domain: spi phys math info

text:
Etienne Bertin

Elliot Brendel

Bruno Hérissé

Julien Alexandre Dit Sandretto

Alexandre Chapoutot

Prospects on solving an optimal control problem with bounded uncertainties on parameters using interval arithmetic

Keywords: Optimal Control, Pontryagin's Principle, Interval Arithmetic, Bounded Uncertainties, Penalization

An interval method based on Pontryagin's Minimum Principle is proposed to enclose the solutions of an optimal control problem with embedded bounded uncertainties. This method is used to compute an enclosure of all optimal trajectories of the problem, as well as open loop and closed loop enclosures meant to validate an optimal guidance algorithm on a concrete system with inaccurate knowledge of the parameters. The differences in geometry of these enclosures are exposed, and showcased on a simple system. These enclosures can guarantee that a given optimal control problem will yield a satisfactory trajectory for any realization of the uncertainties. Contrarily, the probability of failure may not be eliminated and the problem might need to be adjusted.

Introduction

Optimal control of aerospace systems is performed by modeling the considered system by dynamics depending on multiple uncertain parameters (for example, aerodynamic coefficients, mass,...). Usually, optimal control problems are solved for nominal values of these parameters and the robustness of the solution is demonstrated by dispersing the parameters around nominal values with Monte Carlo simulations [START_REF] Brendel | Optimal guidance for Toss Back concepts of Reusable Launch Vehicles[END_REF]. Another approach is to solve a chance constrained optimal control problem in order to guarantee a probability of success [START_REF] Caillau | Solving chance constrained optimal control problems in aerospace via kernel density estimation[END_REF]. These methods do not exclude the eventuality of failure, which can be problematic on critical systems. In

||u(t)|| u x (t) u z (t)
Figure 1: Optimal trajectory, velocity space trajectory and control for Goddard's re-entry problem in Cartesian Geo-centered coordinates 0 . In green the boostback burn, in thin blue the ballistic phase and in red the landing burn. Burn phases are characterized by a saturated control (||u|| = 1) and a dash in velocity space, while ballistic phases are characterized by no control (||u|| = 0). The model, the parameters and the initial and final positions are taken from [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF]. More details on the model and boundary conditions are given in Section 5. addition these problem-solving methods often introduce numerical approximations, which question the validity of the results.

Interval arithmetic has shown their ability to address several control problems [START_REF] Jaulin | Applied Interval Analysis: With Examples in Parameter and State Estimation[END_REF], providing validated solutions while dealing with method uncertainties (numerical approximations) as well as with model uncertainties (unknown but bounded parameters). For instance the reachable set of a dynamic system can be overapproximated. This over-approximation can demonstrate that a system will remain out of a critical region, thus guarantee its safety [START_REF] Althoff | Reachability Analysis and its Application to the Safety Assessment of Autonomous Cars[END_REF]. These methods can also be used to design robust optimal control. For instance, [START_REF] Rauh | Interval methods for optimal control[END_REF][START_REF] Rauh | Interval techniques for design of optimal and robust control strategies[END_REF] propose algorithms to compute a control that stirs a system to a desired state for any realization of a bounded noise while achieving the lowest upper bound on the cost.

Pontryagin's Maximum Principle [START_REF] Trélat | Contrôle optimal : théorie et applications[END_REF] (PMP) provides necessary optimality conditions for the resolution of optimal control problems by transforming an optimal control problem into a root-finding problem. Derived methods have proven their efficiency and their precision compared to direct methods [START_REF] Trélat | Contrôle optimal : théorie et applications[END_REF][START_REF] Trélat | Optimal Control and Applications to Aerospace: Some Results and Challenges[END_REF], but their convergence strongly depends on their initialization and a prior knowledge of the solution structure is needed.

Our goal is to address an aerospace challenge highlighted by [START_REF] Blackmore | Autonomous precision landing of space rockets[END_REF], which is to enclose the trajectory of a reusable launch vehicle subject to external perturbations. The trajectory during the landing burn is computed online so the control is not known beforehand. Computing an enclosure could guarantee that the vehicle lands safely on the landing platform.

To that end, the return version of Goddard's problem [START_REF] Brendel | Optimal guidance for Toss Back concepts of Reusable Launch Vehicles[END_REF] is considered. It consists in performing the landing of the first stage of a rocket while minimizing its fuel consumption. The optimal trajectory of this system for nominal values of the parameters and the associated control are presented on Figure 1. Uncertainties on the parameters will be added into the model in the form of intervals and deterministic enclosures will be computed by combining interval arithmetic and the necessary optimality conditions given by the application of PMP. Although this method cannot be applied to Goddard's problem yet, this paper exposes our reasoning on simplified problems.

Firstly, Section 2 introduces the problem considered. Section 3 presents the interval and optimal control methods that will be later used. Section 4 proposes three enclosures that can be utilized to discuss the reliability of an optimal control problem with uncertainties. Section 5 showcases these enclosures on a simplified problem. Lastly, Section 6 proposes some uses of these enclosures and the remaining work to apply these methods on Goddard's problem.

Notations

Interval variables and interval vectors are always enclosed in brackets: [x]. Any variable that is not enclosed in bracket is a real or vector variable. Interval valued functions are enclosed in brackets:

[f ] : [x] → [f ]([x]).
The bounds of an interval [a] are noted a, a. Its middle point is noted mid ([a]). (.) such as in y(.) denotes a time function: y(.) : t ∈ [0, T ] → y(t) ∈ R n . y(τ ) and y τ are different, the later being a vector unrelated to function y(.). A hat as in ξ denotes a variable that is related to an estimation.

Control problem with uncertainties

In this section, the optimal control problem and the incorporation of uncertain parameters are presented. Then three enclosures meant to discuss the relevance of an optimal control problem will be introduced.

Optimal Control Problem (OCP)

The following non-interval optimal control problem is considered min

u(.)∈U T 0 (y(t), u(t))dt + Ψ(y(T )) s.t.    ẏ(t) = f (y(t), u(t), ξ(t), t), y(0) = y 0 T is fixed.
(1) Interpretation : the system, a launcher for instance, is characterized by the following data:

• A state y(t) e.g. position, velocity and mass of the launcher. The time function y(.) is called a trajectory.

• An initial state y 0 .

• Dynamics ẏ(t) = f (y(t), u(t), ξ(t), t) e.g. the time derivative of position is velocity, the derivative of velocity is the forces divided by mass etc. . . .

• A control input u(.) e.g. the reactor thrust vector.

• Parameters ξ(.) e.g. aerodynamic coefficients, maximum thrust. . . .

• A continuous cost (y(t), u(t)) e.g. fuel consumption.

• A final cost Ψ(y(T )) e.g. the distance between the final state and the target.

System (1) is considered controllable. Hence for any initial state y 0 and specific parameter function ξ(.), an optimal control u(.) can be found. As a consequence, a control can be defined implicitly as the solution of an OCP, as opposed to explicit methods such as time series or gain tables. For instance, the Model Predictive Control (MPC) approach defines the control as the solution of an optimal control problem; the controller solves this optimal control problem at each time step to determine the control to be applied. Some parts of a launch vehicle trajectory are also defined by an OCP: [START_REF] Blackmore | Autonomous precision landing of space rockets[END_REF] states that the control during the landing burn is the optimal solution of a problem that is solved autonomously during the flight.

Bounded uncertainties on the initial state and the parameters will be added into this model. The initial state y 0 and parameter function ξ(.) take their values in interval enclosures: y 0 ∈ [y 0 ] and ξ(t) ∈ [ξ], ∀t, where [ξ] and [y 0 ] are intervals of the form ξ, ξ or boxes (see definition in Section 3.2.1).

Since they are challenging to simulate with validated simulation, optimal control problems with varying time horizons and state dependant transitions are not considered in this paper. However, it is worth noting that time dependent transitions, or switches, are possible (see Section 3.2.3). This assumption excludes Goddard's problem. The bang-bang transition from a zero control to a saturated control seen in Figure 1 is a state dependent transition and cannot be simulated with our method, unless a precise switch time is forced. Section 6.2 discusses the steps to be taken to apply this method to Goddard's problem.

Note also that the presented method requires additional regularity conditions which will be presented in Section 4.

Representation of the uncertainties in the OCP

Parameters are bounded functions of time: ∀t ∈ [0, T ], ξ(t) ∈ [ξ]. It is also assumed that each function ξ(.) is infinitely differentiable. This ensures that dynamics have all the needed mathematical properties to be simulated.

The time dependency establishes a general framework without invalidating the important assumptions of the optimality criterion used (see Section 3.1.2). It covers indeed more scenarios than the approach considering that the parameters are unknown constant values, which is an implicit assumption in [START_REF] Brendel | Optimal guidance for Toss Back concepts of Reusable Launch Vehicles[END_REF] in which parameters are picked randomly at the start and stay constant during the simulations.

The most encompassing framework would consider that parameters are also functions of the state as a whole. However that would arise the need for more assumptions (bounded spatial derivatives. . . ).

Contributions

The motivation is to assess whether a given OCP outputs an adequate trajectory for any realization of the initial state and parameter functions. A trajectory is characterized by two realizations of the parameter function: the actual parameter function ξ(.) and the parameter function estimated by the system ξ(.). Both are taken amidst the same bound [ξ] but they may differ as the actual parameters are often unknown in practice.

Hence, the following three enclosures of the solution of (1) are proposed:

• An anticipative enclosure containing trajectories whose control is computed with the perfect knowledge of the parameter function: ξ(.) = ξ(.).

• An open loop 1 enclosure containing trajectories whose control is computed once at the start with an inaccurate parameter function ξ(.) = ξ(.) and then followed blindly.

• A closed loop 2 enclosure containing trajectories whose control is computed several times online with accurate measurements of the state but an inaccurate parameter function ξ(.) = ξ(.).

Measures of the state are considered accurate, contrarily to parameters that are unknown. The reason is that knowing parameter in advance is fundamentally impossible because it requires seeing into the future, while having accurate measurement is only a matter of hardware. Bounded errors on measurements will be considered in future works.

These three enclosure can be used to validate all solutions of an OCP and assess risks. Contrarily to the approach in [START_REF] Rauh | Interval methods for optimal control[END_REF], which optimizes a single control that works for all realizations of the bounded uncertainties, the proposed method does not output a control directly. Instead, it can guarantee that a given optimal control 1 also known as feed forward 2 also known as feedback problem will produce a satisfactory control for all realizations of the uncertainties on the initial state and parameter function.

To take the example of launch vehicles [START_REF] Blackmore | Autonomous precision landing of space rockets[END_REF], an OCP is solved right before the landing burn to compute a guidance trajectory. This OCP is designed to provide a trajectory to the landing platform with any possible dynamics and from any possible state the launcher may be in right before the landing burn. The proposed approach is to use a conservative enclosure of all possible dynamics and of all possible states before the landing burn to derive enclosures of all trajectories that are solutions of a given OCP with these initials states and dynamics. If every element in the enclosures satisfy the final constraint of finishing on the landing platform, then the OCP is guaranteed to output a satisfactory trajectory in practice and it may be used on an actual launcher. This is only a necessary condition. Since the enclosures are conservative, the fact that the enclosure does not satisfy the constraints does not prove the existence of a realization of the uncertainties that will cause the launcher to violate the constraints. Nevertheless, on such critical systems, the possibility of failure is not tolerated and may justify changing the OCP.

Mathematical preliminaries

In this section, a numerical method for optimal control is presented as well as interval-based validation methods. Those methods will be combined in Section 4.

Numerical resolution of a control problem

As a first step, properties of dynamical systems will be recalled and the flow and resolvent notations will be introduced. Then these notations will be used to formalize the indirect method known as Pontryagin's Minimum Principle.

Flow and resolvent of an ordinary differential equation (ODE)

Consider an ODE in the form:

ẋ(t) = g(x, ξ(t), t) x(t 0 ) = x 0
To simplify notations and since parameters only depend on time, g(x, ξ(t), t) is denoted by g(x, t) whenever doing so does not impede comprehension.

A solution can be approximated by numerical methods, such as Euler's method, or more efficient Runge-Kutta methods [START_REF] Butcher | Coefficients for the study of runge-kutta integration processes[END_REF]. Integrating the dynamics between two boundary times τ and T can be seen as a flow function Φ τ,T . The function Φ τ,T takes an initial state, simulates it from τ to T and returns the final state. If g is twice differentiable, the flow Φ τ,T has a spatial derivative in x τ which is the resolvent R τ,T (x τ ) of the linearized system [START_REF] Coron | American Mathematical Society. Control and Nonlinearity. Mathematical surveys and monographs[END_REF].

To illustrate these notations, consider two boundary times τ and T . Let x(.) be the solution of the ODE:

ẋ(t) = g(x, t) x(τ ) = x τ Φ τ,T (x τ ) is the state of the solution at time T , Φ τ,T (x τ ) = x(T ).
If z(.) is the solution of the same ODE with an initial perturbation δx:

ż(t) = g(z, t) z(τ ) = x τ + δx then its final state will be z(T ) = x(T ) + R τ,T (x τ ) • δx + o(||δx||).
The resolvent of the linearized system enables a first order approximation of the final error caused by an initial perturbation, and as such, it is the spatial first derivative of the flow.

The resolvent can sometimes be computed analytically, notably for linear time invariant systems:

if ẋ = A • x then R τ,T (x τ ) = exp((T -τ )A). If no analytic formula is available, R τ,T (x τ
) can be computed by integrating the following ODE:

         ẋ(t) = g(x) Ṙτ,t (x τ ) = ∂g ∂x (x(t)) • R τ,t (x τ ) x(τ ) = x τ R τ,τ (x τ ) = I n . (2)
In the following sections, the spatial arguments will often be removed to simplify notations. Hence R τ,T means R τ,T (x τ ).

Optimal control

An OCP is considered as in Section 2

min u(.)∈U T τ (y(t), u(t))dt + Ψ(y(T )) s.t.    ẏ(t) = f (y(t), u(t), ξ(t)), ∀t ∈ [τ, T ], y(τ ) = y τ T is fixed.
This control can be computed by multiple methods that mainly fall in two categories:

• direct methods: the control is discretized, thus turning the infinite dimensional problem into a high dimensional Euclidean optimization problem.

• indirect methods: using a characterization of the optimal trajectory and control, the infinite dimensional optimization problem is transformed into a simpler problem.

The second approach is considered in this article.

To characterize the optimal solution, a new variable is added, the co-state p(.) which is analog to the dual multiplier in constrained Euclidean optimization. p(.) is a vector valued time function on the time range [0, T ] and the vectors p(t) are the same dimension as the vectors y(t).

The main result is the following theorem. This is a simplified version that is sufficient for the problem tackled. It does not explicit the domain of each function and does not address abnormal cases. A complete theorem can be found in [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF][START_REF] Trélat | Contrôle optimal : théorie et applications[END_REF].

Pontryagin's Minimum Principle2 (PMP): If (y(.), u(.)) are a normal optimum, then there exists a non trivial co-state p(.) such that (y(.), p(.), u(.)) satisfy the following equations:

H(y, p, u, t) = (y, u) + p.f (y, u, ξ(t))      ṗ(t) = - ∂H ∂y (y(t), p(t), u(t), t) p(T ) = ∂Ψ ∂y (y T ) u(t) ∈ arg min u H(y(t), p(t), u, t)∀t ∈ [τ, T ],
where H is called the pre-Hamiltonian. The optimal control u(.) is defined implicitly as minimizing the pre-Hamiltonian at every time.

In many control problems this implicit definition yields an explicit expression. That is, there is a function µ : y, p, t → µ(y, p, t) such that:

u(t) ∈ arg min u H(y, p, u, t) ⇐⇒ u(t) = µ(y, p, t). (3)
For instance, [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF] shows that in Goddard's problem, the minimization condition implies that the control (i.e. the thrust vector) has the same orientation as a part of the co-state and that the magnitude of the control tends to be either saturated or zero based on a switching function that depends solely on the state and costate (which leads to a bang-off-bang control as seen on Figure 1). Thus, the control in Goddard's problem is determined by the state and the co-state. In the double integrator problem defined in Section 5, the control minimizes the pre-Hamiltonian H(y, p, u) = ||u|| 2 /2 + p v • u, which yields the explicit expression u(t) = µ(y, p, t) = -p v . Noting g y (y, p, t) = f (y, µ(y, p, t), t) and g p (y, p, t) = -∂H/∂y(y, µ(y, p, t), p, t). The equations of the PMP can be combined with those of Problem (1) to form the following two point boundary value problem:

         ẏ(t) = g y (y(t), p(t), t) ṗ(t) = g p (y(t), p(t), t) y(τ ) = y τ , p(τ ) is free y(T ) is free, p(T ) = ∂Ψ ∂y (y T ) (4)
This is a problem of finding the value of p(τ ) and y(T ) such that integrating from the state at time τ leads to the state at time T and vice versa. In many applications, it is enough to solve the Initial Value Problem (IVP), that is finding p(τ ). By splitting the flow of Section 3.1.1, note φ y τ,T (y τ , p τ ) the flow function that returns the final state y(T ) and φ p τ,T (y τ , p τ ) the flow function that returns the final co-state p(T ). For a given y τ , the IVP can be written as follow:

find p τ such that C(y τ , p τ ) = φ p τ,T (y τ , p τ ) - ∂Ψ ∂y φ y τ,T (y τ , p τ ) = 0. ( 5
)
The IVP ( 5) is solved with a shooting method. First a guess of the initial costate is taken. Then the system is shot, i.e. the ODE is integrated until the final time. Then the initial guess is corrected depending on how far the system landed from the target. This is repeated until a satisfactory co-state is found.

A shooting method can be complemented with a continuation method3 when the problem is hard to initialize. See [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF][START_REF] Brendel | Optimal guidance for Toss Back concepts of Reusable Launch Vehicles[END_REF] for the application of this method to a launcher system.

It is worth noting that the more general PMP states that if the dynamics do not depend on time, then the pre-Hamiltonian is stationary. This is not the case under our assumption since the dynamics depend on parameters that depend on time. It follows that the method used in [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF][START_REF] Brendel | Optimal guidance for Toss Back concepts of Reusable Launch Vehicles[END_REF] to compute singular arc of the control, which is based on the stationarity of the pre-Hamiltonian, cannot be used under our assumption. Hence we currently have no method to enclose singular arcs, apart from a very crude over-approximation.

First derivative of the IVP

One way to solve the OCP is to apply a zero finding method to Equation [START_REF] Brendel | Optimal guidance for Toss Back concepts of Reusable Launch Vehicles[END_REF]. It can be useful to compute the derivatives of φ y τ,T (y τ , p τ ) and φ p τ,T (y τ , p τ ). Let the variable x(.) = (y(.), p(.)). This variable is subject to an ODE ẋ = g(x, t) as seen in System (4). It follows that if g is twice differentiable, a resolvent R x τ,T can be computed for this system using Equation (2). The resolvent R x τ,T is split in four square Matrices:

R x τ,T = * R y τ,T * R p τ,T
.

The two square matrices marked by * are related to the final deviation caused by a variation of the initial state and are not useful here. R y τ,T (resp. R p τ,T ) is the final state (resp. co-state) deviation caused by a variation of the initial co-state and is the spatial derivative of φ y τ,T (y τ , p τ ) (resp. φ p τ,T (y τ , p τ )). By splitting the boundary condition R x τ,τ = I d and taking the upper right and bottom right corner, boundary condition R y τ,τ = 0 and R p τ,τ = I d are obtained. Similarly, splitting the dynamic Ṙτ,t (x τ ) = ∂f /∂x(x(t)) • R τ,t (x τ ) yields the dynamics of R y τ,T and R p τ,T . Hence ODE (6) is deduced.

                             ẏ(t) =
y(τ ) = y τ p(τ ) = p τ R y τ,τ = 0 R p τ,τ = I d , (6)
By simulating ODE [START_REF] Butcher | Coefficients for the study of runge-kutta integration processes[END_REF], the value C(y τ , p τ ) and first derivative ∂C/∂p(y τ , p τ ) of the IVP are obtained.

Validation methods

First, the general idea of interval arithmetic is recalled. Then two methods based on this arithmetic are presented: Krawczyk method that encloses the solution of a vector-valued equation and validated simulation that encloses the evolution of a dynamical system.

Set representation using interval arithmetic

An interval [a] is a convex subset of R that contains all reals between a lower bound a and an upper bound a. With a, a ∈ R ∪ {-∞, +∞}. Interval arithmetic can be used as an alternative to floating-point arithmetic to obtain an enclosure of the solution of a problem, rather than an approximation [START_REF] Moore | Interval analysis[END_REF].

For This definition has an inclusion rather than an equality because the set on the left might not be an interval. Moreover, finding the minimal enclosure of the actual set is difficult. The tightness of the results depends on the effort put into their computations. For this reason, there may be multiple inclusion functions for a single real valued function.

Interval vectors, or boxes, are an axis-aligned rectangular set in a finite dimensional space. They are an inexpensive representation of a high dimensional set (compared to polytops) but may induce a wrapping effect during computations [START_REF] Jaulin | Applied Interval Analysis: With Examples in Parameter and State Estimation[END_REF].

In lower dimension, an accurate enclosure of a set can be achieved with a paving of boxes, a set of mutually disjoint boxes which together cover the whole set. A simple paving is illustrated on Figure 2. A paving in which boxes are all the same shape and aligned can be called a tiling. Such a representation is potentially very precise, but the computational cost grows very fast in high dimension. Indeed, to double the precision, the number of tiles has to be doubled in each direction, which means 2 n as many tiles, where n is the dimension of the vector space. Hence the complexity of this representation is exponential in the dimension of the state, which makes it ill-suited for many practical cases.

Krawczyk contractor

Krawczyk's method is an interval based quasi-Newton algorithm. The formulation used is inspired by [START_REF] Jaulin | Applied Interval Analysis: With Examples in Parameter and State Estimation[END_REF]. This method encloses the solutions of an equation C(a) = 0. It uses a contractor [K], an operator which takes an input box and outputs a smaller box when possible. This contractor is built using [∂C/∂a] ([a]), an enclosure of the first derivative of C on [a] in the form of an interval matrix.

Let mid ([a]) the middle point of [a],

[K] is defined as follows:

[K]([a]) = mid ([a])-M •C(mid ([a]))+ I d -M • ∂C ∂a ([a]) •([a]-mid ([a])), ( 7
)
where M is an invertible real matrix, typically

M = mid ([∂C/∂a] ([a])) -1 . If a ∈ [a] is such that C(a) = 0, then a ∈ [K]([a]). If [K]([a]
) is contained in the interior of [a] then there is a unique solution in [a] or none.

To sum up, Krawczyk contractor is the function:

[a] → [a]∩[K]([a]), Krawczyk's method is made of the following steps: i) Initialize [a] with the search area. ii) Repeat [a] ← [a] ∩ [K]([a]) until convergence. Its output is a box containing all solution of F (a) = 0 in [a].

Validated simulation

A set membership ODE is considered:

ẋ(t) ∈ [f ](x, t) x(t 0 ) ∈ [x 0 ].
Validated simulation encloses every solution of this system in a sequence of boxes (it encloses the solution for any realization of the initial state x 0 ∈ [x 0 ] with any realization of the dynamics f (x) ∈ [f (x)]).

To that end, the time range [0, T ] is discretized in (t i ) i∈0..N , t 0 = 0, t N = T . Starting with an enclosure [y i ] of the systems at time t i , an enclosure of the system on the whole time range [t i , t i+1 ] (called a Picard box) is built. Then an enclosure of y(t i+1 ) is computed. In [START_REF] Dit Sandretto | Validated explicit and implicit Runge-Kutta methods[END_REF], interval Runge-Kutta method are used coupled with inflating terms that enclose the truncation error of the method. Indeed, if the dynamics are sufficiently differentiable, the truncation error can be bounded by evaluating the Lagrange remainder of the difference between the Taylor series of the actual solution and the Taylor series of the Runge-Kutta approximation. For instance, if dynamics are four time differentiable, the truncation error of Runge-Kutta 4 may be enclosed.

The output of validated simulation resembles Figure 10, with the plain boxes being state boxes and the dashed boxes being Picard boxes.

Validated simulation is akin to simulating multiple systems between two common time stamps. A time switch is a time horizon shared by all systems, hence all systems can be simulated by doing a first simulation up until the switch and another simulation starting right after the switch. Contrarily, a variable time horizons or a state dependent transitions differs from one system to another. As a consequence, there is no shared time stamp to use as a duration for the simulation, which makes validated simulation challenging.

Computation of three informative enclosures

In this section, the main contribution is presented. An interval valued OCP solver is proposed and is used to build enclosures of the trajectory of a concrete system using the OCP as a controller.

Interval OCP solver based on the PMP

As per our assumption, there are multiple possible initial states and multiple possible dynamics which means infinitely many possible solutions. Hence the following set membership method to solve the OCP is proposed.

Assumption The presented method requires the dynamics g of the IVP (4) to be at least twice differentiable and to be k + 1 times differentiable if a validated simulation method of order k is used. Indeed, Section 3.1.3 states that if g is twice differentiable, then a first derivative of the IVP (5) may be computed with ODE [START_REF] Butcher | Coefficients for the study of runge-kutta integration processes[END_REF]. As ODE (6) involves ∂g/∂x, if g is k + 1 times differentiable, then ODE (6) is k times differentiable and a method of order k may be used. This assumptions hold if f , µ and are infinitely differentiable, as will be the case in Section 5.

Method Based on the statements of Section 3.1.2, at a given time τ any couple (y τ , p τ ) that satisfies the optimality condition C(y τ , p τ ) = 0 is considered optimal. They will be referred as optimal (state, co-state) couples.

For a given enclosure of the state [y τ ], a Krawczyk operator of the function p τ → C(y τ , p τ ) is built using Formula (7):

K([p τ ]) = mid ([p τ ]) + M • [C]([y τ ], mid ([p τ ])) + I d -M • ∂C ∂p ([y τ ], [p τ ]) • ([p τ ] -mid ([p τ ])), (8)
where:

• M = mid ∂C ∂p ([y τ ], [p τ ]) -1
,

• [C]([y τ ], [p τ ] m ) = [φ] p τ,T ([y τ ], [p τ ] m ) - ∂Ψ ∂y [φ] y τ,T ([y τ ], [p τ ] m ) , • ∂C ∂p ([y τ ], [p τ ]) = R p τ,T ([y τ ], [p τ ]) - ∂ 2 Ψ ∂ 2 y ([y τ ], [p τ ]) • R y τ,T ([y τ ], [p τ ]).
The enclosure R y τ,T ([y τ ], [p τ ]) and R p τ,T ([y τ ], [p τ ]) of the resolvents can be computed with an analytic formula if it exists, or by integrating ODE [START_REF] Jaulin | Applied Interval Analysis: With Examples in Parameter and State Estimation[END_REF], which is an enclosure of ODE [START_REF] Butcher | Coefficients for the study of runge-kutta integration processes[END_REF].

                               ẏ(t) ∈ [g y ](y(t), p(t), [ξ]) ṗ(t) ∈ [g p ](y(t), p(t), [ξ]) Ṙy τ,t ∈ ∂g y ∂y (y, p, [ξ]) • R y τ,T + ∂g y ∂p (y, p, [ξ]) • R p τ,T Ṙp τ,t ∈ ∂g p ∂y (y, p, [ξ]) • R y τ,T + ∂g p ∂p (y, p, [ξ]) • R p τ,T y(τ ) ∈ [y τ ] p(τ ) ∈ [p τ ] R y τ,τ = 0 R p τ,τ = I d , (9)
The enclosure [φ] t0,t1 ([y 0 ], mid ([p 0 ])) of the flow can be computed by integrating:

       ẏ(t) ∈ [g y ](y(t), p(t), [ξ]) ṗ(t) ∈ [g p ](y(t), p(t), [ξ]) y(τ ) ∈ [y τ ] p(τ ) = mid ([p τ ]) . ( 10
)
Our OCP solver is an implementation of Krawczyk's method. Algorithm 1 computes the Krawczyk operator, either with analytic formulae or by integrating System [START_REF] Jaulin | Applied Interval Analysis: With Examples in Parameter and State Estimation[END_REF] and System [START_REF] Moore | Interval analysis[END_REF]. Algorithm 2 is a Krawczyk's method that outputs ∅ if there is no solution to C((y τ , p τ ) = 0. The convergence criterion is Hausdorff distance, which is a distance between sets particularly adapted to boxes [START_REF] Jaulin | Applied Interval Analysis: With Examples in Parameter and State Estimation[END_REF].

τ ] ([p τ ]) k ← ([p τ ]) 0 Compute [K](([p τ ]) k ) using algorithm 1 while D Hausdorff (([p τ ]) k , ([p τ ]) k ∩ [K](([p τ ]) k )) > precision do if 0 / ∈ C(([y τ ], [p τ ])) then Return ∅ else ([p τ ]) k ← [K](([p τ ]) k ) ∩ ([p τ ]) k Compute [K]([p]) using algorithm 1 end if end while
Algorithm 2 needs to be initialized with a box ([p τ ]) 0 . This initial box depends on the type of enclosure built. As explained in Section 4.3, the anticipative, open-loop and closed-loop enclosures are computed using Algorithm 2 with different initialization.

Three enclosures to analyze the problem

For each realization of the initial condition parameters, there are an optimal control and a trajectory that are solution of the OCP. As a consequence, an OCP with interval uncertainties defines infinitely many controls and trajectories.

Similarly to many algorithms in the interval arithmetic literature that search a thin box containing all possible solutions of a problem, the anticipative enclosure contains all trajectories that are solution of an OCP. ξ(.) corresponds to both the actual realization of the parameters and the estimation made by the system. It is an ideal case in which the actual parameters are known with perfect accuracy. As such, this enclosure gives little information on the trajectory of a concrete system. In practice, parameter uncertainties will cause the system to deviate from its optimal trajectory and exit this enclosure.

The following two enclosures are proposed so as to inform on the behavior of the system in practical cases. They are meant to enclose the trajectory of a concrete system that computes its control by solving the OCP with an inaccurate model.

Open loop enclosure : It encloses a system which follows blindly an initial solution that was computed with inaccurate parameters. For any couple of parameters functions ξ(.), ξ(.) : [0, T ] → [ξ], and any initial state y 0 ∈ [y 0 ], the open loop enclosure contains the trajectory:

ẏ(t) = f (y(t), û(t), ξ(t)) y(0) = y 0 ,
with û(.) the solution of the OCP:

min û(.)∈U T 0 (ŷ(t), û(t))dt + Ψ(ŷ(T )) s.t. ẏ(t) = f (ŷ(t), û(t), ξ(.)(t)) ŷ(0) = y 0 .
ξ(.) corresponds to the actual realization of the parameters, which is unknown. Hence the system has an inaccurate estimation ξ(.). This is close to worst case analysis, as the worst case being the application of a control tailored for an extreme scenario to the opposite extreme scenario. This enclosure emphasizes the worst under or over-shooting possible.

It is possible to make a less pessimistic enclosure by taking into account the fact that the system can correct its trajectory using sensor data.

Closed loop enclosure : It encloses a system that uses a perfect measure of its state to recompute the solution of the OCP online (but with inaccurate parameters). Consider a list of recomputation time (τ k ) k∈0..K , 0 = τ 0 < τ 1 .. < τ K = T . For any couple of parameters functions ξ(.), ξ(.) : [0, T ] → [ξ], and any initial states y 0 ∈ [y 0 ], the closed loop enclosure contains the trajectory of the piecewise-defined system:

ẏ(t) = f (y(t), ûk (t), ξ(t)), t), ∀t ∈ [τ k , τ k+1 ] y(0) = y 0 ,
with ûk (.) the solution of the OCP:

min û(.)∈U T τ k (ŷ(t), û(t))dt + Ψ(ŷ(T )) such that ẏ(t) = f (ŷ(t), û(t), ξ(t)) ŷ(τ k ) = y(τ k ).
The overall control is made of pieces ûk (.) that are computed with an accurate measurement of the state y(τ k ). As in the open-loop enclosure, ξ(.) corresponds to the actual realization of the parameters, which is unknown, and ξ(.) is an inaccurate estimation. This encloses the actual operation of a system with an optimal control regulator. The system does not have access to the value of the parameters, but it compensates using measures of its state.

In this paper, a finite set of recomputation time (τ k ) k∈0..K is considered. Under this assumption the closed loop enclosure is a sequence of open loop enclosures. Future works will investigate scenarios in which the control is recomputed at every time, which causes the closed loop to have a more unique geometry.

The computation of these enclosures is presented in Section 4.3.

The underlying geometry of these enclosures

A simpler system is considered to emphasize the geometry of the enclosures presented in Section 4.2. The following figures are depiction of the OCP:

min u(.)∈U T τ u 2 2 dt + K (y(T ) -y T ) 2 2 s.t.    ẏ(t) = ξ(t)u, y(τ ) = y τ , T is fixed.
with ∀t, ξ(t) ∈ [ξ], an uncertain parameter. As the state is one dimensional, state and co-state can be drawn on a single graph to showcase the important sets.

y τ y τ y T state space co-state space Open loop enclosure. As seen in Section 3.1.2, in the real case, the optimal control and trajectory are found by first solving the IVP (5) then integrating System (4).

To build the open-loop enclosure, an enclosure of the solution of the IVP is computed, then the evolution of System (4) is enclosed. If the initial state is such that y τ ∈ [y τ ], then the (state, co-state) couples can be anywhere in [y τ ] × R n , the orange vertical strip on Figure 3. The (state, co-state) couples that are solution of the IVP (5) are enclosed in the blue cone. The solutions of the OCP lie in the intersection (in red) of these two sets. If Algorithm 2 is initialized with a box ([p τ ]) 0 that is large enough to contain this intersection, then it computes a box enclosure of this intersection.

Once [p τ ] has been computed, the evolution of System (4) is enclosed using validated simulation during a time range dt. This integration of System ( 4 Closed loop enclosure. If the system recomputes its control at τ + dt using accurate measurements from the sensors, the OCP is solved again with an initial state in [y τ +dt ]. This is illustrated in Figure 5, which is similar to Figure 4. An important point is that the red set of recomputed (state, co-state) couples is not contained in the purple set obtained by integrating System (4) from a prior step. This is due to the fact that the prior co-states were computed with the wrong parameters. Replacing them is a correction. Hence, these trajectories do not respect the equations of the PMP: applying a control that is optimal for a faulty model results in a non optimal trajectory. Anticipative enclosure. To characterize the actual optimal trajectories, the following criterion is applied. If a trajectory is optimal from A to B and C is an intermediate state of that trajectory, then the trajectory is optimal from A to C and from C to B. If A is the state at time τ , B is the state at time T and C is the state at time τ + dt, then its (state, co-state) couple at time τ + dt satisfies two properties.

• Optimality condition between A and C: the couple lies in the purple polyhedron on Figure 6, which is the set obtained by integrating System (4) from step τ .

• Optimality condition between C and B: the couple is in the blue cone on Figure 6, which is the set of solutions of the IVP (4) with the initial time τ + dt.

By taking the intersection of these two sets, the set of optimal (state, co-state) couples is refined. This refinement can be done by initializing Algorithm 2 with ([p τ +dt ]) 0 the projection of the purple polyhedron on co-state space.

Experimentations

The computation of the three enclosures is showcased on a double integrator with quadratic cost.

A naive implementation

y τ y τ y T state space co-state space Since Krawczyk's Algorithm 2 operates with boxes, the state representation has to be a box or a set of boxes. Representing the set of possible states by a box is computationally efficient but causes crippling wrapping effects. Instead, we propose a naive implementation that encloses the state at time t in a tiled bounding box, such as in Figure 2.

To compute an enclosure of state at time t+dt, for each tile that might contain a possible state, the OCP is solved (see Figure 7) and then the system is integrated. A first round of OCP resolution and validated integration is done to compute a bounding box of the state at time t + dt, then this bounding box is tiled uniformly and a second round finds which tiles might contain a solution.

Each tile requires several calls to Algorithms 1 and 2 , and since the complexity of the tiling is exponential in the dimension, this naive method is not adapted to complicated systems.

Numerical results on a double integrator

Consider a double integrator in a uniform gravity field and with a reactor thrust. It is subject to a quadratic continuous cost and a quadratic penalization on the final state. min

T 0 ||u|| 2 2 dt + K v ||v(T )|| 2 2 +K r ||r(T ) -r T || 2 2 such that    ṙ(t) = v v(t) = -C m u -Ge 2 r(0) = r 0 , v(0) = v 0 , ( 11
)
where G is the normalized gravity field, e 2 is the unit vector of the vertical axis, C is the maximum thrust and m the mass of the system, which is assumed to be constant.

By using the method in Section 3.1.2, this OCP turns into the following two point boundary value problem:

       ṙ(t) = v v(t) = -p v C m 2 -Ge 2 ṗr (t) = 0 ṗv (t) = -p r r(0) = r 0 , v(0) = v 0 p r (T ) -K r r(T ) = 0, p v (T ) -K v v(T ) = 0. (12)
The values used are inspired by the take-off problem in [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF]. The parameter nominal values are the same as in [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF]. The initial position of the system corresponds to the final position of the take-off mission. The initial velocity has been chosen to be coherent with a re-entry mission.

The solution of the nominal case is presented on Figure 8.

The initial position has an uncertainty of around 5 km, which is about 2% of the span of the trajectory. The initial speed has a relative uncertainty of 1.7% and the parameter ratio C/m has a relative uncertainty of 0.2%. However, the mission duration is greater than the optimal duration so as to emphasize the differences between the enclosures. The longer the mission, the more the system has to fight gravity, the more uncertainties are accumulated along the vertical axis.

Figures 9, 10 and 11 depict enclosures of the double integrator computed using Algorithm 2 coupled with a tiling of the state enclosure (as shown on Figure 7). The initialization of Algorithm 2 depends on the type of enclosure, as explained in Section 4.3.

Since System ( 12) is linear time invariant with a strictly triangular matrix, there are analytic formulae for the flow and the resolvents. We use these analytic formulae in Algorithm 1. We have also simulated Systems ( 9) and [START_REF] Moore | Interval analysis[END_REF] with DynIbex [START_REF] Dit Sandretto | Validated explicit and implicit Runge-Kutta methods[END_REF]: a C++ library for validated simulation. The precision of the results of the simulations matches those of the analytic formulae. However, these simulations drastically increase computation time, which in turn significantly restrict the precision of the tiling. As a consequence, Figures 9, 10 and 11 were done with analytic formulae rather than validated simulation. The anticipative enclosure on Figure 9 starts wide because of initial uncertainty but gets thinner as all the system converge to the target. As a result, the final box is very thin.

r 0 r T z coordinate r 0 r T x coordinate v 0 v T z velocity v 0 v T x velocity 0 T time 0.0 0.5 control (thrust vector) ||u(t)|| u x (t) u z (t)
[r 0 ] r T [r 0 ] r T [v 0 ] v T [v 0 ] v T
Contrarily, the open-loop enclosure on Figure 10 becomes wider over time and the final box is very big. Systems over-shoot or under-shoot the target because of their lack of correction.

Lastly, the closed loop enclosure on Figure 11 is somewhere in between the two other enclosures. These systems deviate from their optimal trajectory but correct it, leading to a bulge on the middle and end of the trajectory that gets abruptly smaller at the end.

[r 0 ] r T [r 0 ] r T [v 0 ] v T [v 0 ] v T

Discussion

In this section, we propose some uses of the enclosures, then list what needs to be done to address more complex problems.

Risk assessment and indicating the relevance of an OCP

These enclosures are similar to those developed in [START_REF] Althoff | Reachability Analysis and its Application to the Safety Assessment of Autonomous Cars[END_REF]. As such, they can have the same application. Consider the example of final constraints: the final state must be in a safe zone. The final position must be in the recuperation platform and the final velocity must be sufficiently small for the system to land. A similar reasoning can be held for a state constraint, or other requirements of a mission.

If the final state box of the closed loop enclosure (Figure 11) is contained in the safe zone, then a system using this OCP in closed loop is guaranteed to satisfy the final constraint. Similarly, it can be guaranteed that applying the solution of the OCP in open loop will satisfy the mission (although this is clearly not the case on Figure 10). Both are strong assessment of robustness. The cost can be enclosed as well, which gives a worst performance index.

When using a floating-point control, the anticipative enclosure will provide an upper bound to the distance to the actual optimal control. As a consequence, this provide a criterion to discuss the validity of a floating-point control. To minimize this criterion, the middle point of the control box of the anticipative enclosure can be used as a control.

These enclosures could be used to design the model or the concrete system. For instance, if the final enclosure of the velocity of the anticipative enclosure (Figure 9) is not entirely contained in the safe zone, that means that there might be a realization of the initial states and the parameters such that the system will crash by following the optimal trajectory defined by the OCP.

The existence of such a realization is not guaranteed as those enclosures are conservative, nevertheless, for critical systems, the mere possibility of the existence of a failure may require the OCP to be changed.

In Problem [START_REF] Rauh | Interval methods for optimal control[END_REF], this could mean increasing the penalization coefficient K r and K v , thus making the launch more costly but decreasing the uncertainty on the final state and maybe void the possibility of failure.

Contrarily, if the open-loop or closed-loop enclosures final boxes respect the final constraint by a significant margin, then the penalization could be decreased to achieve better performances on other criteria. Alternatively, the uncertainty could be altered. One could assess the maximum acceptable magnitude of the uncertainties and use it as specification for the creation of an engine or sensors. Lastly, if the open loop enclosure respects all the constraints, then it might not be necessary to elaborate a closed loop regulator.

Of course, these statements hold true as long as the model with uncertainties is accurate. If parameter uncertainties have been under-estimated or if the model has neglected a phenomenon, then there is no guarantee that the system will remain in the enclosures.

Generalizing to Goddard's problem

The double integrator with quadratic cost was considered because it is used to initialize a continuation method to solve Goddard's problem [START_REF] Bonnans | Singular arcs in the generalized goddard's problem[END_REF][START_REF] Brendel | Optimal guidance for Toss Back concepts of Reusable Launch Vehicles[END_REF].

In a similar manner to a continuation method, there are two steps to bridge the gap between the double integrator and Goddard's problem. First non linear systems have to be addressed. We need to simulate non linear problems and their resolvent, which is costly. Then state dependent transitions and variable time horizons will need to be modeled and enclosed. This means being able to enclose the evolution of a hybrid system, if possible in a manner that does not induce a significant over-approximation. This also makes the computation of the resolvent more challenging.

In addition to that, Goddard's representation of a spacecraft is in dimension seven. As a consequence, tiling may no longer be used to improve the precision of Algorithm 2. An entirely new method may be needed.

Conclusion

In this paper, a resolution tool was proposed for optimal control problems with embedded uncertainties. This resolution tool is used to compute three enclosures, an anticipative enclosure containing all possible solutions of the optimal control problem, and open-loop and closed loop enclosures that bounds the trajectory of a concrete system that uses this problem as a controller. These enclosures can show the deviation caused by the uncertainties, identify the critical zone that could be crossed or more generally assess the relevance of a given optimal control problem.

In later works, we will try to improve our method until it can reliably compute these enclosures for a problem as complex as Goddard's.

  each real valued function f : a → f (a), one can create an inclusion function [f ] : [a] → [f ]([a]) following a set-membership principle: the result of the interval function [f ] is an interval that contains each possible value of f on [a] [f ] ([a]) ⊃ {f (a)|∀a ∈ [a]} .

Figure 2 :

 2 Figure 2: Outer tiling of a set. The ellipsoidal set is enclosed by the union of red boxes.

Algorithm 1 Algorithm 2

 12 Computes [K]([p τ ]) for given [y τ ] and [p τ ] Require: [y τ ] and [p τ ] Compute R y τ,T [y τ ], [p τ ] and R p τ,T ([y τ ], [p τ ]) Compute [φ] τ,T ([y τ ], mid (p τ )) Compute [K]([p τ ]) using Formula (8) Computes a thin enclosure ([p τ ]) k of the optimal co-states corresponding to a given state enclosure [y τ ] Require: An initial searching area ([p τ ]) 0 and [y

  Anticipative enclosure: For any parameter function ξ(.) : [0, T ] → [ξ] and any initial state y 0 ∈ [y 0 ], the anticipative enclosure contains the trajectory: ẏ(t) = f (y(t), u(t), ξ(t)) y(0) = y 0 , with u(.) the solution of the OCP: min u(.)∈U T 0 (y(t), u(t))dt + Ψ(y(T )) s.t. ẏ(t) = f (y(t), u(t), ξ(t)) y(0) = y 0 .

Figure 3 :

 3 Figure 3: Geometric resolution of the OCP at time τ .

Figure 4 :

 4 Figure 4: Simulation of the System (4) from time τ to time τ + dt.

) is shown on Figure 4 .

 4 This yields a set of possible state, co-state) couples at time τ + dt, which in turn gives an enclosure of the possible states [y τ +dt ]. By going on until the final time T , the open-loop enclosure is computed.

Figure 5 :

 5 Figure 5: Closed-loop: recomputation of the optimal co-states at time τ + dt.

Figure 6 :

 6 Figure 6: Anticipative: refining of the optimal couples at time τ + dt.

Figure 7 :

 7 Figure 7: Paving the state enclosure to get a thinner approximation.

Figure 8 :

 8 Figure 8: Optimal trajectory, velocity space trajectory and control for the a double integrator with quadratic cost. Unlike Goddard's problem on Figure 1, the control changes continuously.

Figure 9 :

 9 Figure 9: Anticipative enclosure of position and velocity.

Figure 10 :Figure 11 :

 1011 Figure 10: Open loop enclosure of position and velocity.

Also known as Pontryagin's Maximum Principle. There are multiple formalization of this principle in the literature.

also known as homotopic method.

Acknowledgment

This work was partially supported by the "Chair Complex Systems Engineering -Ecole polytechnique, Thales, DGA, FX, Dassault Aviation, Naval Group Research, ENSTA Paris, Télécom Paris, and Fondation ParisTech"


Jean-Louis Kraus
email: jean-louis.kraus@univ-amu.fr

Artificial Intelligence applied to the production of high added-value Dinoflagellates Toxins

Keywords: Artificial Intelligence, Biotoxins, Dinoflagellates, Drug Discovery, Neural networks

Trade in high value-added toxins for therapeutic and biological use is expanding.

These toxins are generally derived from microalgae belonging to the dinoflagellate family.

Due to the difficulties to grow these sensitive planktonic species and to the complexity of methods used to synthesize these molecules, which are generally complex chemical structures, biotoxin manufacturers called on artificial intelligence technologies. Through the development of specific learning neural networks applied to each phases of biotoxin production: photo-bioreactors operating at optimal yield; -new chemical synthesis research processes;

-toxin biosynthetic research pathways offering short-cut possibilities, manufacturing processes have been greatly improved.

Introduction

Hundreds of thousands of microalgae species are distributed on the surface of the globe in marine, freshwater or brackish.The long-lasting interest in bioactive molecules (toxins) produced by microalga has risen in recent years [START_REF] Sun | Microalgae as a source of lutein: chemistry, biosynthesis and carotenogenesis[END_REF][START_REF] Khan | The promising future of microalgae: current status, challenges, and optimization of a sustainable and renewable industry for biofuels, feed, and other products[END_REF] ). Microalgae Market is valued at USD 54.64 Million in 2018 and expected to reach USD 76.37 million by 2025 with a compound annual growth rate of 4.9% over the forecast period. (Brandessence Market Research 2019). Microalgae could be seen as ''Cell mills producing recombinant commercial molecules'', which have encouraged the creation of numerous start-ups, specialized in the production of microalgae species (Green algae, Dinoflagellates, Red algae, Euglenoids) mainly in the western countries (Cadoret 2008). The reason for the growing interest of microalgae species is their ability to synthesize a wide variety of complex molecules recognized for their biological features with great potential for use as pharmaceuticals therapies and/or biological probes. Among myriads of microalgae species, the family of dinoflagellates is of particular interest since they exhibit wide varieties of natural compounds. The main known toxins extracted from dinoflagellate microalgae species with strong biological credit profiles are given in -Due to the complex chemical structure of biotoxins of interest, their direct chemical synthesis through multi-steps synthesis requires design of efficient retrosynthetic schemes with focus on the most performing. Besides the possibility of their direct chemical synthesis, hemisynthetic routes can be also envisaged. Numerous examples of hemisynthesis of natural complex compounds are described in literature (Shabir 2015, Ten Lohuis 1995). To process biotoxin production through hemisynthetic technology, the most suitable metabolite involved in the metabolic pathway should be first selected, before to be included in the retrosynthetic scheme envisaged for the targeted biotoxin synthesis. Such pathway selection implies the total knowledge of the metabolic pathways involved in the metabolism of this biotoxin, before finalized a possible retrosynthetic scheme.

-To circumvent quantities limitations of the targeted toxin, and in order to enhance the yields, genetic and metabolic engineering [START_REF] Segler | Planning chemical syntheses with deep neural networks and symbolic AI[END_REF]) could be ways to overcome those constraints. For this purpose, the knowledge of the different metabolism pathways specific to each dinoflagellate species, is required. Assuming that such pathways are not currently available, data mining have to be first undertaken before handling genomic experiments.

After having analyzed difficulties relative to the production of these high-added value biotoxin production from dinoflagellate microalgae, one can ask the question:

'' How Artificial Intelligence (AI) helps biotoxins manufacturers to overcome the whole problems raised in the previous paragraphs?''

AI is a valuable tool that can contribute to the achievement of each important phases (chemical synthetic processes, photo bio -reactor engineering, biological pathway identification) involved in the biotoxin cycle production.

1-AI in dinoflagellate culture production

A deep-learning neural network that considered the whole physical parameters (temperature, pH, irradiance, turbulence airflow, bioreactor capacity, light energy sources), as well as biological parameters (composition media, nutritional requirements) involved in the production of dinoflagellates, helps to define the optimized conditions which allow to get the best dinoflagellate biomass yield for a given dinoflagellate specie; optimized conditions for the preparation of a targeted biotoxin. This algorithm which predicts the best conditions to produce a given dinoflagellate specie, is part of a learning machine, adapted to a photo-bioreactors devoted to the automatic dinoflagellate production.

Optimization of production can be achieved by using a wide range of AI techniques from classical "one-factor-at-a-time" to modern statistical and mathematical techniques, viz.

2-AI and biotoxins chemical synthesis and hemisynthesis.

Synthesis of molecules remains one of the most important challenges in organic chemistry.

Standard approaches based on the development of specific retrosynthetic schemes are repetitive and time consuming task. Through the use of machine learning, substantial progresses in organic chemical synthesis have been made over the past decades (Coley 2016 ,Ahneman 2018, Klucznik 2018). For some some dinoflgellates biotoxin molecules, de novo synthesis could be considered as feasible (saxitoxin, tetrodotoxin, akadaic acid) but for other biotoxins such as palytoxin, yessotoxin, brevetoxin, gamberiol, de novo synthesis is considered exceedingly complex and economically unfeasible.

-AI and de novo dinoflagellate biotoxin chemical synthesis

Using finger prints derived from a neural algorithm, Knowing the metabolic pathways involved in the production of natural compounds, which chemical structure is quite complex, it is sometimes possible to isolate a particular intermediate metabolite, which is part of the metabolic pathway of the targeted natural drug. Starting from this metabolite, are carried specific chemical reactions in order to get the desired final drug in less steps and better yields [START_REF] Shabir | Hemisynthesis of a naturally occurring clinically significant antitumor arglabin from ludartin[END_REF]. In this perspective, AI technologies have been used at different levels -saxitoxin biosynthesis pathway determination (Tsuchiya 2016, Hunter 1993 ).

-hemi-biosynthesis of saxitoxin analogues starting from a specific intermediate named Artificial Intelligence has made faster, cheaper and more accurate DNA sequencing, and has allowed gain perspective on the particular genetic blueprint that orchestrates the whole activities of a given organism. In this feature, it has been reported (Kellmann 2008) that saxitoxin biosynthetic pathway is encoded by more than 35 kb and comparative sequence analysis assigns 30 catalytic functions to 6 proteins. Using in silico functional inferences based on saxitoxin open reading frames combined with liquid chromatography-tandem mass spectrometry analysis, the description of the saxitoxin biosynthetic pathway has been revised, and a saxitoxine biosynthetic gene cluster (stx) has been identified and shown to be involved in saxitoxin production. Knowing the right saxitoxin metabolic pathway, it could be possible to construct an algorithm which satisfies not only each constraint but also allows several alternative pathways that by -pass key enzymes. Recently, new methods in artificial intelligence which offer the possibility to integrate chemistry and biology into complex reaction schemes to provide access to complex molecules not achievable with either alone, have been reported [START_REF] Lin | Retrosynthetic design of metabolic pathways to chemicals not found in nature[END_REF]). These authors concluded that in the next future to build a xenobiotic molecule of a scale of a natural product, will require integrated computer-aided design packages that combine retrosynthesis, metabolic flux analysis, protein engineering and genetic circuit design automation. Such AI technologies have been already applied for systems biology with a high research impact (Cevora 2019).

Conclusion

Dinoflagellates is one of the rich biotechnological source of biotoxins, with interesting biological activities not only in pharmacological and medical fields, but also as promising tools for chemical biology. Despite such recognized value, scarcity of such biotoxins remains a major issue, that new marine natural products start-ups have to face economically and in terms of viability. The problem becomes even more complex since these high value biotoxins are mainly found in dinoflagellates species which are extremely fragile microalgae and which culture required high tech bioreactors. To circumvent those problems, available AI technologies based on learning neural networks, could be applied at each phase of biotoxin production: chemical synthesis and hemisynthesis, biotoxin structural identification, bioreactor engineering systems, biological pathways identification through marker-passing algorithm. AI clearly appears as a promising tool to help new start-uppers to jump in the restricted biotoxin market, in proposing not only biotoxins at reasonable prices but also allowing the discovery of new drugs, considering that dinoflagellates marine organisms, are the sources of several thousand drugs of interest, which remain to be discovered. (Kobayashi 2016). Of course to generalize AI technique to other dinoflagellate species to produce other high value toxins , we will need to generate enough data to train a deep neural network.

Nevertheless, our perspectives are to use the knowledge we acquired with the use of artificial intelligence networks for the optimization of the production of new high added value toxins from new species of dinoflagellates, known to be difficult to grow.

Figure 1 :

 1 Figure 1: Learning Neural Network applied to biotoxin Dinoflagellate production

  Figure 2

'

  'shunt product ( C')'' derived from biosynthetic precursors involved in saxotoxin pathway (Cho 2017) -detection and relative toxicity of saxitoxin analogues nonsulfated, monosulfated and disulfated (Lukovski 2019 ) 3-AI and saxitoxin genomic studies.

table 1 :

 1 Table1-Biotoxins of potential therapeutic use produced by dinoflagellates species (for a complete list of dinoflagellates biotoxins see references Cadoret

2018 and Assunçao 2017)

	Amphidinolide	Amphidinium genus	Cytotoxic macrolide
	Most of these dinoflagellate drugs have complex chemical structures, which development
	requires efficient chemical synthesis processes through the use of new designed
	retrosynthetic schemes. The case of palytoxin is a significant example of high market value
	since its price is around 700 US $ 100g. This molecule has 64 stereogenic centers, its
	synthesis requires more than 140 steps (Hoffmann 2012, Newhouse 2009). Start-ups, which
	economic business model is based on the production and sales of dinoflagellate bioactive
	toxins with high added value have to face three major concerns:
	Biotoxin -Dinoflagellate in general grow slowly and are shear-sensitive species. It is the reason why Dinoflagellate species biotoxin producers Therapeutic use
	dinoflagellate are difficult to culture in photo-bioreactors compared to other classes of
	Saxitoxin microalgae. Consequently chemical-physical conditions (temperature, pH, media Alexandrium ( minutum, tamarense, catenella) Treatment of pain
	Gymnodinium (catenatum) Pyrodinium (bahamense) composition, light, stirring conditions) to find economical and viable options regarding
	Karlotoxin bioreactor design and culture strategies have to be optimized. Moreover, taking into Amphidinium genus. Haemolytic activity
	account those drastic dinoflagellate culture conditions, scale-up biotoxin production
	Palytoxin requests	Ostreopsis (siamensis, mascarenensis, the development of high capacity	Wide spectrum of biological bioreactors with a view to
	lenticularis, ovata, fattorussoi) overcoming dinoflagellate biomass weight limitations (Assunçao 2017) effects
	Tetrodotoxin	Alexandrium ( tamarense)	Nerves and muscles action
				potential blocker
	Okadaic acid	Prorocentrum genus (lima, concavum,	Potent tumor promoter
			belizeanum, maculosum )
			Dinophysis genus (acuta, acuminate, fortii)
	Brevetoxin		Karenia brevis (formerly known as	Powerful Neurotoxic
			Gymnodinium breve or Ptychodiscus brevis)
	Yessotoxin		Protoceratium reticulatum,	Biological and
			Lingulodinium polyedra	pharmacological probes
			Gonyaulax spinifera
	Gambierol		Gambierdiscus toxicus	Modulatory action upon
				voltage gated K+ channels

  Chematica algorithm able to propose optimized synthetic pathways for the design of any molecules of interest. For example, a standard de novo synthesis of saxitoxin, a toxin used in the treatment of chronic pain, has been achieved in 10 steps with a final yield < 5% ,[START_REF] Fleming | A Synthesis of (+)-Saxitoxin[END_REF] in 2006. As represented on

	Figure 2, using deep learning neural networks, medicinal organic chemists can use valuable
	retrosynthetic schemes allowing total drug synthesis in better yields, reducing the number
	of steps, and predicting compound stereochemistry, structure identification (NMR, UV, IR),
	some physical-chemistry properties .
	.
	Figure 2: Learning Neural Network predicting probability of reaction types
	applied to biotoxin synthesis
	-AI and saxitoxin hemisynthesis.
	Wei et al. (Wei 2016), have identified
	chemical reaction types involved in compounds chemical synthesis with more than 80%
	accuracy. Since the development of this algorithm, new deep learning neural networks,
	which involved around 12 millions of known single-step reactions, and allowed the

prediction of reactions which can be used in any single step of multi-step synthesis, has been developed (Coley 2016). More recently was developed by Klucznik et al.(Klucznik 2018), an highly effective Chematica Computer Program capable of designing novel efficient synthesis of medicinally relevant drugs, making

Acknowledgements.

Thanks to Planktovie S.A for helpful discussions. Aix Marseille University (France) and A. Le Bivic and L. Kodjabachian, IBDM-CNRS Marseille directors, are greatly acknowledged for facilities offered.


Fran De Aquino

Deceasing of Gravitational Mass of the Magnesium subjected to an Alternating Magnetic Field of Extremely Low Frequency

Keywords: Gravitational Mass, Magnetic Field of Extremely Low Frequency, Magnesium

Here we propose an experiment in order to check the decreasing of Gravitational Mass of the Magnesium subjected to an alternating magnetic field of Extremely Low Frequency. In addition, we show how the Inertial Properties, of a Spacecraft with Mg core, can be strongly reduced. Gravity Multipliers made of Mg are presented, and then it is show its use in several devices.

INTRODUCTION

In this paper it is suggest an experiment in order to check the decreasing of Gravitational Mass of the light metal Magnesium subjected to an alternating magnetic field of Extremely Low Frequency. Also we show how the Inertial Properties, of a Spacecraft with Magnesium core, can be strongly reduced. In addition, Gravity Multipliers of Mg are presented. It is then shown its use in Gravitational Motors, Gravitational Thruster of Fluids, production of Microgravity environments, and a Cooling and Heating Gravitational System.

THEORY

In a previous paper [START_REF] De Aquino | of the Relativistic Theory of Quantum Gravity[END_REF] we shown that there is a correlation between the gravitational mass, , and the rest inertial mass , which is given by g m 0 i m ( ) is the variation in the particle's kinetic momentum;

1 1 1 2 1 1 1 2 1 1 1 2 1 2 2 2 2 0 2 0 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Δ + - = =
is the electromagnetic energy absorbed or emitted by the particle;

is the index of refraction of the particle; W is the density of energy on the particle ; The instantaneous values of the density of electromagnetic energy in an electromagnetic field can be deduced from Maxwell's equations and has the following expression ( )

2 2 2 1 2 2 1 H E W μ ε + = where t E E m ω sin = and t H H ω sin =
are the instantaneous values of the electric field and the magnetic field respectively.

It is known that

H B μ = , r k B E ω = [2] and ( ) ( ) 3 1 1 2 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = = = ωε σ μ ε κ ω r r
( ) ( ) 4 1 1 2 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + + = = ωε σ μ ε r r r v c n Equation (3) shows that v r = κ ω . Thus, v k B E r = = ω , i.e., ( )
5 H v vB E μ = =
Then, Eq. ( 2) can be rewritten as follows ( ) ( ) 6) and (5) into Eq. ( 1) gives ( )

6 2 2 2 1 2 2 2 1 2 2 1 2 2 2 2 1 H H v H H H v W μ μ μ ε μ μ μ ε = + = = + = For ωε σ >> , Eq. (3) gives ( ) 7 2 2 2 2 2 c v c n r ω μσ = = Substitution of Eqs. (
8 1 4 1 2 1 4 2 2 3 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = = H c f m m i g ρ π σ μ χ Note that if t H H m ω sin = .Then,
9 1 4 1 2 1 1 4 1 2 1 4 2 2 4 2 2 4 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = = ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + - = = rms rms i g B c f H c f m m μρ π σ ρ μ π σ μ χ

SUGGESTED EXPERIMENT

Let us now consider an experiment where the light metal Magnesium (Mg), whose characteristics are given by: ; , is subjected to an alternating magnetic field with Extra-low frequency,

. According to Eq. (9), we have that m S / 10 2 . In 2018, physicists from the Institute for Solid State Physics at the University of Tokyo, Japan, have recorded the largest magnetic field ever generated indoors -a whopping T 200 , 1 [START_REF] Nakamura | Record indoor magnetic field of 1200T generated by electromagnetic flux-compression[END_REF]. This means that the experiment suggested in this work will can be carried out in the very near future.

INERTIAL PROPERTIES

Now, we will show how the Inertial Properties of a Spacecraft can be strongly reduced.

Consider the schematic diagram of a spacecraft shown in Fig. 1. At the center of the spacecraft there is a Magnesium Core, subjected to an alternating magnetic field with Extralow frequency, * rms B f . According to Eq. ( 10), the gravitational mass of the Magnesium core, , is expressed by the following equation:

gC m ( ) 14 1 10 1 . 5 1 2 1 0 4 12 C i rms gC m f B m ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - = - In the equation (14),
is the rest inertial mass of the Magnesium Core. Then, the total gravitational mass of the spacecraft, , can be expressed by means of the following expression:

(total gS M ) ( ) ( ) 15 gC gS total gS m M M + =
where is the total gravitational mass of the spacecraft without the gravitational mass of the Magnesium core. Assuming that density of external electromagnetic energy in is negligible, then we can write that

gS M gS M S i gS M M 0 ≅ , where
is the rest inertial mass of the spacecraft (without the Magnesium core). Thus, Eq. ( 15) can be rewritten as follows:

S i M 0 ( ) ( ) 16 1 10 1 . 5 1 2 1 0 4 12 0 0 C i rms S i gC S i total gS m f B M m M M ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - + ≅ = + ≅ - Therefore, for ( ) 1 / 10 1 . 5 4 12 >> × - f B rms
, we get ( ) ( )

17 10 1 . 5 0 4 12 0 C i rms S i total gS m f B M M ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ × - ≅ - For example, if ; ; kg M M S i gS 000 , 10 0 = ≅ kg m C i 000 , 5 0 = Hz f 1 . 0 =
and, T B rms 529 ≅ , then Eq. ( 17) yields

( ) kg M total gS 8 < .
This means that the weight of the spacecraft becomes less than 80N.

Mach's principle predicts that inertial forces acting on a particle are the result from the gravitational interaction between the particle and the other particles of the Universe. Thus, the inertial forces acting on a particle are proportional to gravitational mass, , of the particle, i.e., ii

F g m i g ii a m F = [1].
This fact shows that the inertial effects upon a spacecraft can be strongly reduced because, as we have seen, the gravitational mass of the spacecraft

(total gS M
) can be strongly reduced (

). In practice, it means that will be possible to become quasinull the inertial properties of the spacecraft.

( ) i total gS ii a M F =
Under these circumstances, the spacecraft can describe incredible trajectories, and to make super accelerations and super decelerations in a very short time interval (<1s), without be destructed (See The Gravitational Spacecraft [START_REF] De Aquino | The Gravitational Spacecraft[END_REF]).

GRAVITY MULTIPLIER

In a previous paper [START_REF] De Aquino | Gravity Control by means of Electromagnetic Field through Gas at Ultra-Low Pressure[END_REF] it was shown that, when the gravitational mass, , of a plate (very thin plate) is reduced by the factor is the gravity acceleration before the plate (See Fig. 2). g Fig. 2 -The gravity acceleration after the plate is g g

1 1 χ =
where g is the gravity acceleration before the plate. The perpendicular axis of the plate can be in any direction.

g 1 = χ 1 g g m g1 χ 1 . . ⊥ Axis
Consequently, after a second plate, with gravitational mass, , the gravity becomes:

2 g m g g g 2 1 1 2 2 χ χ χ = =
, where

02 2 2 i g m m = χ
. In a generalized way, we can write that after the nth plate the gravity, , will be given by n g ( )

18 ... 3 2 1 g g n n χ χ χ χ = If 1 ,..., 2 1 - < = = = = χ χ χ χ n , and
is odd then, the gravitational forces, , between a body B before the first plate and another body A after the nth plate are repulsive (See Fig. 3), and given by

n F ( ) ( ) ( ) 19 2 2 r m M G r M G m g m g m F gA gB n gB n gA n gA n gA χ χ χ + = = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - = = =
Fig. 3 -The gravity after a battery of plates

g n = χ n g g m g1 m g2 m gn χ 1 χ 2 χ n m gA M gB F F B A
In this case, the plates have the same dimensions (with the same inertial mass ), and they are made of Magnesium, subjected to an alternating magnetic field, , with Extra-low frequency, . If the gravitational masses of the plates are, , then, according to Eq. (10), we can write that Thus, this system multiply the gravity, g , by times 6 . 97 . We can use the Gravity Multipliers to convert gravitational energy into rotational kinetic energy by means of a Gravitational Motor, which design can be similar to the Internal Combustion Engine. In that Gravitational Motor, the pistons can be designed as shown in Fig. 4.

P i m 0 rms B f gP m ( ) 20 1 10 1 . 5 1 2 1 4 12 0 ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ - × + - = = - f B m

ANOTHER GRAVITATIONAL MOTOR

In Fig. 5, we show a schematic diagram of another type of Gravitational Motor, with different characteristics to the type previously proposed. Now the Gravitational Motor has 4 Gravity Multipliers (GM), which can be made with plates of Magnesium, subjected to an alternating magnetic field, , with Extra-low frequency, .

rms B f

The Gravity Multipliers, GM1, GM2 and the GM3 are placed below the rotor (See Fig. 5); GM1 and GM2 on the right and GM3 on the left.

Above the GM1 the local gravity, g r

, is intensified for g N g g r v r + = = ′
( ) ( ) ( ) 27 2 1 2 1 r g m g m r F F T g g r r r r r r ′ + ′ ′ - = = × ′ + ′ ′ - = ( 0 i g m m ≅
is the mass of the rotor), and the rotor spins with angular velocity ω . Then, the average power, P , of the gravitational motor is given by ( ) Note that this electrical energy is produced without the use of any type of fuel, because the energy, which moves the Gravitational Motor comes from Earth's gravitational field, i.e., the Gravitational Motor converts directly energy from the Earth's gravitational field into rotational mechanical energy.

Thus, the Gravitational Motors are similar to the turbines of the hydroelectric plants. While the turbines convert energy from the Earth's gravitational field into rotational mechanical energy, by means of water of the rivers, this type of Gravitational Motors convert energy from the Earth's gravitational field directly into rotational mechanical energy, by using the GMs.

Finally, note the small volume of the rotor of this type Gravitational Motor, it shows that the total volume of the motor can be smaller than 1m3.

Massive Rotor r r R

1 2 1 - = - = χ χ N g N g g r r r + = = ′ 2 1 χ χ N - = 3 χ g N g g r r r - = = ′ ′ 3 χ g''' =χ 4 g'' = g g'''' =χ 5 g' = g N - = 4 χ N + = 5 χ GM5 GM1 GM2 GM4 GM3 g r g r
Fig. 5 -Schematic diagram (cross-section) of another type of Gravitational Motor.

GRAVITATIONAL THRUSTER OF FLUIDS

The Gravity Multipliers can be used to make particles acquire enormous accelerations. In practice, this can lead to the conception of a Gravitational Thruster of Fluids (See Fig. 6). In this case, the gravity acceleration after the nth plate, , for In Fig. 7 it is shown a schematic diagram of a thruster system -using a Gravitational Thruster of Fluids, for spacecrafts in the Earth's atmosphere. This system can be used to propeller the spacecraft in any direction.

MICROGRAVITY ENVIROMENTS

In a previous paper [START_REF] De Aquino | Activation of Autophagy Process in Cancerous Cells at Quasi-Zero Gravity Environments[END_REF] we described a way to produce microgravity environments at level of the Earth's surface, in order to "activate" the cellular autophagy process. After an infection, autophagy can destroy bacteria and viruses. Based on the theory here explained, it easy to see that, microgravity environments can be also produced using an Mg plate, subjected to an alternating magnetic field with Extra-low frequency, (See Fig. 8). ) of that experienced at Earth's surface (1g). Consequently, a microgravity environment is one where the acceleration induced by gravity has little or no measurable effect.

COOLING AND HEATING GRAVITATIONAL SYSTEM

Consider the system shown in Fig. 9. It has two hollow spheres A and B connected by a tube; inside this system there is a liquid with density ρ . Bellow sphere A there is an Mg plate (in red Fig. 9), subjected to an alternating magnetic field with Extra-low frequency, .

rms B f

The pressure at point a (See Fig. 9) is By increasing the magnitude of the magnetic field , it is possible to make rms B χ negative (See Eq. ( 20)), and also to increase its magnitude χ .

In this case, g ′ will be expressed by g g

Mg plate Magnetic Coil

  and c is the speed of light.

  electrical conductivity in S/m). From Eq. (3), we see that the index of refraction v c n r = is given by

Fig. 1 -

 1 Fig.1 -Schematic diagram of an Ellipsoidal Spacecraft

Fig. 4 -For 4

 44 Fig. 4 -Gravitational Piston.( gP m is the mass of the piston.)

3 χ.

 3 The function of GM4 and GM5 (See Fig.5), is only to revert the gravity down to values very close to g .As the gravity acceleration on the left half of the rotor becomes acceleration on the right half of the rotor becomes on the rotor is

  By substituting (30) into (28) we obtain the expression of the average power of the gravitational motor, i.e., and are respectively the mass density, the h radius and the height of the rotor then for and

Fig. 6 - 6 Fig. 7 -

 667 Fig. 6-Gravitational Thruster of Fluids

  experienced by a body in a microgravity environment, by definition, is one-millionth (

Fig. 8 -

 8 Fig. 8 -Activation of cellular Autophagy process in Human bodies.

.

  Equation above shows that the pressure inside the sphere A can be reduced by reducing χ . The decreasing of the pressure causes the decreasing of the temperature, , in sphere A, In this case, the system shown in Fig9, it can works like a Cooling Gravitational System.

.Fig. 9 -g

 9 Fig.9 -Schematic Diagram of one element of Gravitational System for Cooling and Heating.

* Note that, another possibility is to apply the magnetic field through the entire spacecraft. In this case the coil must be, obviously, positioned in the edges of the spacecraft.


est destinée au dépôt

Experimental Myocardial Infarction Elicits

  Time-Dependent Patterns of Vascular Hypoxia in Peripheral Organs and in the Brain Hélène David, Aurore Ughetto, Philippe Gaudard, Maëlle Plawecki, Nitchawat Paiyabhroma, Emma Zub, Pascal Colson, Sylvain Richard, Nicola Marchi, Pierre Sicard To cite this version: Hélène David, Aurore Ughetto, Philippe Gaudard, Maëlle Plawecki, Nitchawat Paiyabhroma, et al.. Experimental Myocardial Infarction Elicits Time-Dependent Patterns of Vascular Hypoxia in Peripheral Organs and in the Brain. Frontiers in Cardiovascular Medicine, 2021, 7, 10.3389/fcvm.2020.615507. hal-03143078