), λ 0 , u(·)) satisfying (7)-(10)- (11), and extremal control the control u associated to this extremal trajectory. As t f is free, the Hamiltonian is zero along an extremal trajectory. Following [1], one can prove that λ s is always non-zero (it is therefore of constant sign from the adjoint equation), and that λ 0 < 0 (hence we take λ 0 = −1 in the following). Next, let us define the switching function φ associated to the control u by: We obtain from (11) that any extremal control satisfies the following control law: We say that t 0 is a switching point if the control u is non-constant in any neighborhood of t 0 which implies that φ(t 0 ) = 0. Whenever the control u switches either from 0 to 1 or from 1 to 0 at time t 0 , we say that the control is bang-bang around t 0 . We will write B 0 a bang arc u = 0 and B 1 a bang arc u = 1. Whenever φ is zero on a non-trivial interval I ⊂ [0, t f ], we say that u is a singular control, and that the trajectory contains a singular arc. We will write S a singular arc defined on a time interval. The sign ofφ is fundamental in order to obtain the optimal synthesis. By taking the derivative of φ, we get: Moreover, we can show that λ s < 0 (see e.g. [1,12]). This implies that any extremal trajectory satisfies the property: Now, if an extremal trajectory contains a singular arc on some time interval I := [t 1 , t 2 ], then we have φ =φ = 0 on I, hence we have µ ′ (s) = 0 and s = s on I, hence the singular locus is the line segment S := {s} × [0, v m ]. By solvingṡ = 0, we obtain the expression of the singular control given by (9), see e.g. [2]. Now, along a singular arc, one has easily:φ As λ s < 0 and µ ′′ (s) < 0, one obtains from (15) ). This shows that Legendre-Clebsch necessary condition ∂ ∂u is satisfied along a singular trajectory (see e.g. [15] or [11]). Again using that µ ′′ (s) < 0, one obtains immediately that det(g(z(t)), [g, [f, g]](z(t))) > 0 and that det(f (z(t)), g(z(t))) < 0 which shows that the singular arc is hyperbolic or time minimizing [4,3].

Remark 1.
In other words, the singular arc is a turnpike [8] (this can be also verified using the clock form argument).
Finally, one obtains the time of a singular trajectory by integrating the ODĖ v = u s (see e.g. [1]). Let t s (v 0 , v 1 ) be the time of a singular extremal steering (s, v 0 ) to (s, v 1 ), v 0 < v 1 . One obtains: 3. Optimal synthesis in the general case. In this part, we provide a description of optimal trajectories for problem (4) when (8) is not satisfied. We first introduce a partition of D that will allow us to describe where optimal trajectories have a switching point.

Partition of the domain D. In view of (8) that can be equivalently written
By definition of η, we have: Now, if we define a point v * by v * := η(s), the singular arc is admissible provided that v * ≥ v m . In the following, we make the following assumption on v * : (H1) We suppose that v * < v m .
Following [5,8,15], the point (s, v * ) is usually called saturation point. It follows that the singular arc is admissible only if the volume is less than v * i.e. the admissible part of the singular arc is {s} × [0, v * ]. Indeed, for v > v * equality (9) no longer defines a control in [0, 1]. Next, we will consider the two following cases: Remark 2. Case 1 means that the singular arc is never admissible over (0, v m ]. As the function η can take negative values, v * can be negative. We now introduce curves that are solutions of (7) with u = 1 that will provide a partition of initial states.
Definition 3.1. We defineĈ, resp.Č as the restriction to the set D of the orbit of system (7) with u = 1 that passes through the point (s, v m ), resp. (s, v * ).
Hence,Ĉ is the graph of a C 1 -mapping v −→γ(v) that is the unique solution of the equation: The curvesĈ andČ will play a major role in the optimal synthesis contrary to the case where the singular arc is admissible (see Fig. 3 and Table 1 for parameter values). In fact, they will indicate sub-domains where optimal trajectories have a switching point, and the existence of a frame point [8] depending on the value of M (see Hypothesis (H2) in section 3.3).
Proof. Let us first prove (i). For v ∈ (v, v m ], we can rewrite (17) as follows: On the other hand, we find η ′ (γ(v 2 )) dγ dv (v 2 ) = 0, which gives a contradiction. Therefore, there exists at most one value v 1 for which dγ dv (v 1 ) = 0, andγ(v 1 ) ∈ (0, s in ). Also, by derivating (17) and using the fact that dγ dv (v 1 ) = 0 we get: which is non-zero. In fact, we have seen thatγ(v 1 ) > 0. Moreover we haveγ(v 1 ) = s in from (17) We deduce that at point v 1 , the monotonicity ofγ is changing. The conclusion of (i) follows. Let us prove (ii). By definition of v * , we have dγ dv (v * ) = 0. By a similar argument as for (i), one can prove that v * is the unique    thatγ(v * ) = s, see Fig. 1. From the definition of v * , the volume v * necessarily satisfies 0 < v * < v * . In fact, for any volume value v such that v * < v ≤ v m , one has ds dt |u=1 (s) < 0, thusĈ necessarily intersects the singular arc below v * . 3.2. Optimal synthesis in the case v * ≤ 0. When the singular arc is never admissible, we have the following optimality result (see also Fig. 3).
Proposition 2. Suppose that v * ≤ 0 (case 1). Then, given initial states (s 0 , v 0 ) ∈ D, optimal controls satisfy the following: Proof. Consider an optimal trajectory (s(·), v(·), u(·)) starting at an initial point (s 0 , v 0 ) ∈ D. In the present case, the control u can only take the value 0 or 1 from the PMP (the singular arc is not admissible in D). First, assume s 0 ≤ s. Given the assumption v * ≤ 0, we can show that s(t) ≤ s for all t. We thus have u = 1 in a neighborhood of t = 0. Otherwise, we would have u = 0 together with φ(0) ≤ 0, and from (14) we would have for all t, φ(t) < 0 which is not possible (as the trajectory would not reach the target). It follows that we have u = 1 in a neighborhood of t = 0. The same argument shows that the trajectory cannot switch to u = 0 before reaching v m . This proves the first item.
Assume now that s 0 >s. If φ(0) < 0, then we have u = 0, and the trajectory necessarily switches to u = 1 before reaching s (otherwise we would have a contradiction by the previous case). Now, we have u = 1 on some time interval [t 0 , t 1 ].
Again, the previous case shows that the trajectory cannot switch to u = 0 at some time t ′ such that s(t ′ ) ≤ s with v(t ′ ) < v m . As φ(t 0 ) ≥ 0 andφ(t) > 0 whenever s(t) > s (recall (14)), we obtain that the trajectory cannot switch to u = 0 at some time t ′′ such that s(t ′′ ) > s. Therefore, we have u = 1 until v m , and the conclusion follows.
Remark 4. If s 0 is such that s 0 >γ(v 0 ), then we have u = 0 on some time interval [0, t 0 ] with t 0 > 0. Otherwise we would have φ(0) > 0 implying φ(t) > 0 for all t (recall (14)). As the trajectory necessarily has a switching point when reaching v = v m , we have a contradiction.
The previous proposition allows to define a switching curve C ⊂ D where the control is switching from 0 to 1. Such a curve is the locus of conjugate points where the extremal trajectory ceases to be optimal [3]. Hence, for each v ∈ (0, v m ] there exists at most one point s c (v) such that C is parameterized by the mapping v −→ s c (v). Proof. Let (s 0 , v 0 ) ∈ D and consider an extremal trajectory with u = 1 from (s 0 , v 0 ) until reaching v m . Recall thatγ is a solution of (7) backward in time with u = 1 starting from (s, v m ). From Proposition 2, we must have s 0 ≤γ(v 0 ).
Suppose now that the switching curve crosses v = v m at some point (s ′ , v m ) with s ′ >s. Consider an extremal trajectory with u = 1 until v = v m originating from C. If the initial condition of this extremal is sufficiently close to (s ′ , v m ), then it reaches v m at a substrate value which is greater thans. This is a contradiction with (14). Hence, s ′ ≤s. Now, suppose that s ′ <s. Take an initial condition (s 0 , v 0 ) above C in the plane (s, v) and such that s 0 <s, v 0 > v a . From Proposition 2, one has u = 0 until reaching C. But, any extremal trajectory starting with u = 0 at an initial condition s 0 <s and v 0 < v m is not optimal. Hence, we obtain a contradiction with (14) which proves the result.
The curve C is depicted on Fig. 3. Propositions 2 and 3 allow to obtain an optimal feedback control of the problem in this case.
Theorem 3.2. Suppose that v * ≤ 0 (case 1). Then the optimal feedback steering any initial state in D to the target is given by: Proof. The result follows from Proposition 2 and the presence of the switching curve C.
The optimal synthesis in this case is depicted on Fig. 4.

Remark 5.
When the singular arc is never admissible, optimal controls are of type B 0 B 1 B 0 or B 1 B 0 depending on the initial points. Optimal trajectories contain at most two switching points.

3.3.
Optimal synthesis in the case v * ∈ (0, v m ). Throughout the paper, we only consider case 2, that is v * is such that 0 < v * < v m . We make the following assumption on M : (H2) The constant M is such that there exists a unique v * satisfying 0 < v * < v * andγ(v * ) =s. One can see that for initial conditions (s 0 , v 0 ) such that v 0 > v * , optimal controls are given by proposition 2. Indeed, the admissible part of the singular arc is only defined for v 0 ≤ v * . Therefore, we consider initial states such that v 0 < v * . The next Proposition is illustrated on Fig. 3.

Proposition 4.
Suppose v * ∈ (0, v m ) (case 2) and that (H2) is satisfied. Then, given initial states (s 0 , v 0 ) ∈ D such that v 0 < v * , optimal controls satisfy the following: (1) If s 0 ≤γ(v 0 ), then, there exists t 0 > 0 such that we have If s 0 ≥ s, and v 0 > v * , then, the optimal control is one of the following types: Proof. The proof of the first item is the same as the first one of the previous Proposition. Now, whenγ(v 0 ) < s 0 <γ(v 0 ) and s 0 ≤ s, the trajectory cannot switch from u = 1 to u = 0 before reaching s = s. Therefore, we have two cases when the trajectory reaches s = s: the trajectory either crosses the singular arc, or the control becomes singular. In the latter, the trajectory switches to u = 1 before reaching v * and we have u = 1 until v = v m (otherwise we would have u = 0 at the point (s, v * ) and the trajectory would not reach the target from (14)). Notice that t 1 = t 0 is possible. This means that the time interval where the trajectory is singular can be zero.
Ifγ(v 0 ) < s 0 < s, the proof is the same as for the second item except that the trajectory cannot leave the singular arc with u = 1 before v * (otherwise the trajectory reaches v = v m with u = 1 and φ > 0, and the trajectory cannot switch to u = 0 at v = v m from (14)).
The proof of the fourth item is the same as the third one except that the trajectory starts with u = 0 until reaching the singular arc. Similarly as in the previous item, the trajectory cannot switch to u = 1 before reaching s = s.
The last region is given by initial conditions such that s 0 ≥ s, and v 0 > v * . The same arguments as before can be used except that Pontryagin's Principle is not sufficient to exclude two type of trajectories. First observe that we have u = 0 on some time interval [0, t 0 ] as before (with s(t 0 ) <γ(v 0 ), otherwise the trajectory would not reach the target from (14)). When the trajectory crosses the curveγ, we have two sub-cases. Either the trajectory switches to u = 1 before reaching s = s (as in Proposition 2), either the trajectory switches to the singular arc for s = s. After the first switching times, the behavior of the trajectory is exactly as for the second item, and we can conclude from the other cases.
Remark 6. Proposition 4 shows that optimal controls are of type B 1 B 0 , B 0 B 1 B 0 or B 0 SB 1 B 0 with at most three switching points depending on the initial point.
We now investigate the loss of optimality of the singular arc. More precisely, we show that when the singular arc saturates at v * , it is no longer optimal until v * (see e.g. [8,9,20,23,24] and [5,7,15] for a study of the saturation phenomena). To be self contained, we provide a proof of this result in our setting.

Proposition 5.
Suppose v * ∈ (0, v m ) (case 2) and that (H2) is satisfied. Then, any optimal trajectory containing a singular arc leaves this singular arc between v * and v * (bounds not included).
Proof. If the trajectory leaves the singular arc for a volume value less than v * , then, we have u = 1 until reaching v = v m . From Proposition 4, we have φ > 0 at v = v m in contradiction with the fact that the trajectory switches to u = 0. Hence, it leaves the singular arc for a volume value v > v * . Notice that if a singular trajectory reaches v = v * at a time t ′ , then we have φ(t ′ ) =φ(t ′ ) = 0. For t > t ′ , we have s(t) <s for any control u (this follows from the definition of v * ). Hence, we haveφ(t) < 0, and we deduce that φ(t) < 0 for t > t ′ . Hence we have u = 0 using (11). As the trajectory necessarily has a switching point in order to reach the target, we obtain a contradiction.
We denote by v a the maximal volume value above which a singular arc is not optimal. From the synthesis of singularities in [8], the point (s, v a ) is a frame point of type (CS) 2 which means that it is at the intersection between a singular arc and a switching curve denoted by C. The index 2 means that the switching curve C is originating from the singular arc. Hence, for each v ∈ (v a , v m ] there exists at most one point s c (v) such that C is parameterized by the mapping v −→ s c (v). C connects (s, v a ) and (s, v m ).

Proposition 6. The switching curve
Proof. Let (s 0 , v 0 ) ∈ D and consider an extremal trajectory with u = 1 from (s 0 , v 0 ) until reaching v m . Recall thatγ is a solution of (7) backward in time with u = 1 starting from (s, v m ). From Proposition 2, we must have s 0 ≤γ(v 0 ).
Suppose now that the switching curve crosses v = v m at some point (s ′ , v m ) with s ′ >s. Consider an extremal trajectory with u = 1 until v = v m originating from C. If the initial condition of this extremal is sufficiently close to (s c , v m ), then it reaches v m at a substrate value which is greater thans. This is a contradiction with (14). Hence, s ′ ≤s.
Now, suppose that s ′ <s. Take an initial condition (s 0 , v 0 ) above C in the plane (s, v) and such that s 0 <s, v 0 > v a . From Proposition 2, one has u = 0 until reaching C. But, any extremal trajectory starting with u = 0 at an initial condition s 0 <s and v 0 < v m is not optimal. Hence, we obtain a contradiction with (14) which proves the result.
The switching curve C is depicted on Fig. 3. Propositions 4, 5 and 6 imply the following.
and that (H2) is satisfied. Then, the optimal feedback steering any initial state in D to the target is given by: Proof. From Proposition 6, we know that an optimal trajectory does not contain a singular arc for v ≥ v a . It follows from Proposition 4 that the optimal feedback control is 0 or 1 depending on the position of the initial point w.r.t. the switching curve. Moreover, when v = v m , the only possibility for u is zero. For v < v a , the result follows from Proposition 4.
The optimal synthesis in this case is depicted on Fig. 4. We see that for initial conditions such that v < v a , the optimal feedback is as in Theorem 2.2 whereas for v > v a , the feedback control is either 0 or 1 depending where the point (s, v) is located w.r.t. the switching curve v −→ s c (v).

Numerical simulations.
4.1. Determination of the frame point. We start by computing the optimal volume value v a ∈ (v * , v * ) above which the singular arc is no longer optimal. As we know the structure of optimal controls from Proposition 4, we proceed as follows.
For v 0 ∈ [v * , v * ], consider the strategy u = u s from v * to v 0 , u = 1 until v m and then u = 0 until s ref . The time t a (v 0 ) of this strategy is (recall (16), see also [1]): where s † (v 0 ) is the substrate concentration when this trajectory reaches v = v m : it is defined as the value for v = v m of the solution of (17) that starts at s =s with v = v 0 . Hence, s † (v 0 ) can be computed after solving the Cauchy problem: We now show that v 0 −→ t a (v 0 ) admits a minimum v a ∈ [v * , v * ] that we will characterize hereafter. First, (19) can be equivalently written as ds dv = g(v, s) where g is the right hand-side of (19). By the classical dependence of the solution of an ODE on parameters, we denote by s(v, s, v 0 ) the unique solution of (19). It is Consequently, it admits a minimum on this interval. By differentiating s(v, s, v 0 ) w.r.t. v 0 , we get: using a classical result on the dependance w.r.t. initial conditions of a solution of an ordinary differential equation. Hence ds † dv0 (v 0 ) = ∂s ∂v0 (v m , s, v 0 ). Now, we know from the PMP that v 0 = v * and v 0 = v * are not admissible (see also Proposition 4), hence v a necessarily satisfies dta dv0 (v a ) = 0. So, if we put θ(v 0 ) := vm v0 ∂g ∂s (w, s(w, s, v 0 ))dw, we obtain by taking the derivative of (18) By solving dta dv0 (v 0 ) = 0 using the previous equation, we obtain numerically the volume v a ∈ (v * , v * ) above which extremal trajectories stop to be optimal. We find that t a (v) has a unique minimum for v a ≃ 1.67 (see Fig. 2 and Table 1 where ξ ∈ [s,γ(v 0 )] and ξ † is the solution of (17) that starts at point (ξ, v 0 ) evaluated at v = v m . Numerical simulations indicate that for each value of v 0 , the point s c (v 0 ) where t b is minimal is unique. This allows us to define a curve C whose graph is the mapping v 0 −→ s c (v 0 ), represented in green on Fig. 3. Moreover, we find that for v 0 < v a , we have s c (v 0 ) =s, while s c (v 0 ) >s for v 0 ∈ (v a , v m ). To conclude, based on the optimal synthesis (Proposition 4) and the computations of the mapping and v a , optimal trajectories for various initial conditions (s 0 , v 0 ) are represented on Fig. 3. Table 1. Parameter values (arbitrary units) of simulations for the optimal synthesis when the singular arc is partially admissible (see Fig. 3  5. Discussion on the optimal synthesis. 5.1. Implicit equation of the switching curve. We present a convenient way to obtain the switching curve C as a solution of an implicit equation using the ODE satisfied by the switching function φ. From (13) and the conservation of the Hamiltonian along an extremal trajectory, we get: where ρ(s, v) := (sin−s)µ ′ (s) µ(s)v and (s, v) is a solution of (7) with a control u ∈ U . Define the mapping (w, s 0 , v 0 ) ∈ (0, v m ] × D −→ s(w, s 0 , v 0 ) as the unique solution of (17) such that s(v 0 , s 0 , v 0 ) = s 0 . From Theorem 3.3, an optimal trajectory of type B 0 B 1 B 0 starting at some point (s 0 , v 0 ) with v 0 < v m satisfies u = 0 until reaching C, and then u = 1 until reaching the volume v m . These conditions together with (20) will imply conditions on the point (s 0 , v 0 ). More precisely, we have the following result. Proof. By considering φ as a function of v (recall that dv dt = 1 on [0, t 0 ]), one obtains:

5.2.
Continuity of the value function. In this part, we explicit the value function T (s, v) over D which is associated to the minimal time problem, and we show that it is continuous. We suppose that hypotheses (H1)-(H2) are satisfied and we recall that C is the switching curve connecting (s, v a ) and (s, v m ). We define a curve ζ as the union of the singular arc (until v a ) and C: For (s, v) ∈ D and 0 < s ′ < s, we introduce the following functions: .
The function τ 1 is the time of an optimal trajectory starting on the singular arc at a volume value v < v a (recall (16)). It is clearly continuous over D. The function τ 2 is the time of an extremal trajectory with u = 0 connecting the points (s, v) and (s ′ , v) with s ′ < s. Clearly, it is continuous w.r.t. (s ′ , s, v) ∈ [0, s in ] × D.
Let us now consider an extremal trajectory with u = 1 starting from a point (s, v) ∈ D: • If the trajectory reaches the concentrations before the volume v m , we denote by v 1 (s, v) the volume value of this extremal when the concentration equals s. • If the trajectory reaches the volume v m before the concentrations, we denote by s m (s, v) the substrate concentration of this extremal when the volume equals v m . By the regularity of an ODE w.r.t. initial conditions, these two functions are continuous on D. Finally, consider the restriction to the set D of the mapping s −→ v 2 (s) which is the unique solution of (7) backward in time starting at (s, v a ). The value function T (s, v) can be now defined as follows (recall Theorem 3.3): Proposition 8. The function T is continuous in D.
Proof. One can easily verify that T is continuous at any point (s, v) outside the curve ζ. Hence, we only verify its continuity on this curve. First case. Suppose v 0 < v a , and let us show that T is continuous at (s, v 0 ). If s <s and v ≤ v 2 (s), one has v 1 (s, v) → v 0 whenever s goes tos and v goes to v 0 . So, we have T (s, v) = v 1 (s, v) − v + τ 1 (v 1 (s, v)) → τ 1 (v 0 ). On the other hand, if s >s and v ≤ v a , one has τ 2 (s, s, v) → 0 when s goes tos. Hence, we also obtain that T (s, v) → τ 1 (v 0 ) when s goes tos and v goes to v 0 , s >s. When v < v 2 (s) and s <s, one also has T (s, v) → τ 1 (v a ). Finally, we obtain the same limit for T when (s, v) goes to (s, v a ) with s >s. This ends the proof.
Remark 8. (i) The expression (21) allows to show that the value function T is of class C 1 whenever the singular arc is always admissible i.e. v * ≥ v m . This result is in line with the same result in the impulsive setting [1,14]. The expressions of ∂T ∂s and ∂T ∂v are delicate to handle in particular along an arc u = 1, so we have not detailed this point for brevity. (ii) When the singular arc is no longer admissible i.e. v * < v m , we can show that T is not differentiable along the switching curve C (but it is right and left differentiable on C).

5.3.
Regularity of the optimal synthesis. Given the feedback controls u i , i = 1, 2 (see Theorem 3.2 and 3.3), we would like to know if trajectories corresponding to this feedback are unique [21]. In the case where the singular arc is always admissible [19], then one can use the clock form [4,8] to conclude directly on the uniqueness of an optimal trajectory starting at some point (s 0 , v 0 ) ∈ D. In fact, this argument shows directly that any other trajectory has a greater cost as the optimal one (with equality only if both associated controls are equal a.e.). Hence, optimal trajectories corresponding to the feedback u M and u H are unique (Theorems 2.1 and 2.2).
This method no longer applies in the case where the singular arc is non admissible. Nevertheless, Pontryagin's Principle allows to exclude any extremal trajectory that has a switching point outside the singular arc {s} × [0, v a ] and the switching curve, before reaching the maximal volume (see Propositions 2 and 4). These trajectories exactly correspond to the solution of (7) with the feedback control u i . 6. Conclusion. We have studied an optimal control problem for a fed-batch bioprocess that exhibits a singular arc with a saturating point. Thanks to Pontryagin's Principle and the exclusion of extremal trajectories that are not optimal, we have obtained an optimal synthesis of the problem. We have pointed out the existence of a frame point on the singular arc above which any singular trajectory is not globally optimal. Moreover, we have provided an explicit way for computing numerically the switching curves and the frame point. The present situation can arise for example when the initial biomass concentration in the reactor is high. In this case, one should take advantage of the feedback control law for the practical implementation.
A more detailed insight into the determination of the switching curves (for instance using the theory of conjugate points [3]) could be the basis of future works.
Also, it could be interesting to study an optimal synthesis of the problem with multiple saturating turnpikes [1] (this can happen in practice when there exist more than two non-limiting substrates in the reactor). This question appears to be quite challenging.