Pierre Martinon
email: pierre.martinon@inria.fr

Joseph Gergaud
email: joseph.gergaud@enseeiht.frrtn°0354

Shoot-1.1 Package -User Guide

Keywords: Boundary value problem, discontinuous right hand side, optimal control, shooting method, singular arcs

This package implements a shooting method for solving boundary value problems, for instance resulting of the application of Pontryagin's Minimum Principle to an optimal control problem. The software is mostly Fortran90, with some third party Fortran77 codes for the numerical integration and nonlinear equations system. Its features include the handling of right hand side discontinuities (such as caused by a bang-bang control) for the integration of the trajectory and the computation of Jacobians for the shooting method. The particular case of singular arcs for optimal control problems is also addressed.

Package Shoot-1.1 -Guide Utilisateur Résumé : Ce paquetage implémente une méthode de tir pour la résolution de problèmes aux deux bouts, tels que ceux obtenus par application du Principe du Minimum de Pontryaguine à un problème de contrôle optimal. Le logiciel est principalement écrit en Fortran90, et utilise des codes externes en Fortran77 pour l'intégration numérique et la résolution de systèmes non linéaires.Il prend notamment en compte les discontinuités du second membre, du point de vue de l'intégration de la trajectoire ainsi que du calcul de Jacobien pour la méthode de tir. Le cas particulier des arcs singuliers pour les problèmes de contrôle optimal est également traité.

Mots-clés : Problème aux deux bouts, second membre discontinu, contrôle optimal , méthode de tir, arcs singuliers

Introduction

This package of Fortran 90 routines implements several variants of indirect shooting methods for optimal control problems, with an embedded homotopy approach (discrete continuation). Special care is given to the numerical integration of the shooting function and the precise computation of its Jacobian, especially in the discontinuous case. Originally designed for the resolution of bang-bang low thrust orbital transfers for the CNES3 , the software has been since then extended to handle problems with singular arcs. The integration of state constraints is currently under development.

Shooting method

We begin with a brief presentation of the single shooting method, which is part of the indirect methods, and is based on Pontryagin's Maximum Principle (we refer readers interested in these methods to [?, ?, ?, ?] for instance). We recall that direct methods, on the other hand, typically involve the partial (control) or total (control and state) discretization of the problem, and then use various approaches (SQP and interior point techniques for instance) to solve the resulting optimization problem. Direct methods are thus supposed to be robust, but the counterpart is a relatively low precision, and a huge problem size depending on the discretization stepsize used. This makes these methods ill-suited to some particular cases, such as problems with a bang-bang control structure and a huge number of commutations.

Back to the indirect methods, single shooting consists in finding a zero of the shooting function associated with the original problem. There is no discretization, even if the method still involves an integration of the system in some way. It is a fast and high precision method, but often requires a good initial guess: as they typically consist in applying a (quasi-)Newton solver to the shooting function, the convergence radius may be quite small, depending on the problem. This is particularly true for problems involving commutations, or worse, singular arcs, and often leads to the use of continuation (homotopy) techniques to obtain a suitable initial point.

We consider a general optimal control problem in the Bolza form (P )

           M in g(t 0 , x(t 0 ), t f , x(t f )) + t f t0 l(t, x, u) dt Objective ẋ = f (t, x, u) Dynamics u ∈ U Admissible Controls ψ 0 (t 0 , x(t 0 )) = 0 Initial Conditions ψ 1 (t f , x(t f )) = 0
T erminal Conditions Notation/Remark: for clarity, the time t will often be omitted in the formulas, except in ambiguous cases.

We use here and in all the following the notations: x ∈ R n for the state, u ∈ R m for the control, U the set of admissible controls, and f for the state dynamics. We assume that the initial and final times t 0 and t f are fixed. Free final time problems can always be reformulated as fixed time problems using the classical time transform t = s.t f with s ∈ [0, 1] the normalized time and t f an additional state variable. We introduce the costate p, of same dimension as the state x, and define the Hamiltonian by H(t, x, p, u) = l(t, x, u) + (p|f (t, x, u)).

Remark: we thus assume in all the following that we are in the normal case, meaning that the costate p 0 associated to the integral objective l is non zero, and can be made equal to 1, by dividing the whole costate p by p 0 ... Pontryagin's Maximum Principle then states that, under the assumptions:

∃ (x, u) feasible for (P ), with x absolutely continuous and u measurable. f and l are continuous with respect to u and C 1 with respect to t and x. g, ψ 0 , ψ 1 are C 1 with respect to x.

Let (x, u) be an optimal pair for (P ), then (i) ∃ p = 0 absolutely continuous such that we have the Hamiltonian system

   ẋ = ∂H ∂p (t, x, p, u) ṗ = -∂H ∂x (t, x, p, u) (ii) u is solution of M in w∈U H(t, x, p, w) ae in [t 0 , t f ].
(iii) "Transversality conditions": ∃ (µ 0 , µ 1 ) such that

(T C)            ψ 0 (x(t 0 )) = 0 p(t 0 ) = -∂Φ ∂x0 (t 0 , x(t 0 ), t f , x(t f ), µ 0 , µ 1 ) ψ 1 (x(t f )) = 0 p(t f ) = ∂Φ ∂x f (t 0 , x(t 0 ), t f , x(t f ), µ 0 , µ 1 ) with Φ : (t 0 , x 0 , t f , x f , µ 0 , µ 1 ) → g(t 0 , x 0 , t f , x f ) + (ψ 0 (t 0 , x 0 )|µ 0 ) + (ψ 1 (t f , x f )|µ 1 )
Remark: this is why indirect methods are sometimes referred to as necessary condition methods.

Now we denote y = (x, p) and ϕ the state-costate dynamics derived from the Hamiltonian system. We assume here that the expression of the optimal control given by the necessary conditions (Hamiltonian minimization) is actually a function, noted γ, ie u(t) = ArgM in w∈U H(t, x, p, w) = γ(t, y). Solving (P ) is equivalent to solving the following Boundary Value Problem

(BV P )    ẏ = ϕ(t, y, γ(t, y)) ae in [t 0 , t f ] c 0 (t 0 , y(t 0 )) = 0 Boundary Conditions at t 0 c 1 (t f , y(t f )) = 0
Boundary Conditions at t f Note: these Boundary Conditions c 0 and c 1 correspond to the Transversality Conditions mentioned above, that contain the Initial and Terminal conditions

INRIA

of (P ) in addition to the constraints on the costate p.

It is possible to integrate y = (x, p) if we set the value of y(t 0 ), and we then obtain the following Initial Value Problem

(IV P ) ẏ = ϕ(t, y, γ(t, y)) ae in [t 0 , t f ] y(t 0 ) = ζ Initial V alue.
We introduce now an application called the shooting function, which basically maps the initial value ζ to the value of the Boundary conditions at t f for the corresponding solution of (IV P ). In practice, the initial conditions ψ 0 of the problem (P ) already give a part of y(t 0 ), so the unknown of the shooting function is reduced to the "missing" part, that we note z. A frequent situation is when the initial conditions determine the initial state x(t 0 ), therefore z is actually the initial costate p(t 0 ). Then the value of the shooting function is given by the boundary conditions at t f for the solution y(•, z) of (IV P ) corresponding to the initial value y(t 0 , z) = (x 0 , z)

(Shooting f unction) S : z → c 1 (y(t f , z)).
Finding a zero of the shooting function S is then equivalent to the resolution of (BV P ), and therefore also gives a solution of (P ). The "shooting method" thus consists in solving the equation S(z) = 0, as summarized below:

Problem (P ) -→ Boundary Value Problem (BV P ) -→ Initial Value Problem (IV P )
and Shooting function S

An interesting particular case is when the optimal control is discontinuous and presents switchings (or commutations), for instance when the command law is bang-bang. More precisely, the Hamiltonian minimization gives

u = γ 1 (t, y) , if ψ(t, y) < 0 u = γ 2 (t, y) , if ψ(t, y) > 0 u = Γ(t, y)
, if ψ(t, y) = 0, with Γ(t, y) a subset of U , and ψ the switching function, whose zeros correspond to the commutations of the optimal control. We assume that the set of switching times t such that ψ(t, y) = 0 is finite. We then obtain y(•, z) as solution of the initial value problem with a discontinuous right hand side

(IV P ) disc    ẏ = ϕ 1 (t, y) , if ψ(t, y) < 0 ẏ = ϕ 2 (t, y) , if ψ(t, y) > 0, y(t 0 , z) = (x 0 , z)
As mentioned earlier, one of the main practical difficulties for the shooting method is to find a suitable initial point, due to the small convergence radius of the newton method applied to the shooting function. In our case, we use a continuation approach (homotopy), as described below.

Continuation

It is well known that shooting methods are quite sensible to the initial guess, and it is often quite hard to achieve convergence for non trivial problems because of this. An interesting way to obtain a suitable starting point is to use RT n°0354 continuation (or homotopy) approaches. The principle of continuation is to find a certain homotopy that connects the original problem to an easier one, and then to follow the zero path of this homotopy, from a solution of the easy problem to a solution of the original one.

In the following, we will often parametrize the original problem (P ) by a certain λ ∈ [0, 1], and obtain a problem family (P λ ) such that (P 1 ) = (P ). Moreover, we expect that we are able to find a solution of (P 0 ). If we note S λ the shooting function associated to (P λ ), we can define the homotopy

H : (z, λ) → S λ (z).
Assuming we know a zero z 0 of H(•, 0) = S 0 , all we have to do is to follow the zero path of H from λ = 0 to λ = 1, at which point we have obtained a zero of H(•, 1) = S 1 = S, and therefore a solution of (P ).

There are several ways to perform this path following, such as Predictor-Corrector methods (or differential homotopy) that treat the zero path as a differentiable curve, Piecewise Linear (or simplicial) methods that build a PL approximation of the path over a triangulation of the space. We refer the interested readers to [?] for a general overview of continuation methods, and [?] in the context of Newton methods for nonlinear problems.

A more basic approach is just to try to solve a sequence of problems (P λ k ), k = 1, . . . , N such that λ 0 = 0 and λ N = 1, by using the solution found for (P λ k ) as initial guess for (P λ k+1 ), or with a basic linear prediction step. The stepsize h k for the homotopic parameter (such that λ k+1 = λ k + h k ) can be set to a certain fixed value, or something a little more refined. For instance we can introduce a tolerance tol on the norm of the shooting function that indicates a successful convergence of the shooting method. Then we can start with a stepsize h 0 = 1 and try to solve (P 1 ) directly; if the attempt fails, we divide the stepsize h k by 2, and try again, until we reach λ = 1. It is then safer to also stop the continuation if a certain maximal number of iterations has been reached, or if the stepsize h k goes below some minimal value. This method is implemented in the Shoot package, and can be quite effective for well chosen homotopies.

2 Using the Shoot package

Compilation notes

Requirements: Fortran 95 compiler. On Unix/Linux platforms, just use the make command to build the executables.

Note: the package has been tested with the following fortran compilers: -ifort (Intel Fortran compiler) -gfortran (gcc.gnu.org/fortran/) -g95 (www.g95.org)

User supplied subroutines and input files

In order to solve an optimal control problem with the Shoot package, the user has to complete the following subroutines, located in ProblemFuns.f90, and described below:

InitPar: performs specific problem initializations, if any.

Control: optimal control and switching function evaluation.

Dynamics: state, costate and objective dynamics.

Switch, SwitchDot: optional, compute the switching function and its time derivative (needed for switching detection or singular arcs).

Then for each problem, two input files are used as well:

Problem initialization file .in: problem data and shooting method settings.

Continuation file .cont: discrete continuation settings.

After execution, the code will produce the output file: Solution file .sol: state, costate, control (and switching function) from the initial time to the final time, for the solution of the shooting method (can be visualized with the provided scripts).

Continuation Path file .contpath: the sequence of solutions generated during the discrete continuation (can be visualized with the provided scripts).

Note: for each problem, a single prefix name is associated to all input and output files. Thus if the input file are named Demo1.in and Demo1.cont for instance, the corresponding output files will be Demo1.sol and Demo1.contpath. This makes it easier to keep track of which problem the files correspond to.

INRIA

Once user-supplied subroutines and input files are completed, just run the executable. The program will ask the name (without the extension) of the input files, for instance "Problem1" for Problem1.cont and Problem1.in. Keep in mind that all input and output files associated to a given problem have the same name, only with different extensions. Thus it is quite easy to keep track of several problems at the same time, each one having its set of files with the same prefix.

At the end of the execution, a solution file .sol is generated, that contains the solution at the end of the continuation. The script sol.m traces the state, costate and control with respect to time, as well as the switching function optionally. A second output file .contpath contains the continuation path, that can be traced with the contpath.m script.

InitPar

This subroutine performs the various initializations required by the problem, and can be left empty. It is called at the beginning of the program, and then each time the shooting function is computed.

The input variables are -mode: 0 for first call, 1 for subsequent calls -z: shooting function unknown -lambda: homotopic parameter used for continuation

Subroutine interface

Subroutine InitPar(mode, z, lambda) implicit none integer, intent(in) :: mode real(kind=8), intent(in) :: lambda real(kind=8), dimension(n), intent(in) :: z

Control

This subroutine provides the value of the optimal control, according to necessary conditions (Hamiltonian minimization)

The input variables are -lambda: homotopic parameter used for continuation -t: time -x: state (dimension ns) -p: costate (dimension nc)

And the output variables -u: optimal control (dimension m)

The external parameter controlmode indicates what the subroutine should compute at each call: -0: compute standard control -1: compute bang-bang control according to external parameter switchf lag = sign(ψ).

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This mode is used only when switching detection is enabled. Typical values for switchflag are -1 and 1, and the control should be set accordingly. An incoming value of 0 indicates that switchflag must be initialized to match the sign of the switching function, in addition to computing the control.

-2: compute the exact expression of the control over a singular arc. This mode is used only when one or more singular arcs are specified in the control structure (see 14) and the singular control mode is set to use the exact expression (see 15). For problems with singular arcs, the first time derivative of the switching function is also required. The subroutine SwitchDot must then be completed as well (same interface as Switch).

Subroutine interface

Continuation file .cont

Problem dimension

Integer: same as the Unknown dimension in the intialization file Initial and final value for homotopic parameter Real -Real: Initial value should be the same as λ 0 in the intialization file; the discrete continuation will try to reach the final value set here for λ.

Tolerance, max iterations and iterations output frequency Real -Integer -Integer: tolerance indicates the norm threshold on the shooting function for a continuation step to be accepted. The two integers set the maximum number of iterations for the continuation, and the output frequency (a value of n means displaying info every n iteration.

Debug and verbose flags

Integer -Integer [0..2]: debug should be set to 0 for normal use; the verbose flag can be set to 1 or 2 to display some additional output.

Initialization file .in

This file contains problem initializations, such as dimensions, initial and terminal conditions, initial guess, and integrator choices. Here is a brief summary of the parameters set in this file.

Homotopy choice and problem class

Integer [1+]: tells which homotopy is to be used by the program. Set to 1 by default, this permits to use several different homotopies for a given problem with the same ProblemFuns.f90 file. Integer [1+]: sets the kind of shooting method used. 1: Single shooting 2+: Multiple shooting (under development)

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Unknown, State, Costate and Control dimensions Integer -Integer -Integer -Integer: here are specified the dimensions of the shooting function unknown, and of the state, costate and control of the problem (the state and costate dimensions are usually the same, but can differ if some components are known to be constant, and therefore do not need to be integrated).

Objective and Switch dimension

Integer [1+]: usually set to 1, as the objective is scalar, but additional objectives can also be computed, if the corresponding derivatives are included in subroutine Dynamics. Integer [0+]: like the objective, the switching function ψ has scalar values, but its derivatives can also be computed, if set in subroutine Control (observing the switching function is especially useful when suspecting the presence of singular arcs). For instance, a value of 2 indicates that both ψ and ψ are available.

Number of IVP unknown values

Integer: dimension of IVP unknown.

IVP unknown indices

Integer(): indices of IVP unknown values in (x, p) vector.

Number of initial values

Integer: number of values known at t 0 (given by initial and transversality conditions).

Initial values indices

Integer(): indices of these values in (x(t 0 ), p(t 0 )).

Initial values

Real(): initial values.

Number of terminal values

Integer: number of values known at t f (given by terminal and transversality conditions). The first integer selects the method for the numerical integration of the initial value problem (IV P ) 0: Fixed step -Euler 1: Fixed step -Midpoint 2: Fixed step -Runge Kutta 2nd order 3: Fixed step -Runge Kutta 3rd order 4: Fixed step -Runge Kutta 4th order 5: Variable step -Runge Kutta Fehlberg 4-5th order (rkf45) 6: Variable step -Dormand Prince 8-5-3 (dop853) 7: Variable step -Gragg Bulirsch Stoer extrapolation method (odex) 8: Variable step -Dormand Prince 5-4 (dopri5) 9: Variable step -Radau 5 (stiff) (radau5)

Terminal values indices

The second integer indicates the number of steps for fixed step integrators (ignored for variable step integrators). The two remaining reals set the absolute and relative error tolerances for variable step integrators (ignored for fixed step integrators).

Switching detection Integer

This setting enables or disables the switching detection (cf page 16). Setting this parameter to 0 will disable switching detection (recommended if the right hand side of the IVP is continuous). A value of 1 will test for switchings at the end of each integration steps, higher values mean additionnal tests within each step.

Jacobian mode

Integer [0..1]
This sets the method used to compute the Jacobian of the shooting function (cf page 17). 0: classical finite differences on the shooting function 1: variational equations (with finite differences for the right hand side)

Singular conditions and control

Integer [1..2] -Integer [0..2]
The first integer gives the type of entry/exit conditions for singular arcs: 1: requires ψ = 0 at entry and exit times (ψ is the switching function) 2: requires ψ = ψ = 0 at entry time ( ψ must then be provided)

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The second integer sets the way to compute the singular control: 0: exact singular control is provided 1: uses alternate singular control minimizing ψ 2 2: uses alternate singular control minimizing ψ 2 + ψ2 3 Discontinuities handling

Number of parameters

Switchings detection

Switching detection can be enabled if commutations (control discontinuities, typically for a bang-bang problem) are expected, in order to improve the integration precision and speed. Experiments on an orbital transfer problem (briefly presented below, see [?] for a more detailed study) show that this can significantly improve the convergence and speed of the shooting method.

Letting the integrator deal with the discontinuities of the right hand side on its own is often out of the question with a fixed step integrator, and even with a good variable step method, the cost can be heavy in terms of cpu time and accuracy. Therefore, we try to detect the switchings during the integration of (IV P ). The principle of the detection is described below, and assumes that the integration method used is able to provide a "dense output", ie a cheap approximation of y on each integration interval. This is typically done by some polynomial interpolation, see [?] for a thorough description of dense outputs available for many integration methods.

At integration time t, compute the switching function ψ(t, y).

If the sign of ψ has changed since the previous step, a commutation has occurred (actually, an odd number of commutations, but most often one in practice).

Then we locate the commutation by dichotomy, by using the dense output of the integrator to provide y and ψ. Then we perform the control switch, store the sign of ψ, and restart the integration back from the commutation time.

Else the integration goes on normally. This algorithm is quite simple, but effective. The main issue is the risk of skipping one or several commutations during an integration step. An easy improvement of the detection method is to perform additional sign checks at a fixed number of intermediate points, and not only at the end of the step. If INRIA these additional checks bring some new commutations, then the number of intermediate checkpoints should be increased until no new commutations appear. Besides, with a variable step integrator, we can also expect the stepsize control mechanism to reject a step that would have missed commutations.

Notes:

-switching detection is activated by setting a value greater than 0 for the first parameter of the block Switching detection and variational system in the initialization file (this parameter is actually the number of sign checks for the switching function over each integration interval).

-the switching function ψ MUST be provided by the subroutine Control.

-requires an integrator with dense output (currently rk3, rk4, dopri5 and radau5).

Variational System

The first way to compute the Jacobian of the shooting function is simply to use finite differences. We use for the shooting method the nonlinear solver hybrd, which performs finite differences with a step of

h j = √ |x j |,
where is the error on the shooting function evaluation (for the numerical experiments, we set to the relative error of the integrator). This method is sometimes referred to as "external differentiation" (END). Its drawback is that when used with a variable step integrator, the integration steps can vary at the points where S is evaluated for the finite differences, which can impair the approximation of the Jacobian (see [?], [?] p.200).

Then, another possibility is to use the variational equations to compute the Jacobian of the shooting function. We first recall the smooth case, when we consider the derivative with respect to the initial condition of the system

(IV P )    ẏ(t) = ϕ(t, y(t)) y i (t 0 ) = y 0 i for i = 1, . . . , n y i+n = z i for i = 1, . . . , n.
If we note y(•, z) the solution of (IV P ), we know that ∂y ∂zi (t f , z) is solution of the variational system

(V AR) j Ẏj (t) = ∂ϕ ∂y (t, y(t))Y j (t) Y j (t 0 ) = 0 • • • 1 • • • 0 T
It is then possible to compute the Jacobian of the shooting function by integrating (V AR) along with (IV P ). An interesting possibility is to approximate the right hand side by finite differences (cf [?] p.201):

Ẏ (:, j) ≈ 1 h (ϕ(y + hY (:, j)) -ϕ(y))
The use of the variational equations can be adapted to the discontinuous case, see [?].

RT n°0354

Notes:

-the use of variational equations instead of external differentiation is activated by setting the Jacobian mode parameter in the initialization file to 1.

-if switchings are expected, switching detection MUST be enabled ! -this option uses the hybrj solver instead of hybrd.

Sample problem: Orbital transfer

Problem statement

We study here an orbital transfer problem, originally submitted by CNES4 . We consider a satellite with a mass of 1500 kg and low thrust electro-ionic propulsion (with thrusts ranging from 10 Newtons to 0.1 Newton). We want to transfer it from a strongly elliptic, slightly inclined orbit, to a circular, equatorial, geostationary orbit. The objective is to maximize the payload, i.e., to minimize the fuel consumption during the transfer (this is not a minimum-time problem). This kind of problems was studied for instance in [?, ?], and this specific family of problems in [?, ?, ?].

Model

We consider that the forces applied to the satellite are the Earth attraction (central force) and the engine thrust: r = -µ r r 3 + T m with r the position vector in R 3 , T the thrust also in R 3 (ie the control), m the mass of the satellite, µ the gravitational constant of the Earth.

State

In order to avoid the strong oscillations that would result from the high number of revolutions, we use orbital parameters instead of Cartesian coordinates to define the state variables:

Orbit parameter P Eccentricity vector (e x , e y ), in the orbit plane, oriented towards perigee Rotation vector (h x , h y ), in the equatorial plane, collinear to the intersection of orbit and equatorial planes True longitude L = Ω n + ω + w Mass m

Control

The normalized three-dimensional control u such that T = T max u is expressed in the moving reference frame attached to the satellite, as radial thrust q, transverse thrust s and normal thrust w.

Optimal Control Problem

If we note x = (P, e x , e y , h x , h y , L), with T max the maximal thrust we have the following formulation of the problem:

(P )                    M in t f t0 u(t) dt ẋ = f 0 (x) + Tmax m B(x)u ṁ = -βT max u u ≤ 1 x(t 0 ) = (11625, 0.75, 0, 0.0612, 0, π) m(t 0 ) = 1500 x(t f ) = (42165, 0, 0, 0, 0, f ree) m(t f ) is f ree t 0 = 0 t f is f ixed
with f 0 and B depending on x, and β = 0.05112km -1 s.

Switching function and Control

We note p the costate associated to x, and p m the costate associated to m, and define the switching function ψ:

ψ(x, m, p, p m ) = 1 -βT max p m - T max m B(x) t p .
The application of Pontryagin's Maximum Principle then leads to the following expression of the optimal control

u = -B(x) t p B(x) t p if ψ(x, m, p, p m ) < 0 u = 0 if ψ(x, m, p, p m ) > 0
We can see that this control has a bang-bang structure, as its norm switches between 0 and 1 at zeros of the switching function ψ. The two cases define the two dynamics ϕ 1 and ϕ 2 .

Continuation

We choose here for the "easy" problem the same transfer with minimization of an "energy" criterion, namely t f t0 u(t) 2 dt. For our homotopy, we take the shooting function corresponding to the orbital transfer with the following objective, parametrized by λ ∈ [0, 1]:

J λ = t f t0 λ u(t) + (1 -λ) u(t) 2 dt.
The resulting perturbed problem (P λ ) has a strictly convex Hamiltonian (with respect to u), with a continuous optimal control, and is much easier to solve than (P ) = (P 1 ). As we shall see below, the shooting method indeed converges for (P 0 ) from a simple initial guess. Moreover, with the switching detection enabled, the solution for (P 0 ) is a sufficient starting point for (P ), so the continuation is here reduced to its simplest form. Orbital transfer (10N) -State, costate and control

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We study now the same problem, but with a maximal thrust of only 0.1N . Notice that the transfer time, number of revolutions (i.e. final longitude) and number of switchings are inversely proportional to the thrust, therefore we find values a hundred times greater than for the 10N transfer. Also, the value of the criterion (minimal fuel consumption) remains the same as for the 10N transfer.

We draw once again the solution and trajectory, which have the same structure than for the 10N transfer, but with 1800 switchings, 750 revolutions around the earth, and a final time of 640 days (!). Orbital transfer (0.1N) -State, costate and control, and trajectory

Here is the execution log for the 0.1N transfer (input files are also given in appendix). We notice that the cpu time is also roughly a hundred times higher than for the 10N transfer, which is mainly due to the fact that the integration of (IV P ) to compute the value of the shooting fonction is a hundred times longer. For this family of orbital transfer problems, experiments with maximal thrust ranging from 10N to 0.1N show that the number of Jacobian evaluations needed for the shooting method (both at λ = 0 and λ = 1) is roughly the same for all thrusts, between 3 and 6. Moreover, the discrete continuation always converges directly after 1 iteration. Among optimal control problems, singular arcs problems are interesting and difficult to solve with indirect methods, as they involve a multi-valued control and differential inclusions. In short, singular arcs occur in indirect methods when the necessary conditions from the Hamiltonian minimization fails to determine uniquely the optimal control over a non-trivial interval. In this case, applying the Pontryagin's Maximum Principle leads to a Boundary Value Problem with a differential inclusion.

Fortunately, there exist some well-known additional conditions that allow us to determine the value of the singular control, see [?, ?, ?]. In particular, exploiting the fact that the successive time derivatives of ∂H ∂u must vanish over a singular arc usually give the expression of the singular control (this is in practice often done by derivating the switching function).

The shooting method can be adapted to solve this kind of problems, but this typically requires some a priori assumptions on the control structure (number and approximate location of singular arcs in particular), and a good starting point. So we look for a method able to provide a reliable singular structure detection and initialization.

We resort once again to a continuation approach, and regularize the problem by adding a quadratic (u 2 ) perturbation to the criterion. We limit here ourselves to the case where the Hamiltonian is affine with respect to the control (which is quite common when singular arcs are involved), so that the regularized Hamiltonian is strongly convex with respect to the control. This ensures that the optimal control u * is continuous, without any commutations or singular arcs, which makes the regularized problem much easier to solve.

The aim here is not to perform the continuation until the regularization vanishes, which would be impossible anyway due to the presence of singular arcs in the original problem. We just want to obtain sufficient information concerning the control structure, and an acceptable starting point for the shooting method.

This approach is illustrated below on a 3D version of the famous Goddard problem.

Shooting problem formulation

The shooting problem formulation has to be adapted to take singular arcs into account. The structure of the control must be prescribed by assigning a fixed number of interior junction times between nonsingular and singular arcs. These times (t i ) i=1..n switch are part of the unknowns and must satisfy some junction conditions. Optionaly, the value of the state and costate at the start of each interior arc may also be added to the shooting unknowns. In this case matching conditions must be verified at the junction times (typically state and costate RT n°0354 continuity).

A junction condition indicates a change of structure, which corresponds here to an extremity of a singular arc. Along a singular arc, it is required that the switching function ψ and its time derivatives vanish. In practice, the singular control is often computed using the relation ψ = 0, therefore switching conditions may consist for instance in requiring either ψ = 0 at both extremities of the singular arc, or ψ = ψ = 0 at the beginning of the arc (tangential entry). In our simulations, we choose the latter solution which happens to provide better and more stable results.

Additional unknowns:

interior junction times (t i ) i=1,nswitch optiona state and costate values at interior times Additional equations: switching conditions at junction times optional matching conditions at interior times Summary: shooting function unknown and value layout (here with the starting points for each interior arc) Unknown z IVP unknown at t 0 (x 1 , p 1 ) (x 2 , p 2 ) ... ∆ 1 ∆ 2 ...

-usual IVP unknown at t 0 -values of (x i , p i ) at interior junction times t i -interior junction times (t i ) i=1,nswitch

Value S Struct (z) Junction cond (t 1 ) M atch cond (t 1 ) ... Conditions at t f -junction and matching conditions at interior times -usual terminal and transversality conditions at t f 4.3 Sample problem: Goddard (3D)

Problem statement

Note: the full study of the following problem is described in [?].

For the third example, we consider an extension of the classical Goddard's problem (maximizing the final altitude of a rocket with vertical trajectory, the controls being the norm and direction of the thrust force, see [?, ?]) for nonvertical trajectories. We want here to maximize the final mass (i.e., minimizing the fuel consumption) while steering the system from a given initial point to a fixed final position, the final velocity vector and final time being free.

INRIA

M in

t f 0 u(t) dt ṙ = v, v = -D(r,v) m v v -g(r) + C u m , ṁ = -b u u(t) ≤ 1 r(0) = r 0 , v(0) = v 0 , m(0) = m 0 r(t f ) = r f , v(t f ) f ree, m(t f ) f ree t f f ree
where the state variables are r(t) ∈ R 3 (position of the spacecraft), v(t) ∈ R 3 (velocity vector) and m(t) (mass of the engine). Also, D(r, v) > 0 is the drag component, g(r) ∈ R 3 is the usual gravity force, and b > 0 is a positive real number depending on the engine. The thrust force is Cu(t), where C > 0 is the maximal thrust, and the control is the normalized thrust in R 3 .

Depending on the features of the problem (initial and final conditions, mass / thrust ratio, etc), it is known that control strategies that consist in choosing the control so that u(t) is piecewise constant all along the flight, either equal to 0 or to the maximal authorized value 1, may not be optimal, as a consequence of the high values of the drag for high speed. Optimal trajectories may indeed involve singular arcs, on which the thrust is neither null nor maximal.

The Hamiltonian of the optimal control problem is

H = p r , v + p v , - D(r, v) m v v -g(r) + C u m -(1 + bp m ) u ,
which leads to the switching function

ψ(t) = C m(t) p v (t) -(1 + bp m (t)), if ψ(t) < 0 then u(t) = 0; if ψ(t) > 0 then u(t) = pv(t) pv(t) ; if ψ(t) = 0 on I ⊂ [0, t f ],
then the control u is singular, and

u(t) = α(t) p v (t) p v (t)
a.e. on I Furthermore, on a singular subarc, derivating the switching function twice yields the expression of α via a relation of the form A(x, p)α = B(x, p). The computations are actually quite tedious to do by hand, and we used the symbolic calculus tool Maple. The expressions of A and B are quite complicated and are not reported here.

The free final time problem is formulated as a fixed final time one via the usual time transformation t = t f s, with s ∈ [0, 1] and t f an additional component of the state vector, such that ṫ f = 0 and t f (0), t f (1) are free, with the associated costate satisfying ṗt f = -H. All the graphs in the following will use this normalized time interval [0, 1]. We can now perform the main continuation on the quadratic regularization, using the solution obtained after the atmosphere homotopy as initial guess. We choose to stop the continuation after 250 iterations, which corresponds here roughly to λ ≈ 0.8. Here is the solution at the end of the continuation. The control and switching function graphs strongly suggest the presence of a singular subarc around the time interval [0.1, 0.4]. We show the state, costate, control and switching function for the solution with a singular arc. For the 0.1N transfer, the continuation file is the same as for the 10N transfer. The only differences for the initialization file are the final "time" (actually longitude), the initial guess for t f (first component of z 0 ), the number of integration steps for the RK4 method, and of course the maximal thrust (first of the parameters at the end of the file).

Energy to mass homotopy -transfer-01N.in

  Integer(): indices of these values in (x(t f ), p(t f )). Terminal values Real(): terminal values. Fixed times Integer: number of fixed times (often 2: initial and final time) Real -Real: values of fixed times. Times structure Integer [2+]: number of switching times (see below). Integer() List giving the type of switching times (see sample problems for examples of structures): 0: initial of final time 2: singular arc entry (-2 for light formulation) 3: singular arc exit (-3 for light formulation) Full formulations use interior nodes and matching conditions; light formulations use only the switching times. INRIA Starting point (z0,lambda0) for zeropath Real(n+1): initial guess z 0 for shooting function unknown, and initial value λ 0 for the homotopic parameter λ. Scaling mode Integer [0..1]: selects the scaling mode. 0: no scaling 1: initial point scaling (to [0.1, 1])Integrator choice (type, steps, abstol, reltol) Integer [0..9] -Integer -Real -Real

  Integer [0+]: number of problem specific parameters. Parameters Real(): problem specific parameters, if any.

  end of the continuation for λ ≈ 0.8 Using the information from the previous solution at λ = 0.8, we assume a control structure of the form bang-singular-bang and try the singular shooting, that converges to a solution with the expected singular arc. Sgod Continuation file Problem.cont not found, enter problem files prefix: g3d-shoot Testing Initial Shooting point 3.8612091309731476 Initial Shoot... HYBRD solver returns: 5 NFEV: 499 (NJAC: 8 ) Solution: // (solution removed for more clarity ...) // 1. Shooting function norm at the solution: 0.0002645393058478628 Iterations: 0 Execution time: 3.0875310000000002 Generating solution... switchtime*tf 0. 0.022774715513691285 0.07977732948936873 0.2191378890189921 Switching condition at switch 1 at 0.10392869811626898 is -0.0000019081405440157795 -0.00002932641104428758 Homotopy norm at lambda= 1. : 0.0002645393058478628 Singular shooting objective... Criterion value: 0.4002680426684089

  ) -State, costate, control and switching function The altitude, speed and mass during the flight are represented below.

  Solution with singular arc: altitude, speed and mass RT n°0354

The French space agency RT n°0354

Centre National d'Etudes Spatiales INRIA

RT n°0354

http://www.inria.fr ISSN 0249-0803

Numerical results

We present here briefly the numerical results for the orbital problem, for a maximal thrust of 10N and 0.1N . For both problems we see that the shooting method converges for λ = 0 from the given starting point, as indicated in the "Initial Shoot" output block. Then the algorithm proceeds to the discrete continuation, and in this case directly converges for λ = 1 with the solution from λ = 0, therefore only 1 iteration is needed.

Note: the executable for the orbital transfer problem is named "Orb", and the input files for these two problems are given in Appendix, see page 30 We show the solution (state, costate and control) obtained for λ = 1, where the control switchings (18) are clearly visible on the control and switching function graphs. We also draw the trajectory of the satellite around the earth (about 7 revolutions)

INRIA

Transversality conditions on the adjoint vector yield p v (1) = (0 0 0), p m (1) = 0, and p t f (0) = p t f (1) = 0. The unknown of the shooting function S is therefore

As usual, we regularize the cost function by considering an homotopic connection with an energy,

where the parameter of the homotopy is λ ∈ [0, 1]. The resulting perturbed problem (P λ ) has a strongly convex Hamiltonian (with respect to u) for λ < 1, with a continuous optimal control.

Numerical results

In this case, even solving the regularized problem (P 0 ) is not obvious, due to the aerodynamic forces (drag). For this reason, we introduce a preliminary continuation on the atmosphere density, starting from a problem without atmosphere. This first homotopy can be initialized with a very simple intial guess (see Appendix for the input files) and converges without any difficulties to a solution of the regularized problem.


Dr. Jean-Philippe Malet
email: jean-philippe.malet@unicaen.fr

Prof. Christophe Delacourt
email: christophe.delacourt@univ-brest.fr.

Prof. Olivier Maquaire
email: olivier.maquaire@unicaen.fr.

Dr. David Amitrano
email: david.amitrano@ujf-grenoble.fr.

O Delacourt

D Maquaire

Amitrano

Introduction to the thematic volume Issues in landslide process monitoring and understanding: an introduction

All mountainous countries have been exposed to slope movements throughout their history, reaching often disastrous dimensions. A significant increase in damage effects had to be recognised within the last two decades. The reasons for this steady increase in damage are often relayed on a probable impact of climate change. But people contribute also to, and even may exacerbate, or modify the hazard. The increase of our living standards, the concentration of people, infrastructure and goods at economically privileged but probably hazardous places, additional settlements in disaster prone regions and the enormous increase in mobility on road and train have to be considered as well. The number of victims and the cost of the damage may be high, depending on the duration, spatial extension and magnitude of the processes, and on the vulnerability of the exposed environment.

Even though considerable advances in the last decade have been gained in landslide process understanding (landslide databases and event catalogues, monitoring techniques, investigation tools), there are a series of gaps that should be filled in existing knowledge in order to apply this knowledge for longterm development of the mountain territories and safety of the citizens. This is mainly the consequence of the large kinematic variability of slope movements (slide, flow, fall, spreading) which are dynamic systems, complex in time and space and closely linked to both inherited and current preparatory and triggering controls.

To develop comprehensive hazard assessment procedures, it is therefore important to incorporate time series, 3-D patterns and deformation analyses in the model-building exercise; it is also essential that the physically-based models be improved so that a greater spatial and temporal description can be included. This goal requires first that rapid-varying factors (rainfall, freeze-thaw, meltwater, ground acceleration) and slow-varying factors (tectonic movements, weathering and associated property changes, erosion, deposition, changing confinement and unloading) are properly specified at adequate spatial and temporal resolutions. The influence of these elementary factors for the different landslide types can be identified through new investigation and monitoring techniques, and detailed analyses of event databases.

 [2003][2004][2005]. The project consortium was composed of eight research institutes (Laboratoire de Sciences de la Terre, LST, Lyon; Institut de Physique du Globe de Strasbourg, IPGS, Strasbourg; Laboratoire Environnement, Géomécanique et Ouvrages, Laego, Nancy ; Laboratoire de Géophysique Interne et de Tectonophysique, LGIT, Grenoble ; GeoSciences Azur, Nice ; Laboratoire Interdisciplinaire de Recherche Impliquant la Géologie et la Mécanique, Lirigm, Grenoble ; Laboratoire d'Hydrogéologie de l'Université d'Avignon, LHA, Avignon ; Bureau des Recherches Géologiques et Minières, BRGM, Orléans).

The goals of this project were:

(1) to federate the French community working on landslide characterization and monitoring using remote-sensing, geophysics, geo-and hydrochemistry, hydrology, geotechnics and geomorphology;

(2) to elaborate and test quantitative investigation methods of spatial and temporal heterogeneities of landslides;

(3) to elaborate multiscale protocols suitable to the investigation of several landslide types (rock-and soil-type landslide);

(4) to initiate and extent the collection of multi-source data on some reference sites in terms of geometry, temporal and spatial kinematics, internal structure, rheology and hydrology. Two reference sites have been chosen in order to cover the main types of landslides: one in hard rocks (La Clapière rockslide) and one in soft (reworked) mudrocks (Super-Sauze mudslide). The 3D geometry of the landslides has been investigated by combining various geophysical and remote-sensing data. The multiscale (spatial and temporal) strain fields were analysed from 0.1 second to some years by seismic and remote-sensing methods. The residence time and the origin of water circulating within the landslides were determined from geo-and hydro-chemical data. These techniques are now operational and can be applied to other landslides. The databases are available for the French research community.

THE PRESENT VOLUME

The present volume of 'Bulletin de la Société Géologique de France' brings together a number of studies where landslide characterization and monitoring have allowed to progress in the understanding of landslide mechanisms. State-of-the-art papers are also included in order to synthesize existing knowledge available for landslide hazard assessment. These papers might be regarded as a time-slice, showing us age around 2006. The present collection of papers makes a significant contribution to bringing together the ideas of geophysicians, hydrologists and geomorphologists. Techniques used in these studies for description and analyses are by no means uniform, and if standardisation is desirable there is still a long way to go.

In the introductory paper, Van Asch et al. review the state-of-the-art in the understanding of landslide processes and identify some pressing challenges for the development of modelling capabilities for hazard assessment. Their paper focuses on the special nature of slope movements and the difficulties related to simulating their complex time-dependent behaviour in mathematical, physically-based, models. It analyses successively the research frontiers in the recognition of first-time failures (pre-failure and failure stages), reactivation and the catastrophic transition to rapid gravitational processes (post-failure stage). Subsequently, the paper discusses avenues to transfer local knowledge on landslide activity to landslide hazard forecasts on regional scales.

Delacourt et al. are concerned with the characterization of landslide surface displacements and review the techniques of image analysis (interferometry and optical data correlation). Examples of applications of these techniques in French South Alps are shown. Depending on the landslide characteristics (exposure conditions, size, velocity) as well as the goal of the study (operational or scientific purpose), one or a combination of several techniques and data (characterized by several resolution, accuracy, covered surface, revisiting time) have to be used. The authors demonstrate that investigating landslide surface displacement is a key parameter to access to their geometries and mechanical properties.

Jongmans & Garambois present a current state-of-the-art on the application of surface geophysical methods (2D, 3D and even 4D) to landslide characterization, showing both their interest and their limits.

The authors emphasized the geophysical image characteristics (resolution, penetration depth) which have to be provided for assessing their reliability, as well as the absolute requirements to combine geophysical methods and to calibrate them with existing geological and geotechnical data. They stressed also the link to establish between physical parameters derived from geophysics and the geological and mechanical properties required by geologists and engineers.

Bogaard et al. give an overview of the potential hydrogeochemistry can have for landslide research and analyse the use of hydrogeochemical information to unravel the hydrological processes in landslide triggering. The authors stress the impact hydrogeochemistry has on physical properties of the soil and rock material, and investigate how the identification of the origin of water, the flow paths, the travel times and the soil chemical information are environmental characteristics of paramount importance to identify contribution of different aquifers to the landslide area.

Grandjean et al. combine several geophysical methods (acoustic P-wave velocity and shear S-waves velocity, ρ electrical resistivity) to investigate the structure and rheology of a mudslide. The authors show a correlation between the seismic velocities and electrical resistivity data, confirming that the simultaneous use of both methods gives complementary information: the seismic data provide information on the variations of fissure density and on the presence of deformed material whereas the electrical resistivity data provide information on the variations of water content within the mudslide. A data fusion strategy based on fuzzy subsets theory is developed. The computed fuzzy cross-sections show the possibility of geomechanical hypotheses to be realized in specific areas of the mudslide, and highlight the places where plastic or solid-body deformations could occur.

Méric et al. present a paper on the use of seismic noise based methods to better understand the mechanics and dynamics of two clayey landslides. Their investigations were composed of seismic profiles for surface wave inversions, H/V measurements in order to image shear wave contrasts (slip surfaces), and seismic noise array measurements to provide S-waves velocity profile versus depth. Comparison between the geophysical investigations and geotechnical data proved the applicability of such passive methods in 3D complex structures, and particularly for landslides difficult to access. Indeed, these passive methods require lighter instrumentation and easier processing tools.

Amitrano et al. presents seismic monitoring data acquired on a mudslide, and show that recordable seismicity exists in the frequency range of 0.01 Hz to 10 Hz associated with landslide deformation. The origin of the seismicity is attributed to shearing of the moving mass at the interface with an in-situ stable bedrock. Spectral analysis of the seismic wave reveals that the range 0.1-1 Hz is the most sensitive to the variations in landslide velocity. The results emphasize the strong potential of seismic monitoring for improving our understanding of mechanisms controlling rock mass deformation and gravitational instability.

Binet et al. determine the long-term relationships between groundwater and gravitational instabilities in a landslide prone region of the South French Alps. The authors propose a conceptual model of water flow within unstable rocky slopes using water chemistry and flow records. Variations in infiltrated yield, flow velocity and porosity are observed and linked to collapsed and toppled structures in the upper parts of the hillslopes.

In this assemblage of papers, the sum is larger than the parts, and the collection reveals the range of techniques, processing tools and models, and current ideas on the study of slope movements in a wide range of geomorphic settings, and the benefit of geophysical, geotechnical, hydrochemical and engineering techniques in applied geomorphology.

This special issue of 'Bulletin de


Dr M Guille-Collignon

Dr J Delacotte

Prof F Lemaître

Prof E Labbé

Dr O Buriez Pasteur

Keywords: Analytical Methods, Fluorescence, Fluorescent Probes, Molecular Electrochemistry

This short review is aimed at emphasizing the most prominent recent works devoted to the fluorescence modulation of organic fluorescent or fluorogenic molecules by electrochemistry. This still expanding research field not only addresses the smart uses of known molecules or the design of new ones, but also investigates the development of instrumentation providing time-and space-resolved information at the molecular level. Important considerations including fluorescent/fluorogenic probes, reversible/irreversible fluorescence switch, direct/indirect fluorescence modulation, or environment properties are especially scrutinized in recent works dealing with bioanalysis perspectives.

Introduction

In recent years, the interest in stimuli-responsive molecules encompassing thermal, photonic, electric, pH…external triggers has grown rapidly with the multiplication of applications such as the reversible control of molecular memories, molecular logic gates, ion sensing, bioanalysis and fluorescence imaging. Accordingly, electrochemical commands aimed at modulating the fluorescence of molecules have been particularly explored, notably in analytical and bioanalytical chemistry, considering that many analytes are redox-active and can be detected/mapped with high sensitivity through fluorescence. With this respect, numerous organic and organometallic dyes, dyads, triads, or polymers exhibiting both fluorescence and redox properties are continuously developed and investigated [1,2]. In parallel, instrumental methodologies aimed at synchronizing electrochemical stimulation and fluorescence detection have been developed to provide reliable time-and space-resolved responses, as well as information at the molecular level [3,4]. Two types of electrochemical fluorescence switch can be obtained depending on the redox starting state of the molecule. Hence, a molecule is said fluorescent when it is emissive in its stable/original form and less or non-emissive when oxidized or reduced. Conversely, when a molecule is poorly emissive (or nonemissive) in its starting state but becomes emissive when oxidized or reduced, it is called fluorogenic. From the electrochemical point of view, the switch on/off of fluorogenic/fluorescent molecules may appear reversible, as this is related to the stability of the electrogenerated new redox state. Nevertheless, both reversible and irreversible situations may be interesting as a function of the targeted applications. Whatever the intrinsic photo-redox and chemical properties of these molecules, their fluorescence may also strongly depend on the physico-chemical parameters of their surrounding environments (polarity, pH). This article reviews some essential fluorescent and fluorogenic organic probes used in molecular electrochemistry with a special emphasis on recent works having potential applications in bioanalysis. It does not ambition to provide an exhaustive list of organic electrochemically-activable fluorescent or fluorogenic molecules. Actually, it surveys the most prominent recent works achieved in this field through illustrative examples. The first paragraph deals with the direct electrochemical switch (reversible or not) of some important fluorescent organic molecules in polar and non-polar environments. In the second paragraph, a focus is made on fluorogenic molecules whose fluorescence modulation is electrochemically promoted either directly or indirectly. A specific paragraph is devoted to pH-responsive fluorogenic probes possessing redox properties. It is noteworthy that most of the redox organic fluorescent dyes described in the first paragraph may also behave as fluorogenic species when they are covalently connected to a redox unit. In this case, molecules are generally non-emissive under photonic excitation due to a photoinduced electron or an energy transfer process. Oxidation or reduction may prevent this transfer either by a direct change of the redox state for photoinduced electron transfer or indirectly by shifting the absorption features in the case of energy transfer. Combination of rhodamine and ferrocene [5] or BODIPY (4,4-difluoro-4-bora-3a,4a-diaza-s-indacene) and a fullerene C60 [6] are two typical examples recently investigated. Similarly, fluorescence of BODIPY was quenched when adsorbed at a gold metal surface whereas luminescence was only observed for molecules reductively desorbed and then separated from the electrode surface [7]. For all these developments the readers are invited to refer to recent relevant reviews [1 -4].

Direct electrochemical switch of organic fluorescent molecules

The use of molecules whose luminescent properties are directly related to their redox state is a straightforward way to electrochemically modulate fluorescence. In that respect, a direct electrochemical switch has been evidenced on various families of redox-fluorescent organic molecules. These molecules generally possess a rigid aromatic or -conjugated backbone because molecular motions like rotation or vibration are well-known nonradiative de-excitation pathways. Moreover, fused aromatic rings, possibly incorporating heteroatoms (N, O, S) are recurrent substructures of organic fluorophores (Figure 1). On the one hand, situations where fluorescence modulation can be directly and reversibly achieved by electrochemistry are particularly targeted in the development of electrofluorochromic devices. Such situations require the electrogenerated species to be stable enough to be reduced/oxidized back into the initial stable redox state. On the other hand, a quest for an irreversible electrochemical switch may be relevant notably to discriminate between two different molecular populations. In the following, both reversible and irreversible switches are presented. Rhodamine derivatives such as rhodamine 6G (R6G) or rhodamine 101 (R101) are known as standards for quantum yield determinations. They are typical electrofluorochromic probes whose fluorescence can be directly and reversibly modulated by electrochemistry. Yet, in 2001, first electrochemical investigations of the R6G molecule, at a gold electrode surface and in aqueous solution, showed an irreversible reduction process associated to the strong adsorption of the probe on gold material [8]. Nevertheless, modification of the gold electrode surface with 4,4'bipyridine suppressed the adsorption step and led to a quasireversible one-electron process. Under these conditions, spectroelectrochemistry investigations showed a progressive fluorescence decrease under reduction of R6G while the starting luminescence could be recovered under re-oxidation of the R6G reduced form [8]. More recently, the commercially available R101 has been investigated in a polar organic solvent such as acetonitrile: it was established that the radical form of R101, obtained upon mono-electronic electrochemical reduction, was non-fluorescent. Furthermore, fluorescence emission could be recovered by switching the potential to a value positive enough to allow the re-oxidation of the radical, establishing the electrofluorochromic properties of R101. Interestingly, multiple On/Off fluorescence switch events could be observed during several minutes, demonstrating the reversibility of the redox activation/deactivation process (Figure 2) [9]. Perylene diimides are another family of redox-fluorescent organic compounds which are popular thanks to their i) emission in the visible spectrum, ii) their strong absorption and fluorescence, iii) their good thermal, chemical, and photochemical stability. As a representative example, the electrochemical behavior of perylene dipentylimide displayed in dichloromethane two fully reversible one-electron reduction processes at -0.38 V/SCE and -0.58 V/SCE ascribed to the formation of an anion and a dianion, respectively [10]. Under successive electrochemical reduction / oxidation switches (0 → -0.8 → 0 → -0.8 V/SCE etc…) the fluorescence of perylene dipentylimide (ex = 480 nm; em = 540 nm) decreased and increased accordingly. It was further shown that the quenching of fluorescence coincided with the first reduction peak.

In 2006, it was established for the first time that tetrazine derivatives could undergo reversible electro-fluorescent switches thanks to the design of an electrofluorochromic system allowing fast potential switching, thus limiting the time window required for side reactions to proceed [11]. Accordingly, the starting fluorescent chloro-methoxytetrazine became non-emissive when a negative potential value was applied (electroformation of the corresponding anion-radical and dianion). Importantly, most of the fluorescence was restored through the application of a positive potential value (oxidation) regenerating the starting neutral molecule. Such electro-fluorescence switching was found to be reversible upon repetitive cycling.

Compounds based on the BODIPY skeleton (4,4-difluoro-4bora-3a, 4a-diaza-s-indacene) constitute a large class of fluorophores [12] notably for bioimaging applications. The narrow absorption/emission bands in the visible region and high fluorescence quantum yields of the 4,4-difluoro-4-bora-3a,4adiaza-s-indacene core are key characteristics of BODIPY-based bio-labels and sensors, which also benefit from the synthetic tunability of their optical properties, photostability, and solubility. Generally, BODIPY derivatives can be reversibly oxidized into a non-emissive cation radical [13,14]. Among the most relevant modifications of the BODIPY core, the substitution of carbon by nitrogen between the two pyrrolic units shifted the emission in the Triphenylamine derivatives also constitute a family of very promising candidates for applications in electrofluorochromism.

Similarly to BODIPY, redox potential, fluorescence wavelength and intensity can be tuned according to the nature of the substituent on the triphenylamine core. Triphenylamine derivatives generally exhibit moderate to high fluorescence quantum yields in their neutral state whereas their corresponding electrogenerated cation-radicals are non-fluorescent. Though fluorescence switching could be achieved from both electrodonating and electro-withdrawing substituted triphenylamines, donor groups stabilize the radical cation, a feature recommended for the fabrication of electrofluorochromic materials [16].

Interestingly, some molecules display both electrochromic (EC) and electrofluorochromic (EFC) properties opening avenues in the development of multifunctional optoelectronic devices. The example of carbazole, a polycyclic aromatic hydrocarbon consisting of two six-membered benzene rings fused on either side of a five-membered nitrogen-containing ring is interesting. When combined with dibenzofuran, the original carbazoledibenzofuran derivatives display both EC and EFC properties [17]. From the EFC point of view, it was shown that the fluorescence intensity of these carbazole-dibenzofuran derivatives can be reversibly modulated from their redox states. Accordingly, the starting emissive neutral compounds become non-emissive when reduced in the corresponding radical anions.

Thienoviologen are also compounds displaying interesting EC and EFC properties. The first examples of electrofluorochromic -conjugated ionic liquid crystals based on thienoviologens substituted by long alkyl chains were reported in 2014 [18]. Electrofluorescence emission/extinction could be observed several times upon repeated On/Off switching cycles.

Compared to the above-described molecules, Nile Red is mostly known as a lipophilic stain because it is not emissive in most polar solvents, but can be intensely fluorescent when located in a lipid-rich environment. Moreover, Nile Red is a highly fluorescent solvatochromic dye [19]. As the polarity of the solvent increases, the absorption and emission spectra exhibit a bathochromic shift to longer wavelengths. Moreover, it was established that fluorescence quantum efficiency decreases in solvents prone to form hydrogen bonds [20]. Recently, Nile Red was used to track the attachment of microdroplets (filled with this fluorogenic species) onto carbon fibers [21]. More precisely, it was used to fluorescently and electrochemically label an organic-inwater emulsion, where the organic phase was an ionic liquid / toluene mixture. Under these conditions, the fluorescence intensity of the surface bound particles appeared to switch off and on as a function of the applied potential. The two-electron / twoproton reduction of the starting Nile Red compound led to a nonor weakly-fluorescent product. Conversely, the fluorescence intensity of the droplets was found to increase upon oxidation demonstrating the reversibility of the process.

Though reversible electrochemical fluorescence modulation is particularly interesting and sought in the development of electrofluorochromic devices, irreversible processes can also be relevant notably in the discrimination between species internalized and not internalized in lipid vesicles. This approach was recently developed to quantify amounts of cell-penetrating peptides (CPP) that had crossed vesicular lipid membranes [22]. CPP were labeled with NBD (7-nitrobenz-2-oxa-1,3-diazole), a small organic dye displaying a strong fluorescence in non-polar media such as phospholipids. In a typical experience, NBDlabelled peptides were first incubated with large unilamellar vesicles (100 nm in diameter). Then, the fluorescence emitted by non-internalized peptides was cleanly switched off through the electrochemical reduction of the NBD nitro group whereas internalized NBD-CPP remained fluorescent. Accordingly, the NBD-labeled peptides that had been internalized into the vesicles could be easily quantified from the residual fluorescence. Importantly, this discrimination proved to be efficient not only when the lipid membrane was in gel phase, but also in fluid phase, demonstrating the beneficial contribution of electrochemistry compared to chemical reduction (e.g. using dithionite, S2O4 2-as the chemical reducing agent) [23]. As shown in Figure 3, the use of dithionite when vesicular lipid membranes were under fluid gel indiscriminately led to a switch off of all the NBD-labelled peptides (internalized and externalized) while electrochemistry only switched off the remaining external peptides leading thus to a clear discrimination between both peptide populations. This work demonstrated how electrochemically-controlled fluorescence could unambiguously quantify the proportion of internalized peptides without interferences often observed with reducing agents. Interestingly, this work was further successfully extended to NBD-labelled phospholipids (P) in view of discriminating between outer and inner leaflets of vesicles [24]. Indeed, this distinction is impossible with usual fluorescence techniques such as FRAP (Fluorescence Recovery After Photobleaching) or FLIP (Fluorescence Loss In Photobleaching) due to limitations in optics and poor laser beam focusing resolution. Accordingly, three kinds of vesicles made of 1,2-dioleoyl-sn-glycero-3-phosphoglycerol (DOPG) phospholipids (95 mol%), a classical anionic phospholipid used to prepare endosome models, and of a small fraction (5 mol%) of 1,2-dipalmitoyl-sn-glycero-3phosphoethanolamine-N-(7-nitro-2-1,3-benzoxadiazol-4-yl) (NBD-PE) were investigated as a function of the localization of NBD fluorescent lipid: (a) either on the outer, (b) or on the inner, (c) or on both inner/outer leaflets. After verifying both the integrity of the investigated vesicle and its contact with a biotinylated transparent electrode surface made of ITO (Indium Tin Oxide, 10 nm in thickness), the working electrode was polarized at a potential value (-0.9 V vs. Ag/AgCl) sufficiently negative to reduce the NBD nitro group and turn NBD-PE to its non-fluorescent state (Figure 4). First, fluorescence remained unchanged on vesicles labelled on the inner leaflet (A), confirming that the electrochemical reduction cannot be achieved for probes not in direct contact with the electrode surface (e.g. located inside the vesicle). The slight fluorescence loss (around 10 %) was very likely due to the fact that a small fraction of fluorescent phospholipids may have been incorporated in the outer leaflet of the vesicle during their preparation [START_REF] Pautot | Proc. Natl. Acad. Sci[END_REF][START_REF] Yamada | [END_REF]. Conversely, a significant fluorescence intensity decrease was observed in the case of outer-leaflet NBDlabelled vesicles (C). In this case, almost 80% of the fluorescence was quenched after 600 seconds of electrochemical reduction. As a symmetrical situation of the preparation of inner-leaflet-labeled vesicles, outer-leaflet-tagged vesicles finally presented 20% of the fluorescent phospholipids inserted in the inner leaflet, a proportion consistent with reported values [START_REF] Pautot | Proc. Natl. Acad. Sci[END_REF]. A 50% fluorescence extinction was observed on vesicles where both inner and outer leaflets are labeled with NBD-PE.

In conclusion, and contrary to the FLIP technique, these results confirmed that electrochemical bleaching only concerned fluorescent phospholipids localized on the outer-leaflet of the vesicles, thus allowing a discrimination between internal and external lipid bilayers. Furthermore, simulation of the fluorescence loss observed at the equatorial plane of vesicles established that the fluorescence decay was driven by phospholipid-lateral diffusion towards the electrochemical bleached area (i.e the electrode/vesicle interface). This allowed an estimation of the diffusion coefficient of phospholipids located on the outer-leaflet. By analogy with the FLIP technique (Fluorescence Loss In Photobleaching), the FLIE acronym was therefore introduced to qualify this powerful electrochemical quenching method (FLIE for: Fluorescence Loss In Electrobleaching) [24].

Electrochemical switch ON or/and detection of organic fluorogenic molecules

This section is devoted to organic fluorogenic molecules whose fluorescence can be electrochemically switched either directly or upon electrochemical generation of a pH gradient that is an important parameter notably in bioanalysis. Moreover, a focus will be done on pH-responsive probes also possessing redox properties in view of concomitant optical/electrochemical detections (Figure 6). Interestingly, most of the redox fluorogenic molecules investigated in electrochemistry have been used for in situ imaging in fluorescence microscopy. Accordingly, the most relevant fluorogenic organic molecules investigated through electrochemistry and fluorescence microscopy are presented in the following. Resazurin (RZ) [29] Fluorescein [37][38][39][40][41][42][43][44] FFN102 [48,49]

Direct induced fluorescence modulation

A first relevant example of a fluorogenic probe undergoing direct electrochemical fluorescence modulation is resazurin (RZ), also called Alamar Blue, which is a blue (abs= 640 nm) fairly fluorescent redox dye (fluo = 640 nm, fluo= 0.11) [27]. Upon electrochemical reduction (bi-electronic process), RZ is irreversibly converted into resorufin (RF), the latter species absorbing at a lower wavelength than the starting molecule (abs= 570 nm) and exhibiting a brighter fluorescent than RZ (fluo= 0.42 in aqueous solution at pH 10 [28]) [27,29] (Figure 7). Interestingly, resorufin (RF) is itself reducible into dihydroresorufin (DH), through a reversible 2 electrons process, which neither absorbs visible light nor emits fluorescence. In summary, the reversible reduction of RF to DH is responsible for a fluorescence extinction whereas the irreversible reduction of the poorly emitting RZ to RF is accompanied by an emission enhancement due to the large increase in quantum yield. Based on these properties, the combination of electrochemistry with in situ fluorescence confocal microscopy allowed the reactivity investigation of RF and the reconstruction of its relative concentration profile around the electrode. Thanks to this original approach, surface heterogeneity of the electrode surface could also be potentially studied [29]. Combination of Scanning Electrochemical Microscopy (SECM) and fluorescence microscopy was also successfully achieved using resorufin as fluorophore and redox mediator [30].

Recently, a comproportionation reaction between the nonfluorescent dihydroresorufin (DH) and the oxidized weakly fluorescent RZ species was evidenced and led to the highly fluorescent RF thanks to the combination of electrochemistry and fluorescence confocal laser scanning microscopy which allowed a precise mapping of the fluorescence modulation in the vicinity of an electrode surface [31].

Dihydroresorufin was also used as a fluorogenic indicator in fluorescence-enabled electrochemical microscopy (FEEM) a new electrochemical imaging technique [32]. In this work, the authors correlated the fluorescence intensity of the fluorogenic dye and the electrochemical oxidation current during potential sweep experiments on a bipolar electrode. This concept of FEEM was extended to bipolar microelectrode arrays [33]. In this work, fluorescence microscopy and bipolar electrodes were used to reveal electrochemical and electrocatalytic activity on large electrochemical arrays. More precisely, a bipolar electrode was used to directly couple a conventional oxidation reaction, e.g. the oxidation of ferrocene, to a special fluorogenic reduction reaction. The generation of the fluorescent product on the cathodic pole enables one to directly monitor an electrochemical process with optical microscopy. The same group has also investigated 3D diffusion layers by correlating fluorescence images acquired at the reporting pole of an array of bipolar electrodes with the distance between a microelectrode and the other sensing pole of the array. In other words, fluorescence-enabled electrochemical microscopy was employed to image transient concentration profiles of redox species electrogenerated at ultramicroelectrodes [34].

Indirect induced fluorescence modulation

In some cases, the fluorescence of fluorogenic probes can be indirectly activated. This is particularly the case of pHresponsive probes whose fluorescence depends on the pH of the solution. Actually, fluorescein is a typical molecule used by electrochemists when a production/consumption of protons is triggered at the level of an electrode. Fluorescein, which exhibits strong fluorescence in its basic form [35,36], was notably used for the establishment of pH gradients. The approach was developed in the 1990s to monitor, with a wide field fluorescence microscope, reactions producing or consuming protons at the level of an electrode [37][38][39]. In 2002, following the work reported on the use of confocal Raman microscopy to map molecular diffusion at microelectrodes, [40,41] three-dimensional pH gradients were successfully imaged at electrode surfaces using a confocal laser scanning microscope that improved the axial and lateral resolutions. [42] In this case, the local pH changes caused by the electrochemical reduction of benzoquinone to hydroquinone were evidenced thanks to the presence of fluorescein, which displayed a pH-dependent fluorescent signal. Confocal pH measurements in solution were also performed later in the presence of a scanning electrochemical microscope [43].

Recently, fluorescein, which displays exalted fluorescence under basic conditions, was used in an original sensing concept allowing delocalized detection of biorelevant molecules (1,4naphthoquinone and dopamine) at bipolar electrodes. The approach is based on the generation of local pH gradients which allow the indirect detection of redox-active molecules due to a modulation of the fluorescence intensity in the vicinity of the bipolar electrode. Accordingly, an oxidation is taking place at the anodic pole of the bipolar electrode, while protons are consumed at the cathodic extremity. The consumption of H + locally increases the pH, leading to enhanced fluorescence compared to the background intensity [44].

A similar approach was recently developed for visualizing cancer cell surface glycoprotein [45]. In this case, the device consisted in an anodic sensing cell and a cathodic reporting cell connected together by a screen-printed electrode patterned on poly(ethylene terephthalate) membrane. In the sensing cell, anti-MUC1 antibody immobilized on a chitosan-multiwalled carbon nanotube (CSMWCNT)-modified anodic bipolar electrode channel was used for capturing mucin-1 (MUC1) or MCF-7 cancer cells. Then ferrocene (Fc)-labeled mucin 1 aptamers were introduced through hybridization. Under an applied voltage, the ferrocene was oxidized and the electroactive molecules of 1,4-benzoquinone (BQ) in the cathodic reporting cell were reduced. This produced a strongly basic 1,4-benzoquinone anion radical (BQ •-), which turned on the fluorescence of pH-responsive molecules of (2-(2-(4-hydroxystyryl)-6-methyl-4Hpyran-4-ylidene)malononitrile) (SPM) coexisting in the cathode reporting cell for both spectrophotometric detection and imaging [45].

Resazurin (RZ)

Weakly fluorescent Red.

Red. Ox.

Resorufin (RF)

Highly fluorescent

Dihydroresorufin (DH)

Non-fluorescent

O N O O _ O N O + _ O O _ O N O O H _ _

Fluorogenic coumarin derivatives also possessing redox properties for concomitant optical/electrochemical detection of exocytotic release

The design of redox organic fluorescent or fluorogenic molecules requires a particular attention when bioanalysis perspectives are envisioned. Within this context, important parameters such as (i) the nature of the investigated biological mechanism and (ii) the ability of the fluorescent/fluorogenic probe to be uptaken by the investigated cells must be taken into consideration before synthesizing the probe. The quest for pH-responsive fluorogenic probes displaying independent redox and fluorescent properties is a typical example in the case of vesicular exocytosis [46]. Accordingly, the development of unique dual electrofluorescent probes was carried out to investigate this important biological mechanism through the combination of amperometry and total internal reflection fluorescence microscopy (TIRFM). On the one hand, amperometry provides quantitative information on the amount of secreted electroactive molecules, with sub-millisecond temporal resolution that matches the fast kinetics of exocytotic release. On the other hand, TIRFM is capable of revealing the distribution of secretory vesicles within a single cell, with tens of nanometers spatial resolution, and can track their location during the exocytotic process in real-time. Historically, in the context of this coupling strategy, single exocytosis events were first investigated using independent electrochemical (serotonine) and fuorescent (GFP) probes [47,48]. However, the use of two distinct probes prevented a quantitative analysis of the release ignoring the precise relative and absolute proportion of both probes in secretory vesicles (only 20% of correlated simultaneous optical and electrochemical signals were recorded concomitantly). Accordingly, a unique dual electrofluorogenic probe, possessing a coumarin backbone, was recently introduced [49,50]. This probe called FFN102 belongs to the family of fluorescent false neurotransmitters (FFN) (i.e. a synthesized analogue of a biogenic neurotransmitter). Under physiological conditions (pH = 7.4), optimal excitation and emission wavelengths of this probe lie at 371 nm and 456 nm, respectively. This molecule is a pH-responsive fluorescent/electroactive dual probe and its fluorescence intensity increased with pH especially in a pH 5.3 to pH 7.3 window. Accordingly, during exocytosis, FFN102 diffused out from the acidic intravesicular microenvironment to the neutral extracellular space, leading to fluorescent emissions and to the electrochemical oxidation signals that were simultaneously collected at the transparent ITO electrode surface (Figure 8). However, the high oxidation potential value of FFN102 at ITO electrodes (+ 900 mV vs. Ag/AgCl) induced electrical noise restricting the advantages of this material. Very recently, a new dual electrofluorescent probe called FFN42 was designed [51]. This new electrofluorogenic probe actually acted as an optical/electrochemical dual reporter in the coupling method, especially with longer excitation/emission wavelengths and/or lower oxidation potential. Furthermore, pKas of the three acid functions (hydroxyl groups in positions 6 and 7, respectively; amonium group in position 4) were determined (Figure 6). From the electrochemical point of view, and compared to FFN102, the FFN42 probe is oxidized at +0.45 V vs. Ag/AgCl on ITO electrodes. The electrochemical detection of FFN42 at a less anodic potential value than FFN102 (+ 900 mV vs. Ag/AgCl) is ascribed to the replacement of the chlorine group by a second hydroxy group in position 6 of the coumarin nucleus. This stronger electrondonating group is responsible of promoting easier oxidation, i.e. at lower oxidation potential values, contributing in the decrease of the electrical perturbations.

Summary and Outlook.

This short overview of the electrochemically-promoted fluorescence modulation of organic fluorescent or fluorogenic molecules shows that this emerging field is very active. This is clearly illustrated through the smart uses of canonical molecules or the design of new ones. Within this context, it is also noteworthy that instrumental setups and methodologies aimed at combining/synchronizing electrochemical commands and fluorescence microscopy/spectroscopy play a crucial role to allow the collection of time-and space-resolved information on various systems. The unique sensitivity of luminescent techniques requires they proceed at specific locations and within well-defined time windows and such properties perfectly match the general features of electrochemistry, making the design of electrochemically-controlled luminescent molecules essential in the future.

  Manon Guille-Collignon achieved her Ph.D. in 2005 in the field of physico-chemisty of exocytosis and nitric oxide effects on neurovascular coupling, from the Université Pierre & Marie Curie, Paris 6, France. During her postdoctoral research at University of Warwick in 2005, in Coventry, UK (Marie-Curie Fellowship; direction of Dr. J.V. Macpherson), she worked on the electrochemical properties of single wall carbon nanotubes. She reached an Assistant Professor position in September 2006 in Université Pierre & Marie Curie, now Sorbonne Université. Manon's research interests focus on bioelectrochemistry, microfluidics and bioenergetics via combined fluorescent and electrochemical measurement to study cellular secretion, oxidative stress and photosynthesis. Jérôme Delacotte received his PhD in soft matter physics from Université Paris VII, (Paris, France) in 2009. His PhD work focused on physical chemistry and dynamics of thin liquid films, under the supervision of Dr. Langevin and Dr. Rio. Then, as a postoctoral research fellow at the Laboratoire de Physique Statistique (Paris, France), he opened to biophysics of lipid droplets and lipid membranes. In 2013, he was appointed associate professor in chemistry at the École Normale Supérieure in Paris. Jérôme is interested in multidisciplinary approaches at the physical-chemical-biological interface to study cellular organelles dynamics and photosynthesis. Frédéric Lemaître did a joint thesis between the Université de Bourgogne (France) and the Université de Sherbrooke (Canada). He received a double Ph. D. degree in 2003. His Ph. D. research dealt with electrochemistry of palladium clusters and their reactivity towards alkyl-and acyl halides. After doing postdoctoral research in the electrochemistry group at the Ecole Normale Supérieure de Paris (palladium catalysis -Heck reaction), he has been obtained an Assistant Professor position at the Université Pierre et Marie Curie -Sorbonne Université in 2004. His research works concern bioelectrochemistry and especially the detection of secretion processes at the single cell level. Full professor in Sorbonne Université since 2020, he is also interested in the production of bioelectricity by using photosynthesis. Eric Labbé received his PhD in Analytical Chemistry in 1992 from Université Pierre et Marie Curie (now Sorbonne Université, Paris, France). He studied electrosynthetic mechanisms in the group of Jacques Périchon on nickel and cobalt-catalyzed processes. Since 2006, he is Professor at Sorbonne Université and his research topics concern the electroanalytical determination of reaction mechanisms, drug metabolism and drug vectorization. Olivier Buriez is a Research Director at CNRS. During his PhD at the École Normale Supérieure (Paris, France) and postdoctoral research at the Berkeley lab. (LBNL, Univ. of California, USA), he investigated the structure-activity relationships and mechanistic aspects of (bio)-organometallic compounds by electrochemistry. Since 1999, when he joined the CNRS, his research interests center on the electroanalytical determination of reaction mechanisms to provide answers in fundamental fields such as catalysis, synthesis, analytical chemistry and molecular biology. Within this context, part of his work also includes the development of analytical methodologies and devices combining electrochemical and luminescence techniques. From 2015 to 2020, he was elected Officer of the Molecular Electrochemistry Division of the International Society of Electrochemistry (ISE).

Figure 1 .

 1 Figure 1. Chemical structures of some important redox-organic fluorescent molecules whose fluorescence can be directly modulated by electrochemistry.

Figure 2 .

 2 Figure 2. Direct reversible electrochemical switch of rhodamine 101 (adapted from ref. [9]).

  Such aza-BODIPY derivatives are nice examples of NIR emissive electrofluorochromic compounds[15].

Figure 3 .

 3 Figure 3. Compared to the chemical switching off of NBD-labelled peptides by dithionite (a chemical reducer), the electrochemical procedure allowed a clear and clean discrimination between internalized and non-internalized CPP under fluid phase [22]. (a) With Tat (peptide sequence: YGRKKRRQRRR) 30% of the peptide was considered as internalized (unpublished results).

Figure 4 .

 4 Figure 4. Selective electrochemical bleaching of the outer leaflet of NBDlabelled micrometric vesicles. The fluorescence intensity was monitored at the equatorial plane of the vesicle (i.e. out of the reduction/bleached area) by taking confocal fluorescence microscopy images every 30 seconds. As shown in Figure 5, three different behaviors were obtained depending on the NBD fluorescent probe localization.

Figure 5 .

 5 Figure 5. Fluorescence variations as a function of polarization time (E= -0.9 V/Ag/AgCl) obtained at a biotinylated ITO electrode surface for (•) inner (N=5), (•) inner/outer (N=13), and (•) outer (N=5) leaflet NBD-labeled DOPG GUVs. The numbers N between parentheses are those of the GUV analyzed in each case. The error bars represent the standard error (S.E.).

Figure 6 .

 6 Figure 6. Chemical structures of some relevant organic fluorogenic molecules recently used in molecular electrochemistry.

Figure 7 .

 7 Figure 7. Resazurin-Resorufin: a typical redox-responsive fluorogenic couple (adapted from ref. [29]). Experimental conditions: aqueous solution; pH = 10.

Figure 8 .

 8 Figure 8. Correlation of amperometric and fluorescence information for a single exocytotic event of FFN102-stained BON N13 cells over an ITO microdevice. Top: An exocytotic event appeared as the current spike in electrochemical detection. Bottom: Sequential pseudocolor TIRFM images of a single exocytotic event viewed as a flash of fluorescence. Scale bar = 500 nm. (Adapted from ref. [49]).

Acknowledgements

This work has been supported in part by CNRS (UMR 8640), Ecole Normale Supérieure (PSL Research University), Sorbonne Université, the French Ministry of Research, "Emergences Ville de Paris 2014" Grant and Institut Universitaire de France Junior 2015 Fellowship Program.