ON THE CONTROLLABILITY OF RACING SAILING BOATS WITH FOILS

. The development of foils for racing boats has changed the strategy of sailing. Recently, the America’s cup held in San Francisco, has been the theatre of a tragicomic history due to the foils. During the last round, the New-Zealand boat was winning by 8 to 1 against the defender USA. The winner is the ﬁrst with 9 victories. USA team understood suddenly (may be) how to use the control of the pitching of the main foils by adjusting the rake in order to stabilize the ship. And USA won by 9 victories against 8 to the challenger NZ. Our goal in this paper is to point out few aspects which could be taken into account in order to improve this mysterious control law which is known as the key of the victory of the USA team. There are certainly many reasons and in particular the cleverness of the sailors and of all the engineering team behind this project. But it appears interesting to have a mathematical discussion, even if it is a partial one, on the mechanical behaviour of these extraordinary sailing boats. The numerical examples given here are not the true ones. They have just been invented in order to explain the theoretical developments concerning three points: the possibility of tacking on the foils for sailing upwind, the nature of foiling instabilities, if there are, when the boat is ﬂying and the control laws.

1. Introduction. The control of foiling during the America's cup appeared to be a determinant point in the success of Oracle Team USA (OTUSA). In particular during upwind legs, when the boat had to avoid the waves created by the wake of the preceding boat, the automatic stabilization is a fundamental advantage that OTUSA exploited in a smart way and finally won the competition. Such situations are represented on the Figures 2 taken from TV transmissions during the America's cup in San Francisco (September 2013). Figure 2. Two situations where the controlled foiling could avoid to tack and to loose the race Two movements of the ship are taken into account in this paper : the heaving which is a normal displacement to the surface of the sea, and the pitching which is the rotation around a horizontal axis transverse to the main direction of the ship. Hence, the yawing angle and the rolling are eliminated from our model. Obviously they are meaningful, but according to our mind, not necessary for the understanding of our purpose.
Our aim is to study the controllability of the boat modeling the skipper's action on the foil by a control law. The inclination of the main foil should be manually driven but a hydraulic ram can be used for the control process (rules of the race) using the high pressure collected from a small hole in the foil. Because the system is a second order one (with inertia, damping and stiffness), only a phase control can lead to optimal results. This driven angle is named the rake. It appears, in the numerical simulations, that the control loop strongly depends on the ship velocity. Even if the experimental data that we introduce in our numerical model could be improved, they are sufficient in order to give an idea of how things work.
The rudder is supposed to be fixed, the control being an action only on the main foil. The handling foil is modeled by a rotation around a center O and with an angle δ : this angle is the control variable.
We study several control laws. The first one is an exact control law. From a numerical point of view, the approach consists in an optimal control problem. From a theoretical point of view, the exact controllability will be obtain, under suitable conditions, by an asymptotic analysis from the optimal control problem. This kind of controls is part of the large familly of the theory of controls without constraint. Unfortunately, this is not realistic in our case since it would be not acceptable to obtain an angle at the foil equal to π 2 for example! The second type of control is an optimization problem with constraints : the magnitude of the control, which is still the angle of the main foil, being restricted to 2 or 3 degrees. The time of controllability and the magnitude of the constraints are then connected.
The third and last kind of control law is a proportional feed-back loop system. The law is commonly used by airplanes when they enter in a turbulence zone. This type of control is the one used by OTUSA team in 2013.
Let us notice that this paper rests on a previous one (see [7]) where we gave a simple and precise mathematical model of such boats in a bi-dimensional case which respects the following facts: -existence of the foiling velocity under which the boat can't stand up on its foil, -possibility that the velocity of the boat can be greater than the wind velocity, and the possibility to discuss the stall flutter phenomenon of the foils. We consider the same model and we recall in a first section some results obtained in (see [7]) without any proof.
2. Dynamical model of the boat. The orthonormal basis of R 3 is denoted by (e x , e y , e z ). The velocity of the boat is −ue x where u > 0 and therefore the velocity of the flow of the water in the basis connected to the boat is ue x . As said before, the movement is assumed to be represented by two functions (see Figure 3): the heaving z and the pitching angle γ (rotation in the plan (e z , e x )). The equilibrium is written at an arbitrary point -say O. For sake of convenience, it is chosen to be the center of rotation of the main foil.
The following notations are used: Characteristics of the boat, of the air and of the water: a mass density of the air, e mass density of the water, g = 9.81m/s 2 is the gravity, −ue x velocity of the ship, M is the mass of the ship, G center of mass of the boat, J G is the inertia with respect to the center of mass G in the pitching, J O is the inertia with respect to O in the pitching, M o is the moment of the external forces at point O in the pitching, d s = O S is the length of the stick supporting the steering rudder, O being the anchor point of the rear foil; d f = OF is the length of the foil in the depth direction; S s , S f are respectively the cross sections of the foils at the extremities of the rudder and the main foil; a (respectively b) is the distance between the center of mass and O (respectively Variables for the description of the movement of the boat: z is the heaving γ is the pitching angle For the angles, apparent velocities and forces: α is the angle of attack of the main foil, β is the angle of attack of the rear foil, it is supposed to be fixed, δ is the control angle of the main foil, in fact one has: α = γ + α 0 + δ; where α 0 and β 0 are the nominal positions of the foils (main and rear), c zf and c zs are the hydrodynamic lift coefficients for the main foil and the rear foil. They are continuous functions in their variables, c mf et c ms are the hydrodynamic pitching coefficients at points F and S. They are continuous functions in their variables, L is the length used in the definition of the pitching coefficients, v is the absolute wind velocity: it is in the plane (e x , e y ), V is the modulus of v, absolute wind velocity, θ = (v, −e x ) is the angle between the velocity of the wind and the direction in which the boat is moving forward, V a is the modulus of the apparent velocity of the wind, V as , V af are respectively the apparent flow velocity at the two foils: one on the rudder and the other one which is the main one, supported by the daggerboard; u is the modulus of the velocity of the ship, c f and ξ are respectively the stiffness and the damping coefficient of the system used for the stabilization of the main foil as far as it is not perfectly fixed to the daggerboard.
The forces applied to the ship and implying an evolution of the two functions z and γ, are those due the rear and the main foils. The local hydrodynamical coefficients (c zs and c zf for the lift and c mf a,d c cs for the pitching moment) depend respectively on the apparent local angle of attack of each foil. It is denoted by (β + γ) a for the rear foil and by (α + γ) a for the main one.
The variables are z and γ. Let us recall that β is assumed to be fixed (equal to β 0 ), and the evolution of the pitching angle of the rear foil is only due to the global pitching of the boat -say γ.
We introduce several matrices which are constant with respect to the unknowns z, α and δ of the system (M, C and K) whose coefficients will be explicited in a next step: and the righthandsides B, E are defined by: In the following we have K 1 = K 2 = 0. We proved in [7] that the linearized movement of the boat can be modeled by the following system at the equilibrium with the initial data: X(0) = X 0 ∈ R 2 andẊ(0) = X 1 ∈ R 2 . In fact the control variable δ and its time derivative appear in the model. It implies a hydrodynamical damping. This is why one can say that it improves the stability. In automatic one say that it is a dynamic control. Let us point out that it can be the only realistic control. In fact it has been observed in the numerical tests that the control should imply both δ andδ for a better accuracy. One advantage is that the two values of δ(0) and δ(T ) can be prescribed to zero (space H 1 0 (]0, T [)) and this avoids shocks when applying the control law. For ξ 1 and ξ 2 small enough, we set: The expressions in (3) at the equilibrium point are the following: • the stiffness matrix is :

PHILIPPE DESTUYNDER AND CAROLINE FABRE
where R 1 and R 2 are defined by: and • the coefficients of C are given explicitely in the following table: Coefficient Expression of the coefficients of the matrix C near γ = 0 2 1 Table 1 Expressions of the coefficients C ij near γ = 0 versus α and β • the righthandsides B and E are : with with We now turn to the controllability of the boat. There are two steps: the first one consists in defining an exact control without constraint and the second step takes into account the constraints on the control amplitude. In this second step, appear the links between the control time -say T -and the bounds on the amplitude of the control (ie. the maximum value for δ). Let us set δ = α − α 0 , where α 0 is a nominal position of α. Obviously it is fixed at the begining of the race depending on the average wind velocity.
3. The control of dynamical behavior via the rake of the main foil without constraint. We first consider an optimal control problem depending on a small parameter ε. Exact controllability will be obtain with an asymptotic analysis when ε tends to 0 and it will involve restrictions on the coefficients.
3.1. The optimal control problem. The control variable is the rake variation δ. In this study, we don't discuss the pratical feasibility of the optimal control law that we compute. It is just an indication of what should be done and it can be understood as an educational result, as far as the real driven law is manual. The strategy that we are using consists in defining a control criterion and minimising it with respect to the control variable. This control criterion is a norm of the gap between the measured state variables at a time T (pitching and heaving) and the desired values of these functions. Furthermore we make use of a linearized approximation of the state equation recalled in section 2.
Therefore the control problem consists in defining a criterion depending on the state variables (ie. z and γ) but also on the control δ = α − α 0 and to minimize it with respect to δ. Let us first define a control time -say T -corresponding to the reaction time required in the control process. Our goal is to define a control law such that at time T the ship is back to the equilibrium The value of T is not important in the theoretical developments. It modifies the amplitude of the control depending on the magnitude of the perturbations. But any kind of perturbations can be taken into account. Nevertheless the influence of T is meaningful as far as the amplitude of the control is restricted. In this case one can define a minimum time for which one obtains an exact control. Obviously it depends on the upper bound of the control and on the amplitude of the excitation. In our tests, we have chosen T = 10 sec. but this an arbitrary value considered as an upper bound for T . In fact the rest is generally obtained at a time t ≤ 3 sec. and the perturbations used are realistic.
Let X 0 ∈ R 2 et X 1 ∈ R 2 be the initial data. In fact, there are several controls which can give the same results. Therefore it is worth to select the one which is the cheapest with respect to a given norm. The basic cost of this control is obtained by introducing a norm which takes into account the least square of the amplitude of the control and the one of its time derivative. The ponderation between the two is an engineering choice. Therefore let us consider three coefficients a 0 > 0, b 0 > 0 and ε > 0. The coefficients a 0 and b 0 play a role in the convergence when ε tends to zero. If a 0 is more important than b 0 the engineer chooses to promote the amplitude of the control. If b 0 is larger than a 0 , then the velocity is considered as more important. In our numerical examples, we choose a 0 = b 0 = 1. The criterion for the control problem, is defined as follows: where ω ∈ H 1 0 (]0, T [) and (10) Let us compute de gradient of J ε . Let X be the solution of: and Z be the derivative of X with respect to δ at ω. Z is solution of: We write X = z γ The optimal control is finally defined as the solution of the following problem: Remark 1. The choice of the space H 1 0 (]0, T [) has been done in order to have a finite cost for the control δ. The boundary conditions could be different (at t = 0 and t = T ). For instance, with free edge conditions, one would obtain more degrees of freedom for the control. But the discontinuity at the extremities is not always a good strategy in the practical implementation. Nevertheless, this could be useful in particular case where the controllability condition is not satisfied. This point is discussed in the following.
The optimization problem (14) is very classical (see [3] or [14]): there exists a unique solution δ ∈ H 1 0 (]0, T [) solution of (14). The first point is to formulate the optimality relation which is obtained in this case by writing that J ε (δ) = 0. This is quite classical to introduce the adjoint state P solution of the following differential system. Let then X (solution at the optimum value) and P ∈ H 2 (0, T ) 2 (adjoint state) be the solutions of: and The optimality relation can be written (see [1]): (. denotes the scalar product in 3.1.1. Computation of the optimum (version 1). Let us introduce the hilbertian basis (w n ) n of L 2 (]0, T [): and let us look for and Hence 3.1.2. Computation of the optimum (version 2). We present another way to compute δ which is numerically more stable because it avoids the difficulties connected to the weak convergence of the Fourier series. With the notations of the previous paragraph, and considering (18), we write: We easily get Since δ(0) = δ(T ) = 0, we must have: The second expression of the optimum is : From a numerical point of view, it is strongly suggested to use a gradient method with an optimal step, unless we use the asymptotic method described in the next section which is the one used in the numerical tests.
Our study depends on the small parameter ε. In what follows, we denote by δ ε , P ε and X ε the optimum solution which minimizes the functional J ε and we are now interested in the asymptotic analysis when ε tends to 0.
3.2. Asymptotic analysis. Let us introduce the set of admissible controls: There is no chance to control exactly if (and only if) U ad = ∅. In the contrary, if U ad = ∅, it's a close and convex set in H 1 0 (]0, T [). We prove : Proof. Let δ ε be the minimum argument of J ε in H 1 0 (]0, T [) and X ε be the solution of (15) associated to δ ε . Let ω 0 ∈ U ad (independent on ε.) For every ε > 0: On one hand, we deduce that and on the other hand, We deduce from (22) There exists a subsequence (still denoted by δ ε )) weakly converging in H 1 0 (]0, T [) to a function δ ∈ H 1 0 (]0, T [). By linearity of the model, X ε weakly in H 1 (]0, T [) to the solution X of (15) associated to δ. Furthermore, M does not depend on ε and is invertible thus X ε is then bounded in H 2 (]0, T [). After a second extraction of a subsequence, X ε strongly converges in C 1 ([0, T ]). Assertion (23) leads to X(T ) = t (z 0 , 0) andẊ(T ) = t (0, 0). We have proved that a subsequence of δ ε converges to an exact control δ ∈ U ad . Since , which is valid for every ω 0 ∈ U ad , implies (21). The functional is strictly convex and continuous on U ad (close and convex), its reaches its minima at a unique point. Therefore, the (weak) limit point of the sequence (δ ε ) ε is unique and thus all the sequence converges to δ. Finally, let's take ω 0 = δ in (22). We get: and we deduce the strong convergence of (δ ε ) ε to δ in H 1 0 (]0, T [), unique minima of J on U ad . The theorem is now proved.
Remark 2. The exact control depends on the data and on the control time T . The more T is small the more the amplitude of the control is large. But, in our case, the amplitude of the control is limited by the angle at the foil. This is why, it is particularly interesting to study the controllability with constraint. This is done in the future sections.
We now turn to the study condition U ad = ∅ and the computation of the exact control when it exists. We end this section with numerical simulations.
3.3. The exact controllability and condition U ad = ∅. A formal development with respect to ε leads to an other way to solve exact controllability. This approach explicits the unique continuation property underlying the condition U ad = ∅. Let us set a priori: By introducing a priori these expressions into the system (15), (16) et (17) and by equatting the terms of same power in the resulting expressions, one obtains: • at the order 0: • at the order 1: and others orders are similar to first one. Our first point is to prove that with reasonable assumptions, one has P 0 (t) = 0 on [0, T ]. This is in fact an exact controllability result for z 0 (T ),ż 0 (T ),γ 0 (T ) and γ 0 (T ) because it will imply that z 0 (T ) = z 0 ,ż 0 (T ) =γ 0 (T ) = γ 0 (T ) = 0.
Let us now turn to a controllability result which is a more adapted version of the general Bellman's result [1]. We assume that B, E are linearly independent. Let B * , E * be the dual basis of B, E defined by: Let H, L, Z 1 , Z 2 and Z 3 be the following matrices or vectors: We prove: Theorem 3.2. Assume B, E are linearly independent and that Z 4 .E = 0. Let r i (i = 1, 2) be the roots of: The unique continuation property P 0 = 0 is valid under the hypotheses : 1. if r 1 = r 2 (single roots), and for i = 1, 2 : (25), we get on ]0, T [: and thus (still on ]0, T [): Let us multiply the first equation by Z 4 .E , the second one by Z 4 .B and then add, we get: The solutions are ξ 1 (t) = A i e rit where r i is a root of and: Finally, if none of these relations is true, we must have: ∀t ∈ [0, T ] : ξ 1 (t) = ξ 2 (t) = 0 and thus ∀t ∈ [0, T ] : P 0 (t) = 0.
This ends the proof in the case of single roots.
Case 2. assume that there exists a double roots: r 1 = r 2 . The solutions of (27) are: The following condition is needed for the unique continuation property : The theorem is proved. Let us now turn to special cases for which the condition is simplified.
3.3.1. The case without kinetic coupling. We imagine in this subsection, that the kinetic coupling can be neglected and thus C is chosen equal to zero. This leads also to E = 0 in the system (25). The unique continuation property to establish is then:  This condition is a very simple test to do.
Remark 3. We write the second column of K: As said before, the matrix K is made of two contributions : one of the rudder and one of the foil and the last one is −B. We write −K s the rudder's one, and we obtain: Furthermore, we have Since B.D = 0, we get: The condition K s .D = 0 being equivalent to K s non parallel to B, this ends theorem's proof.

3.3.2.
Kinetic coupling from the rudder. We assume that the kinetic coupling from the foil is null. No hypothesis is made on the rudder coupling. In that case, E = 0. The unique continuation property is The four reals numbers r i (i = 1 · · · 4) introduced are the coefficients of the system and they do not depend on t. In that case, the unique condition is (r 2 4 − r 1 r 4 r 3 + r 2 r 2 3 )ξ(t) = 0 on ]0, T [ and Theorem 3.4 is proved.
We end this section with the explicit computation of the exact control.

3.3.3.
Computation of the exact control. The control δ 0 can be computed in different way as what we have done for the optimal control. A Fourier development leads to with: In order to avoid Gibbs phenomenum, it can be judicious to explicit δ 0 in the following way : Once δ 0 explicit, we write using Fourier decomposition and we look after Φ solution of: This is the way used in the numerical tests.

3.3.4.
Numerical tests for the exact control without constraints. We present numerical tests of exact control both for heaving and pitching movements. The values for parameters a 0 and b 0 are a 0 = b 0 = 1. For each Figure, the upside graph concerns the pitching movement, the middle one concerns the heaving movement and the downside one is the graph of the exact control. Angles are always expressed in radians. Figures 4, 5 and 6 concern the exact controllability of an initial heaving of 10 cm, with zero velocity and null for both the pitching and its velocity; three boat velocities have been tested: 10 m/s, 15 m/s or 20 m/s. We observe that an exact control is acceptable for a 10m/s speed boat but not in the other cases. Indeed, the heaving or the angle at the main foil would be too large for relalistic models. If the speed of the boat is 10m/s, the angle at the main foil can reach 0.4 radian (20 degrees) which seems to us a very extreme case. Figures 7, 8 and 9 concern the exact controllability of an initial pitch, equal to: −2 degrees with null initial pitch velocity and null heaving. The velocity of the boat is equal to 10m/s, 15m/s or 20m/s. The same remarks as before apply in this case. Let us point out that all the remarks formulated for the heaving control are still true for the pitching control. In practice the magnitude of the control is restricted. Therefore the minimisation of the criterion J ε must be performed over the bounded set: where q max > 0.
The new control problem is now: From classical results in optimisation, the optimal solution is characterized by: With (13), we get:     For ε → 0, one can define a limit control problem (see [6]) following the same idea as in the previous section, but with a different strategy. Let us consider the two following possibilities.

4.1.
Case where there is an exact control in K. We assume that there is an exact control -say δ 0 ∈ K for the initial conditions and the time control T . The solution of (15) satisfies at the final time T : X(T ) = t (z 0 , 0) etẊ(T ) = 0. We have δ ε ∈ K and ∀q ∈ K, J ε (δ ε ) ≤ J ε (q).
Let's input q = δ 0 , we get (X ε is the solution de (15) associated to δ ε ): The functions δ ε are bounded in H 1 (]0, T [) and up to a subsequence, we get: and thus δ * ∈ K. In a same way, X ε are bounded in H 2 (0, T ). A subsequence strongly converges in H 1 (0, T [) to the solution X * of (15) with righthandside δ * . Furthermore, with (39), we obtain : We deduce that δ * is an exact control in K and: The function δ * is thus an exact control of minimum norm. Since this minimum is unique, there is a unique limit point and the full sequence converges to δ * . It is easy to see that the adjoint state P ε , which is solution of (16), converges to 0.

4.2.
Assuming that there is no exact control in K. A strategy is to weaken the admissibe control set. Consider the functional where X is solution of: J ε reaches its minimum at δ ε ∈ K 0 and we get: Following the steps of the previous section, we consider X ε the solution of (40) associated to δ ε and the adjoint state P ε solution of: We get: and thus : Since δ ε ∈ K 0 , these functions are bounded in L ∞ (]0, T [). Up to a subsequence, they converge in L 2 (]0, T [)− weak and in L ∞ (]0, T [)− weak-* to δ * ∈ K 0 . Furthermore X ε weakly converges in H 2 (]0, T [) to the solution X ∈ H 2 (0, T ) of (40) associated to δ * and we can assume that the convergence of X ε to X is strong in C 1 ([0, T ]). We deduce that (P ε ) ε strongly converges in C 1 ([0, T ]) to the solution P * of the adjoint limit problem. Finally, we get: Since (J ε ) ε ponctually converges to condition (43) can be written This is the optimality condition of J on K 0 and we deduce that δ * satisfies J(δ * ) = min q∈K0 J(q).
By loss of strict convexity, we loose the unique character of the limit point and of the optimal point. Condition (43) is equivalent to: and this is what we call a bang-bang control : δ * takes only the two extremal values. The lack of convergence and the non uniqueness of the optimal control explain numerical instabilities in this case. Therefore it is worth to choose T in order to ensure that there exists an exact control, otherwise some instabilities could appear in the computation of δ (even if it is unique in the space H 1 0 (]0, T [)), for any ε > 0 small enough.

Remark 4.
Usually the R. Bellman condition [1] and the fact that the real part of the eigenvalues of the characteristic equation of the linear system, are negative (even strictly negative for multiple eigenvalues) are sufficient conditions to ensure that there exists an exact control but with a control time which can be larger than the one used in the computation of the exact control without constraint. The minimum time T min for an exact control is the boundary between the existence and the non-existence of an exact control in the set K. Nevertheless the bang-bang control should be avoided in this case (as for an aircraft) because it is not robust and implies shocks which can be at the origin of unwanted perturbations. Therefore, knowing the maximum amplitude q max of the control, it is possible to compute the minimum control time -say T min -and therefore to ensure that the exact control law is computed with a control time T sufficiently larger than T min . 4.3. Numerical tests. The data of the tests are the same as in the case of exact control without constraint. We authorize foil angles between ±3 degrees. We notice that the control time is longer than before (without bounds). 5. A feed-back system based on the sign ofα.

5.1.
A short description of the model used in this case. As we have pointed out in the previous sections, it is necessary to introduce the termδ in the control law in order to stabilize the model and to avoid shocks at t = 0 and t = T .
A control which only include δ would be unstable because of the vanishing stiffness in the direction of the heaving movement unless we would take into account the unilateral contact between the foils and the surface of the water. But this is a much more complicated problem that we didn't study here. Furthermore, the absolute referential for δ = α − α 0 is not obvious in the practical implementation.
In fact the connection between the foils and the daggerboard is a little bit flexible and the goal of this section is to take into account this aspect and to show that it can be a useful phenomenon in the control process. In fact the pitching velocity is easier to detect as far as a flexible rotation is allowed at the jonction between the foil and the bow of the boat. This is shown on Figure 26.
The flexibility can be stiffened using a spring damped by a hydraulic ram. The control is a simple feed-back loop of the velocity of the rakeδ based on prescribing the rake δ + α 0 through this ram. Let us try to explain why this method, even if it is not optimal, is a good one for the stabilization of the oversea flight.
In this case the flexibility of the foil is ensured around a rotational axis fixed on the bow of the ship or simply on the daggerboard case. Therefore a new degree of Figure 10. Control with contraints and an initial heaving with null initial pitching and velocity u = 10m/s. One can notice that at the beginning the control is bang-bang and finally the adjustement is interior to the upper bounds of the control. Even if the computations are performed with T = 10 sec.; one can see that the exact controllability is quite performed at t = 5 sec. with the values chosen which are realistic. freedom representing the rotation around this axis is introduced. Furthermore, two additional mechanical devices are introduced: one is a damper and the other one is a spring. The damping coefficient is ξ and the stiffness c f . Due to this new degree of freedom (now we have three degrees of freedom (z, γ, α) and the control δ), the equations of the model are a little bit changed. We introduce three new coefficients: one is the mass of the foil -say m f -the second one is the inertia of the foil (alone) with respect to the point o -say J f and the last one is the distance between the point o and the center of mass of the foil (alone) -say d og -; they are represented on Figure 16.
The equations of the movement are now (we use the same notations as before excepted that J o and M are respectively the inertia and the mass of the ship with respect to the point o without the main foil): (44) Figure 11. Control with contraints and an initial heaving with null initial pitching and velocity u = 15m/s.
The equation (44) can therefore be written as follows (with initial conditions): The strong stability of the system is obtained as far as the imaginary part of the solution to the following eigenvalue problem are positive: In fact, the larger is the least imaginary part of the solution to (46), the best is the stabilization of the system. The evolution of the imaginary part of these solutions are plotted on Figure 17 with respect to the velocity of the ship for various values of c f and ξ. We check that the added spring c f is not necessary for obtaining a stabilization (even without control). Furthermore the results are not really dependent on these values. In the report published by the American team for America's cup, they mention that this spring was not used. 5.2. The system used by Oracle Team USA. Let us consider the simple equation, just for explaining the strategy used, but clearly it has to be adjusted at purpose during practice: This is a very classical control strategy which is often used for stabilizing aircraft. It is not an exact control and the result depends on the value of the maximum amplitude α max . But this can be adjusted easily by a computer on an aircraft and in the case of the America's cup boat, it requires a simple education of the helmsman. One can see on Figure 18 how works the so-called input button which drive α max by step of ±.5 0 for each press on the control button. But if this strategy is not used, the results are those plotted on Figure 19. The rest is purely automatic and based on simple mechanical devices. The sign function is detected by the rocker switch (see Figure 26). The solution has been plotted on Figure 18 for six values of α max . One can see that this works perfectly. The time required for the stabilization depends on both the initial data and the maximum magnitude α max of the control.
One could imagine that instead of the control −α max sign(α) one set −α max sign(α). The results are on Figure 18 and show that this doesn't work at all as it is well know in any course in control theory. Figure 18. Several trajectories for different values of α max starting from the same initial condition. One can observe that α max should be adjusted in order to obain the right control. This is why the helmsman has a control box which enables him to adjust α max by steps of ±.5 0 5.3. Comparison between the exact control, OTUSA control and exact control with upper bound. For the same boat and foil-rudder set, we compare the obtained results. The results are given on the Figures 4-9 for exact controls, 10-15 for controls with constraints and 20-25 for OTUSA control. The time of control that we choose in the two first cases is 10 seconds which seems large enough.
The initial data in the case of initial heaving, are: z 0 = 0.1 (10 cm), z 1 = γ 0 = γ 1 = 0. For the control of the initial pitching, they are: z 0 = z 1 = γ 1 = 0 et γ 0 = −2 degrees or γ 0 = −0.035 radian. The desired final data are null. Figure 19. Several trajectories for different values of α max starting from the same initial condition. One can observe that this control is unuseful. This is why the rocker switcher has to detect the sign of the velocity of the rake and not the one of the angle (the pitching angle) We can conclude that exact controllability is not realistic in 10' even for small velocities. Indeed, the angle must be larger than 0.4 rad for heaving control and 0.2 for pitching control. These values are not acceptable.
The control with constraint on the magnitude of the foil angle (≤ 3 degrees) is efficient but some values of the heaving seems dangerous for large velocity (larger than 0.25 cm at velocities larger than 20m/s). This appears in both cases of initial pitching or heaving control.
One can see that the OTUSA system is very efficient at the beginning but the exact control is more precise for small perturbations. The ideal solution is to couple OTUSA control law at the beginning and the exact control when the magnitude of the oscillation are small enough. In this case the mechanical manufacturing of the system can be done using a steering box with an additional button compared to the one of OTUSA. But this is another study not included in this analysis. The use of a system with a mass, a string and a dashpot can lead to a suitable approximation of the exact feed-back. Let us also point out that the matrix leading to the exact control with respect to the perturbation is only dependent on the ship. It is in this study a 4 × 4 symmetrical matrix which therefore depends on only 10 coefficients. A least square method can be applied in order to give a fine approximation of this impedance using a proportional-integral-derivative composed of several mechanical blocks. 6. Conclusion. In this paper, we have suggested a formulation for a simple model in order to check the controllability of flying boats in a reduced configuration. In fact, only two degrees of freedom have been considered. The control law is derived from optimal control for a vanishing cost parameter (ε → 0). The controllability is assumed using the theory of R. Bellman [1]. But a careful study has been necessary because the control system is a part of the stiffness and of the damping matrices. It has been explained and justified that both the rake angle and its time derivative are involved in the control process.
The optimal control laws found in subsections 3.1 are theoretical and can only be exactly applied as far as an electronic device coupled with a mechanical actuator, is used.
The comparison with the control system used by OTUSA shows the theoretical advantages of this exact control. But from a practical point of view, it is not obvious that these advantages would be significant. Because of the regulations in the America's cup, only mechanical devices can be used and the manufacturing of a mechanical system reproducing the exact control law is not discussed in this paper. Therefore the strategy used by the American Team which is fully compatible with the rules, even if it is not optimal, is an interesting alternative. Nevertheless, much better results would be obtained by using slightly different manufacturing of the foil and the rudder. We do not explicit these improvements in this paper which are still to be discussed and checked. They will be discussed in a futur work taking into account a new design of ships as those scheduled for the America's cup 2021. Figure 26. The Oracle USA-Team used this mechanical control to win the last eight races in a row. In particular, but not only, it enables them to navigate in the wake of the challenger mainly at the tacking.