15.4.1Finite-dimensional minimization problem

Let us rewrite our time-optimal control problem as a general optimization problem. First of all, we consider a time control T ̄ (i.e., a time that solves the control problem (1.1)–(1.4)) so that the minimal time T⋆ is lower than T ̄ and we define the cost function J0 : (T, λ) ∈ [0, T̄ ]× BV(0, T̄ )2 ↦ T ∈ ℝ. The optimization problem reads as follows:

According to item 2 of Remark 2.3, any optimal solution λ satisfies |λ(t)| = 1 for almost every t and hence it can be expressed as λ(t) = R(Θ(t))e1 for almost every t, with e1 = (1, 0)⊤. The state variable α is also given by Then,we introduce the new cost function J1 : (T, Θ) ∈ [0, T̄ ]× BV(0, T) → T ∈ ℝ and theabove optimization problem can be simplified to

In order to numerically tackle this problem, we set n ∈ ℕ∗ an integer and t0 < ⋅ ⋅⋅ < tn n + 1 time discretization points. For every i ∈ {0, . . . , n}, we consider Θi ∈ ℝan approximation of Θ(ti). Let us also define a quadrature formula ?(t0, . . . , tk; f0, . . . , fk)approximating

Thus, defining the cost function Jn : (T, Θ0, . . . , Θn) ∈ ℝ+ × ℝn+1 ↦ T ∈ ℝ, the discretized version of (4.1) is

This nonlinear minimization problem can be solved, for instance, with an interiorpoint method, sequential quadratic programming (SQP), or an active-set algorithm (e.g., [7, 9, 13]) as implemented is the Optimization toolbox of MATLAB through the fmincon function. We will see on practical examples that even if the interior point algorithm is robust with respect to the initialization, the better the initialization guess is, the better the result is.

A natural initialization procedure is to use the time-optimal control obtained analytically for the linearized problem with ∇V(0)α in place of V(α).

15.4.2Numerical example

Let us consider the following:

For the computation of the integrals in theminimization problem, we use the trapeze method, that is, for f ∈ C([τ0, τ1], ℝ), we approximate dt by

with

In order to numerically solve the optimization problem (4.2) with (4.3), we use the fmincon function of the Optimization toolbox of MATLAB with a SQP algorithm. A first observation is that if we choose the initial value (T, Θ0, . . . , Θn) = (1, 0, . . . , 0), the algorithm is not converging. On the other hand, when initializing with the optimal solution of the approximate linearized problem(3.1),(1.1c),(1.2)–(1.4) (see Section 15.3), we obtain a reasonable solution to the nonlinear optimization problem(4.2) (Figure 2).

The SQP algorithm with the trivial initialization point, (T, Θ0, . . . , Θn) = (1, 0, . . . ,0), fails to converge because the discrete state trajectory α0, . . . , αn associatedto this control does not fill the state constraint |αk| ≤ c in (4.2). Indeed, for the initial value (T, Θ0, . . . , Θn) = (1, 0, . . . , 0), due to (4.4) we have αk = tke1 for all k ∈ {1, . . . , n} and in particular we get |αn| = |Te1| = T = 1. A way to avoid this is to use the initialization point:

so that the discrete state variables associated to this control are given by αk = 0 (for every k ∈ {0, . . . , n}) and thus the state constraints are satisfied. We compare in Table 1 the solutions obtained for these to different types of initialization values. These results show that the algorithm is more efficient when the optimal solution of the linearized problem is chosen as an initial guess.

In Table 1, the comparison is made with V given by (4.3), the target hf = 1 and the state constraint given by (1.3) with c = 0.3. The number of discretization points is given by n +1, T⋆ is the optimal time obtained, N SQP is the number of iterations inside the SQP algorithm and TCPU returns the CPU time used for computations.

Table 1: Comparison of the results for the two different types of initializations with the SQP algorithm.

(a) Initialization with the initial value (4.5)

(b) Initialization with the optimal control obtained for the linearized problem (3.1)

In addition, the optimal solution for the nonlinear optimization problem (4.2) obtained with the initial value (4.5) does not look as good as the optimal solution obtained from the optimal solution of the linearized problem (3.1) (see in comparison Figures 2 and 3).

15.5.1Modeling and problem formulation

Axisymmetric coordinates

Since we are considering axisymmetric swimmers, we introduce the spherical coordinate system (r, θ, ϕ) ∈ ℝ+× [0, π] × [0, 2π). For every x = (x1 x2 x3)⊤ ∈ ℝ3, the spherical coordinates (r, θ, ϕ) = (r(x), θ(x), ϕ(x)) are such that

with the associated local system of unit vectors (er , eθ, eϕ) given by (see Figure 4)

Swimmer’s deformations

The shape of the swimmer at rest is the unitball ofℝ3 ,which forms the reference shape denoted by S0.We assume that the deformation of the swimmer is axisymmetric with respect to the symmetry axis e1. More precisely, we assume that the deformation X is built from two elementary deformations D1 and D2, that is,

where D1 and D2 are axisymmetric and radial deformations, that is,

with δi ∈ C1([0, π], ℝ) and αi ∈ L∞(ℝ+, ℝ) such that

so that X(t, ⋅) is a C1-diffeomorphism on S0. In practice, we will consider c > 0 small enough such that (1.3) implies (5.3). We define the domain ?(t) occupied by the deformed swimmer at time t in the reference frame attached to the swimmer:

We also assume that the deformation X does not produce any translation. To this end, we introduce the mass density ρ0(x) = 1 in the shape of the swimmer S0 at rest and we assume that the mass is locally preserved during the deformation, that is, to say that the density of the swimmer at any time t ≥ 0 is given by

where JacX(t, ⋅) denotes the Jacobian of the mapping X (t, ⋅). According to (5.1) and (5.2) we have

With this mass density, we have, for all t ≥ 0

and the mass center of the swimmer is given by

Consequently, we assume

so that the mass center of the swimmer does not move with the deformation X.

Finally, let us consider the domain ?†(t) occupied by the swimmer in the fluid at time t ≥ 0, which is given by

Since we have assumed that the deformation X does not introduce any translation,h(t)e1 is the mass center position of the swimmer in the fluid domain at time t. Thedomains S0, ?(t), and ?†(t) are depicted in Figure 5.

In terms of the control theory, α = (α1, α2)⊤ is the system’s input and h is its outputwe aim to control.

Micro-swimmer and fluid flow

It is well known that the system governing the motion of a micro-swimmer is inertia less (see for instance [20, § 5.3] or [11]) and reduced to the following problem coupling:

–the Stokes equation:

where ℱ(t) = ℝ3 ?(t) and with

–the boundary condition (continuity of the velocity at the boundary of the microswimmer):

–quasi-static Newton law:

where pI3 is the Cauchystress tensor.

The well posedness of (5.7) follows from [14], where the limit (5.7c) has to be understood in a weak sense (i.e., More precisely, we have (u, p) ∈ where we have set

This ensures that and consequently, theexpression ∫∂S(t)σ(u, p)n dΓ is seen as a duality product (since σ(u, p)n ∈ (∂?(t))3 and 1 ∈ (∂?(t))).

A geometric control problem

Let us first notice that in the full system (5.7)–(5.9) the time does not appear directly but only through the parameter α(t). Consequently, we define ?(α) as the image of S0 by the map X(α) : x ∈ ℝ3 ↦ x + α1D1(x) + α2D2(x) ∈ ℝ3, for α ∈ ℝ2 such that (5.3) holds. For convenience, we also define the corresponding fluid domain ℱ(α) = ℝ3 ?(α).

For every α ∈ ℝ2 satisfying (5.3), we define the solution of

and for i ∈ {1, 2}, is the solution of:

Then, by decomposing the solution (u(t, ⋅), p(t, ⋅)) of (5.7)–(5.8) in terms of and due to the linearity of the Cauchy stress tensor, relation (5.9) becomes

where for every i ∈ {0, 1, 2} and every α ∈ ℝ2 satisfying (5.3), we have set

with

By Green formula, we have

Consequently, F0(α) ⋅ e1 ≠ 0 and the control problem can be recast as the generalizedBrockett system (1.1) with

This system form has already been established in the pioneer work of Alouges et al. [5].Notice that even if α represents the physical control of the swimming system, it is ofmore convenience for analysis to consider λ = α̇ as the control variable. This will also allow us to control both the shape and the position of the swimmer.

We also set the initial conditions (1.1c) for this system and the target position to be reached in a time T > 0 is given by (1.2). These initial and final conditions mean that the micro-swimmer is the unit sphere located at the origin at initial time and a unit sphere located in hf e1 at final time T.

We point out that a state constraint on α is needed to be able to well define Fi(α(t)) for every time t ≥ 0. This constraint is given by (5.3).

In addition, according to [21, Lemma 1] or [23, Theorem 2.6], there exists c > 0 small enough such that the mapping α ∈ ℝ2 ↦ V(α) ∈ ℝ2 with V defined by (5.14) is of regularity C∞ on B0(2c). Consequently, we chose c > 0 small enough such that this condition holds together with (5.3).

Remark 5.1. From [23], there exists δ1, δ2 ∈ C1([0, π], ℝ) in (5.3) such that ∇V(0) is not symmetric. Moreover, using the arguments of [21], this situation is generic.

15.5.2Numerical computation of a time-optimal trajectory

Let us consider the deformations D1 and D2 similar to the one given in [23], that is, to say that D1 and D2 are given by (5.2) with

where P2 (resp. P3) is the Legendre polynomial of order 2 (resp. 3). With these two elementary deformations, we obtain from [27],

These expressions allow us to compute the explicit form of the time-optimal controls for the approximate linearized system (3.1).

In order to numerically compute the fluid forces Fi(α) ⋅ e1 for i ∈ {0, 1, 2}, we use the spherical harmonics expansion of the exterior Stokes solution given in [8] (see also [15] or [27]). Then, we compute the time-optimal controls by using the direct discretizationmethod explained in Section 15.4. The initial guess for the discrete nonlinear optimization problem is chosen as the explicit optimal solution of the approximate linear control problem (3.1) given in Proposition 3.1. We expect that the approximate optimal deformation for the linearized problem is close to the optimal solution for the nonlinear problem so that the direct method will converge quickly.

We apply this computational method with the deformation D1 and D2 given abovethrough Legendre polynomials and we choose c = 0.3 and The optimal trajectories for h and α are depicted in Figure 6. The optimal time is T⋆ ≃ 48.9.

We numerically observe in Figure 6 that the optimal trajectory for α is mainly periodic. In Figure 7, we also give the optimal trajectory of h during a period in time and finally we plot in Figure 8 the different shapes of the swimmer under the optimal deformation for different instants in the time period.