diff --git a/BipedRobot/ZMPBipedRobot/docs/2. Optimisation Process/objectives_values_Fs50.txt b/BipedRobot/ZMPBipedRobot/docs/2. Optimisation Process/objectives_values_Fs50.txt
deleted file mode 100644
index 1f4f8aee6..000000000
--- a/BipedRobot/ZMPBipedRobot/docs/2. Optimisation Process/objectives_values_Fs50.txt	
+++ /dev/null
@@ -1,3 +0,0 @@
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diff --git a/BipedRobot/ZMPBipedRobot/docs/2. Optimisation Process/solutions_Fs50.txt b/BipedRobot/ZMPBipedRobot/docs/2. Optimisation Process/solutions_Fs50.txt
deleted file mode 100644
index a8fe1322e..000000000
--- a/BipedRobot/ZMPBipedRobot/docs/2. Optimisation Process/solutions_Fs50.txt	
+++ /dev/null
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-0.12675,0.9,0.1,0.05,0.0,0.05
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-0.12675,0.9,0.1,0.15,0.0,0.03
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-0.12675,0.9,0.1,0.2,0.0,0.04
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-0.12675,0.9,0.15000000000000002,0.05,0.0,0.03
-0.12675,0.9,0.15000000000000002,0.05,0.0,0.04
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-0.12675,0.9,0.15000000000000002,0.1,0.0,0.02
-0.12675,0.9,0.15000000000000002,0.1,0.0,0.03
-0.12675,0.9,0.15000000000000002,0.1,0.0,0.04
-0.12675,0.9,0.15000000000000002,0.1,0.0,0.05
-0.12675,0.9,0.15000000000000002,0.15,0.0,0.02
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-0.12675,0.9,0.15000000000000002,0.15,0.0,0.04
-0.12675,0.9,0.15000000000000002,0.15,0.0,0.05
-0.12675,0.9,0.15000000000000002,0.2,0.0,0.02
-0.12675,0.9,0.15000000000000002,0.2,0.0,0.03
-0.12675,0.9,0.15000000000000002,0.2,0.0,0.04
-0.12675,0.9,0.15000000000000002,0.2,0.0,0.05
-0.12675,1.0,0.05,0.05,0.0,0.02
-0.12675,1.0,0.05,0.05,0.0,0.03
-0.12675,1.0,0.05,0.05,0.0,0.04
-0.12675,1.0,0.05,0.05,0.0,0.05
-0.12675,1.0,0.05,0.1,0.0,0.02
-0.12675,1.0,0.05,0.1,0.0,0.03
-0.12675,1.0,0.05,0.1,0.0,0.04
-0.12675,1.0,0.05,0.1,0.0,0.05
-0.12675,1.0,0.05,0.15,0.0,0.02
-0.12675,1.0,0.05,0.15,0.0,0.03
-0.12675,1.0,0.05,0.15,0.0,0.04
-0.12675,1.0,0.05,0.15,0.0,0.05
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-0.12675,1.0,0.05,0.2,0.0,0.03
-0.12675,1.0,0.05,0.2,0.0,0.04
-0.12675,1.0,0.05,0.2,0.0,0.05
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-0.12675,1.0,0.15000000000000002,0.2,0.0,0.05
diff --git a/Project.toml b/Project.toml
index 55b90e7c6..2adcf0692 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "Dionysos"
 uuid = "d92c97cf-b87d-42c1-a9c0-25df00b4d958"
-authors = ["Raphael Jungers <raphael.jungers@uclouvain.be>", "Antoine Aspeel <antoine.aspeel@uclouvain.be>", "Guillaume Berger <guillaume.berger@uclouvain.be>", "Julien Calbert <julien.calbert@uclouvain.be>", "Mahsa Farjadnia <mahsa.farjadnia@uclouvain.be>", "Benoît Legat <benoit.legat@uclouvain.be>", "Zheming Wang <zheming.wang@uclouvain.be>", "Lucas N. Egidio <lucas.egidio@uclouvain.be>", "Adrien Banse <adrien.banse@uclouvain.be>", "Somya Singh <somya.singh@uclouvain.be>"]
+authors = ["Raphael Jungers <raphael.jungers@uclouvain.be>", "Antoine Aspeel <antoine.aspeel@uclouvain.be>", "Guillaume Berger <guillaume.berger@uclouvain.be>", "Julien Calbert <julien.calbert@uclouvain.be>", "Virginie Debauche <virginie.debauche@uclouvain.be>", "Mahsa Farjadnia <mahsa.farjadnia@uclouvain.be>", "Benoît Legat <benoit.legat@uclouvain.be>", "Zheming Wang <zheming.wang@uclouvain.be>", "Lucas N. Egidio <lucas.egidio@uclouvain.be>", "Adrien Banse <adrien.banse@uclouvain.be>", "Somya Singh <somya.singh@uclouvain.be>"]
 version = "0.0.1"
 
 [deps]
diff --git a/docs/src/reference/Optim.md b/docs/src/reference/Optim.md
index 3fe632b67..80734928c 100644
--- a/docs/src/reference/Optim.md
+++ b/docs/src/reference/Optim.md
@@ -1,14 +1,14 @@
 # Optim 
 
-This folder contains all the different solvers that can be used.
+This folder contains all the different (abstraction-based or not) solvers that can be used. Note that all the solvers are defined using the MathOptInterface framework: for each solver, we define a subtype of  [`AbstractOptimizer`](https://jump.dev/MathOptInterface.jl/stable/reference/models/#MathOptInterface.AbstractOptimizer) and implement the [`Optimize!`](https://jump.dev/MathOptInterface.jl/stable/reference/models/#MathOptInterface.optimize!) function.
 
 ## Abstraction-based solvers
 ```@docs
-Dionysos.Optim.Abstraction.LazyAbstraction.Optimizer
+Dionysos.Optim.Abstraction.NaiveAbstraction.Optimizer
 Dionysos.Optim.Abstraction.EllipsoidsAbstraction.Optimizer
 Dionysos.Optim.Abstraction.HierarchicalAbstraction.Optimizer
+Dionysos.Optim.Abstraction.LazyAbstraction.Optimizer
 Dionysos.Optim.Abstraction.LazyEllipsoidsAbstraction.Optimizer
-Dionysos.Optim.Abstraction.NaiveAbstraction.Optimizer
 ```
 
 ## Other solvers
diff --git a/docs/src/reference/Problem.md b/docs/src/reference/Problem.md
index 4815908ed..27e54f217 100644
--- a/docs/src/reference/Problem.md
+++ b/docs/src/reference/Problem.md
@@ -1,9 +1,14 @@
 # Problem 
 
-This folder contains structures that are used to encode which kind of problem you want to solve.
+This folder contains structures that are used to encode which kind of problem you want to solve. Each problem is encoded as a ProblemType.
 
 ```@docs
 Dionysos.Problem.ProblemType
+```
+
+So far, two types of problems have been considered: the reach-avoid optimal control problems and the safety control problems.
+
+```@docs
 Dionysos.Problem.OptimalControlProblem
 Dionysos.Problem.SafetyProblem
 ```
\ No newline at end of file
diff --git a/docs/src/reference/Symbolic.md b/docs/src/reference/Symbolic.md
index 2412b4188..f6e6b2a50 100644
--- a/docs/src/reference/Symbolic.md
+++ b/docs/src/reference/Symbolic.md
@@ -1,6 +1,10 @@
 # Symbolic 
 
-This folder contains different methods to build an abstraction.
+This folder contains the data structures needed to encode the different abstractions.
+
+```@docs
+Dionysos.Symbolic.SymbolicModel
+```
 
 ```@docs
 Dionysos.Symbolic.SymbolicModelList
diff --git a/docs/src/reference/System.md b/docs/src/reference/System.md
index a336fcd47..3c2fbea04 100644
--- a/docs/src/reference/System.md
+++ b/docs/src/reference/System.md
@@ -3,23 +3,36 @@
 This folder contains different ways to define systems, for instance to encode a controller.
 
 ## Control system
+
+Each control system should be implemented as a ControlSystem.
+
 ```@docs
+Dionysos.System.ControlSystem
+```
+
+So far, we have implemented a few examples of control systems: 
+
+```@docs
+Dionysos.System.SimpleSystem
 Dionysos.System.ControlSystemGrowth
 Dionysos.System.ControlSystemLinearized
-Dionysos.System.SimpleSystem
 Dionysos.System.EllipsoidalAffineApproximatedSystem
-Dionysos.System.AffineApproximationDiscreteSystem
-Dionysos.System.SymbolicSystem
 ```
 
 ## Controller 
+So far, the abstraction-based methods that we use define either piecewise-constant or piecewise-affine controllers.
+
 ```@docs
 Dionysos.System.ConstantController
 Dionysos.System.AffineController
 ```
 
-## SHOULD BE REMOVED 
+## Trajectories 
 ```@docs
 Dionysos.System.DiscreteTrajectory
 Dionysos.System.ContinuousTrajectory
+Dionysos.System.HybridTrajectory
+Dionysos.System.Trajectory
+Dionysos.System.Control_trajectory
+Dionysos.System.Cost_control_trajectory
 ```
\ No newline at end of file
diff --git a/docs/src/reference/Utils.md b/docs/src/reference/Utils.md
index a16c9c24a..bf3ade174 100644
--- a/docs/src/reference/Utils.md
+++ b/docs/src/reference/Utils.md
@@ -2,11 +2,6 @@
 
 This folder contains all the auxiliary functions needed.
 
-## Ellipsoids 
-
-```@docs
-```
-
 ## Functions 
 
 ```@docs
diff --git a/src/domain/continuous_domain.jl b/src/domain/continuous_domain.jl
index 4c504a9a8..90c7ff81a 100644
--- a/src/domain/continuous_domain.jl
+++ b/src/domain/continuous_domain.jl
@@ -3,7 +3,7 @@ abstract type ContinuousDomain{N, T} <: DomainType{N, T} end
 """
     ContinuousUnboundedDomain{N,T}
 
-Struct for a basic unbounded continuous domain
+Struct for a basic unbounded continuous domain.
 """
 struct ContinuousUnboundedDomain{N, T} <: ContinuousDomain{N, T}
     orig::SVector{N, T}
@@ -12,7 +12,7 @@ end
 """
     ContinuousBoundedDomain{N,T,B}
 
-Struct for a basic bounded continuous domain
+Struct for a basic bounded continuous domain.
 """
 struct ContinuousBoundedDomain{N, T, B} <: ContinuousDomain{N, T}
     orig::SVector{N, T}
@@ -22,7 +22,7 @@ end
 """
     ContinuousBoundedEllipsoidDomain{N,T,S<:Grid{N,T}}
 
-Struct for a basic bounded continuous domain formed by a finite number of ellipsoids
+Struct for a basic bounded continuous domain formed by a finite number of ellipsoids.
 """
 struct ContinuousBoundedEllipsoidDomain{N, T, B, E} <: ContinuousDomain{N, T}
     orig::SVector{N, T}
diff --git a/src/domain/custom_domain.jl b/src/domain/custom_domain.jl
index 9d54ba218..ca39d607a 100644
--- a/src/domain/custom_domain.jl
+++ b/src/domain/custom_domain.jl
@@ -1,7 +1,7 @@
 """
     CustomList{N,T} <: DomainType{N,T}
 
-Struct for a custom generic domain
+Struct for a custom generic domain.
 """
 
 struct CustomList{N, T} <: DomainType{N, T}
diff --git a/src/domain/domain_list.jl b/src/domain/domain_list.jl
index 2a37edeec..1b15f3959 100644
--- a/src/domain/domain_list.jl
+++ b/src/domain/domain_list.jl
@@ -7,7 +7,7 @@ end
 """
     DomainList{N,T,S<:Grid{N,T}}
 
-Struct for a basic domain based on a `Grid`
+Struct for a basic domain based on a `Grid`.
 """
 struct DomainList{N, T, S <: Grid{N, T}} <: DomainType{N, T}
     grid::S
@@ -17,7 +17,7 @@ end
 """
     DomainList(grid::S) where {N,S<:Grid{N}}
 
-Return a new DomainList
+Return a new `DomainList`.
 """
 function DomainList(grid::S) where {N, S <: Grid{N}}
     return DomainList(grid, Set{NTuple{N, Int}}())
diff --git a/src/domain/general_domain.jl b/src/domain/general_domain.jl
index 6ee72350d..d49126e9c 100644
--- a/src/domain/general_domain.jl
+++ b/src/domain/general_domain.jl
@@ -3,7 +3,7 @@ using StaticArrays, Plots
 """
     RectangularObstacles{VT} <: AbstractSet{VT}
 
-Struct for a rectangular domain with rectangular obstacles
+Struct for a rectangular domain with rectangular obstacles.
 """
 struct RectangularObstacles{VT} <: AbstractSet{VT}
     X::UT.HyperRectangle{VT}
@@ -36,7 +36,7 @@ _fit_grid(elems::Set, grid, nx, fit) = elems
 """
     GeneralDomainList{N,E<:AbstractSet{NTuple{N,Int}},T,S<:Grid{N,T},F} <: DomainType{N,T}
 
-Struct for a rectangular domain with rectangular obstacles
+Struct for a rectangular domain with rectangular obstacles.
 """
 struct GeneralDomainList{N, E <: AbstractSet{NTuple{N, Int}}, T, S <: Grid{N, T}, F} <:
        DomainType{N, T}
diff --git a/src/domain/grid.jl b/src/domain/grid.jl
index f141892d2..79860e6b5 100644
--- a/src/domain/grid.jl
+++ b/src/domain/grid.jl
@@ -113,7 +113,7 @@ end
     GridEllipsoidalRectangular{N,T} <: Grid{N,T}
 
 Uniform grid on rectangular space `rect`, centered at `orig` and with steps set by the vector `h`.
-Cells are (possibly overlapping) ellipsoids defined at each grid point `c` as `(x-c)'P(x-c) ≤ 1`
+Cells are (possibly overlapping) ellipsoids defined at each grid point `c` as `(x-c)'P(x-c) ≤ 1`.
 """
 struct GridEllipsoidalRectangular{N, T} <: Grid{N, T}
     orig::SVector{N, T}
diff --git a/src/optim/abstraction/ellipsoids_abstraction.jl b/src/optim/abstraction/ellipsoids_abstraction.jl
index c29a4201d..84d95d140 100644
--- a/src/optim/abstraction/ellipsoids_abstraction.jl
+++ b/src/optim/abstraction/ellipsoids_abstraction.jl
@@ -16,7 +16,7 @@ const PR = DI.Problem
 """
     Optimizer{T} <: MOI.AbstractOptimizer
 
-Ellispoids abtraction solver
+Abstraction-based solver for which the domain is covered with ellipsoidal cells, independently of the control task.
 """
 mutable struct Optimizer{T} <: MOI.AbstractOptimizer
     concrete_problem::Union{Nothing, PR.OptimalControlProblem}
diff --git a/src/optim/abstraction/hierarchical_abstraction.jl b/src/optim/abstraction/hierarchical_abstraction.jl
index 084f3d7c9..1a86ce8f1 100644
--- a/src/optim/abstraction/hierarchical_abstraction.jl
+++ b/src/optim/abstraction/hierarchical_abstraction.jl
@@ -19,7 +19,7 @@ using ..LazyAbstraction
 """
     Optimizer{T} <: MOI.AbstractOptimizer
     
-Hierarchical abstraction solver
+Abstraction-based solver for which the domain is initially partioned into coarse hyper-rectangular cells, which are iteratively locally smartly refined with respect to the control task.
 """
 mutable struct Optimizer{T} <: MOI.AbstractOptimizer
     concrete_system::Union{Nothing, Any}
diff --git a/src/optim/abstraction/lazy_abstraction.jl b/src/optim/abstraction/lazy_abstraction.jl
index 80ec52859..811cfbaac 100644
--- a/src/optim/abstraction/lazy_abstraction.jl
+++ b/src/optim/abstraction/lazy_abstraction.jl
@@ -16,7 +16,7 @@ const PR = DI.Problem
 """
     Optimizer{T} <: MOI.AbstractOptimizer
 
-Solver from Lazy Abstraction
+Abstraction-based solver for which the hyper-rectangular abstraction and the controller are co-designed to reduce the computation cost of the abstraction.
 """
 mutable struct Optimizer{T} <: MOI.AbstractOptimizer
     concrete_system::Union{Nothing, Any}
diff --git a/src/optim/abstraction/lazy_ellipsoids_abstraction.jl b/src/optim/abstraction/lazy_ellipsoids_abstraction.jl
index b1241c939..6fb7683c6 100644
--- a/src/optim/abstraction/lazy_ellipsoids_abstraction.jl
+++ b/src/optim/abstraction/lazy_ellipsoids_abstraction.jl
@@ -22,7 +22,7 @@ global NI = nothing
 """
     Optimizer{T} <: MOI.AbstractOptimizer
 
-Lazy ellipsoids abstraction solver
+Abstraction-based solver using the lazy abstraction method with ellipsoidal cells.
 """
 mutable struct Optimizer{T} <: MOI.AbstractOptimizer
     concrete_problem::Union{Nothing, PR.OptimalControlProblem}
diff --git a/src/optim/abstraction/naive_abstraction.jl b/src/optim/abstraction/naive_abstraction.jl
index 31cbefa06..21648ae16 100644
--- a/src/optim/abstraction/naive_abstraction.jl
+++ b/src/optim/abstraction/naive_abstraction.jl
@@ -15,7 +15,7 @@ using JuMP
 """
     Optimizer{T} <: MOI.AbstractOptimizer
 
-Naive abstraction solver
+Solver based on the classical abstraction method (used for instance in SCOTS) for which the whole domain is partioned into hyper-rectangular cells, independently of the control task.
 """
 mutable struct Optimizer{T} <: MOI.AbstractOptimizer
     concrete_problem::Union{Nothing, PR.ProblemType}
diff --git a/src/optim/bemporad_morari.jl b/src/optim/bemporad_morari.jl
index c80c4c295..5e327da57 100644
--- a/src/optim/bemporad_morari.jl
+++ b/src/optim/bemporad_morari.jl
@@ -20,7 +20,7 @@ using FillArrays, MathematicalSystems, HybridSystems, JuMP, SemialgebraicSets, P
 """
     Optimizer{T} <: MOI.AbstractOptimizer
 
-Bemporad Morari solver
+Bemporad Morari solver: Optimal control of hybrid systems via a predictive control scheme using mixed integer quadratic programming (MIQP) online optimization procedures.
 """
 mutable struct Optimizer{T} <: MOI.AbstractOptimizer
     continuous_solver::Any
diff --git a/src/optim/branch_and_bound.jl b/src/optim/branch_and_bound.jl
index 0f41d3c19..353cb283a 100644
--- a/src/optim/branch_and_bound.jl
+++ b/src/optim/branch_and_bound.jl
@@ -16,7 +16,7 @@ using HybridSystems
 """
     Optimizer{T} <: MOI.AbstractOptimizer
 
-Branch and bound solver
+Branch and bound solver: Optimal control of hybrid systems via a predictive control scheme combining a branch and bound algorithm that can refine Q-functions using Lagrangian duality.
 """
 mutable struct Optimizer{T} <: MOI.AbstractOptimizer
     continuous_solver::Any
diff --git a/src/problem/problems.jl b/src/problem/problems.jl
index cec96e279..f99d26d5c 100644
--- a/src/problem/problems.jl
+++ b/src/problem/problems.jl
@@ -13,12 +13,12 @@ The structure
     OptimalControlProblem{S, XI, XT, XC, TC, T}
 
 encodes an optimal control problem where 
-`S` is the system,
-`XI` is the initial set, 
-`XT` is the target set,
-`XC` is the state cost,
-`TC` is transistion cost and
-`T` is the number of allowed time steps
+- `S` is the system,
+- `XI` is the initial set, 
+- `XT` is the target set,
+- `XC` is the state cost,
+- `TC` is transistion cost and
+- `T` is the number of allowed time steps
 """
 struct OptimalControlProblem{S, XI, XT, XC, TC, T <: Real} <: ProblemType
     system::S
@@ -35,10 +35,10 @@ The structure
     SafetyProblem{S, XI, XS, T}
 
 encodes a safety problem where
-`S` is the system,
-`XI` is the initial set, 
-`XS` is the safe set and
-`T` is the number of allowed time steps
+- `S` is the system,
+- `XI` is the initial set, 
+- `XS` is the safe set and
+- `T` is the number of allowed time steps
 """
 struct SafetyProblem{S, XI, XS, T <: Real} <: ProblemType
     system::S
diff --git a/src/symbolic/lazy_symbolic.jl b/src/symbolic/lazy_symbolic.jl
index 1d8de6b2c..7149948c6 100644
--- a/src/symbolic/lazy_symbolic.jl
+++ b/src/symbolic/lazy_symbolic.jl
@@ -1,7 +1,7 @@
 """
     LazySymbolicModel{N, M, S1 <: DO.DomainType{N}, S2 <: DO.DomainType{M}, A} <: SymbolicModel{N, M}
 
-TO ADD
+is one implementation of the `SymbolicModel` type for the lazy abstraction-based methods, i.e. when a subset of the domain is partitioned/covered.
 """
 mutable struct LazySymbolicModel{N, M, S1 <: DO.DomainType{N}, S2 <: DO.DomainType{M}, A} <:
                SymbolicModel{N, M}
diff --git a/src/symbolic/symbolicmodel.jl b/src/symbolic/symbolicmodel.jl
index 25ce7a48d..15eb98e7c 100644
--- a/src/symbolic/symbolicmodel.jl
+++ b/src/symbolic/symbolicmodel.jl
@@ -1,11 +1,16 @@
 using Plots, Colors
 
+"""
+    SymbolicModel{N, M}
+
+is the abtract type which defines a symbolic model.
+"""
 abstract type SymbolicModel{N, M} end
 
 """
     SymbolicModelList{N, M, S1 <: DO.DomainType{N}, S2 <: DO.DomainType{M}, A} <: SymbolicModel{N, M}
     
-TO ADD
+is one implementation of the `SymbolicModel` type for classical abstraction-based methods, i.e. when the whole domain is partitioned/covered.
 """
 mutable struct SymbolicModelList{N, M, S1 <: DO.DomainType{N}, S2 <: DO.DomainType{M}, A} <:
                SymbolicModel{N, M}
diff --git a/src/system/controller.jl b/src/system/controller.jl
index b415a981f..d541a1f13 100644
--- a/src/system/controller.jl
+++ b/src/system/controller.jl
@@ -8,7 +8,7 @@ end
 """
     ConstantController{T, VT}
 
-encodes a constant state-dependent controller of the κ(x) = c
+encodes a constant state-dependent controller of the κ(x) = c.
 """
 struct ConstantController{T <: Real, VT <: AbstractVector{T}} <: Controller
     c::VT
@@ -24,7 +24,7 @@ end
 """
     AffineController{T, MT, VT1, VT2}
 
-encodes an affine state-dependent controller of the κ(x) = K*(x-c)+ℓ
+encodes an affine state-dependent controller of the κ(x) = K*(x-c)+ℓ.
 """
 struct AffineController{
     T <: Real,
diff --git a/src/system/controlsystem.jl b/src/system/controlsystem.jl
index 7e64a4f6e..ad3ec4b62 100644
--- a/src/system/controlsystem.jl
+++ b/src/system/controlsystem.jl
@@ -1,4 +1,10 @@
+"""
+The structure
+
+    ControlSystem{N, T}
 
+is the abstract type that defines a control system.
+"""
 abstract type ControlSystem{N, T} end
 using IntervalArithmetic
 function get_f_eval(sys)
@@ -23,7 +29,7 @@ end
 """
     ControlSystemGrowth{N, T, F1 <: Function, F2 <: Function, F3 <: Function} <: ControlSystem{N, T}
 
-TO ADD
+is one implementation of the `ControlSystem` type for which we have a growth bound function.
 """
 struct ControlSystemGrowth{N, T, F1 <: Function, F2 <: Function, F3 <: Function} <:
        ControlSystem{N, T}
@@ -71,7 +77,7 @@ end
 """
     ControlSystemLinearized{N, T, F1 <: Function, F2 <: Function, F3 <: Function, F4 <: Function, } <: ControlSystem{N, T}
 
-TO ADD
+is one implementation of the `ControlSystem` type for which we have linearized the system map.
 """
 struct ControlSystemLinearized{
     N,
@@ -164,7 +170,7 @@ end
 """
     SimpleSystem{N, T, F <: Function, F2} <: ControlSystem{N, T}   
 
-TO ADD
+is one implementation of the `ControlSystem` type. 
 """
 struct SimpleSystem{N, T, F <: Function, F2} <: ControlSystem{N, T}
     tstep::Float64
@@ -183,7 +189,7 @@ end
 """
     EllipsoidalAffineApproximatedSystem{}   
     
-TO ADD
+is a system whose dynamics is a noisy constrained affine control discrete system whose cells are ellipsoids, with a bound on the Lipschitz constant.
 """
 struct EllipsoidalAffineApproximatedSystem{}
     dynamics::Dict{UT.Ellipsoid, NoisyConstrainedAffineControlDiscreteSystem}
@@ -243,11 +249,7 @@ end
 ############################################
 # Affine discrete time system approximaton with its Lypschitz constant and the region of validity
 # of the Lipschitz constant
-"""
-    AffineApproximationDiscreteSystem      
-    
-TO ADD
-"""
+
 struct AffineApproximationDiscreteSystem #<: ControlSystem
     constrainedAffineSys::NoisyConstrainedAffineControlDiscreteSystem
     L::Any
@@ -266,11 +268,6 @@ end
 ############################################
 ############################################
 
-"""
-    SymbolicSystem{}  
-    
-TO ADD
-"""
 struct SymbolicSystem{}
     fsymbolicT::Any
     fsymbolic::Any
diff --git a/src/system/trajectory.jl b/src/system/trajectory.jl
index ea73b4c7a..ab3fa3378 100644
--- a/src/system/trajectory.jl
+++ b/src/system/trajectory.jl
@@ -9,8 +9,7 @@ using Polyhedra
 """
     DiscreteTrajectory{Q, TT}
 
-`q_0` is the starting mode and `transitions` is a sequence of discrete
-transitions in the system
+`q_0` is the starting mode and `transitions` is a sequence of discrete transitions in the system.
 """
 struct DiscreteTrajectory{Q, TT}
     q_0::Q
@@ -36,8 +35,7 @@ end
 """
     ContinuousTrajectory{T, XVT<:AbstractVector{T}, UVT<:AbstractVector{T}}
 
-`x` is a sequence of points in the state space and `u` is a sequence of points
-in the input space
+`x` is a sequence of points in the state space and `u` is a sequence of points in the input space.
 """
 struct ContinuousTrajectory{T, XVT <: AbstractVector{T}, UVT <: AbstractVector{T}}
     x::Vector{XVT}
@@ -46,11 +44,21 @@ end
 
 struct ContinuousTrajectoryAttribute <: MOI.AbstractModelAttribute end
 
+"""
+    HybridTrajectory{T, TT, XVT <: AbstractVector{T}, UVT <: AbstractVector{T}}
+
+`discrete` is the discrete trajectory of type `DiscreteTrajectory` and  `continuous` is a the continuous trajectory of type `ContinuousTrajectory`.
+"""
 struct HybridTrajectory{T, TT, XVT <: AbstractVector{T}, UVT <: AbstractVector{T}}
     discrete::DiscreteTrajectory{TT}
     continuous::ContinuousTrajectory{T, XVT, UVT}
 end
 
+"""
+    Trajectory{T}
+
+provides the sequence of some elements of a trajectory.
+"""
 struct Trajectory{T}
     seq::Vector{T}
 end
@@ -62,6 +70,11 @@ get_elem(traj::Trajectory, n::Int) = traj.seq[n]
     return UT.DrawTrajectory(traj.seq)
 end
 
+"""
+    Control_trajectory{T1, T2}
+
+provides the sequence of states and inputs of a trajectory.
+"""
 struct Control_trajectory{T1, T2}
     states::Trajectory{T1}
     inputs::Trajectory{T2}
@@ -76,6 +89,11 @@ get_elem(traj::Control_trajectory, n::Int) = (get_state(traj, n), get_input(traj
     return traj.states
 end
 
+"""
+    Cost_control_trajectory{T1, T2, T3}
+
+provides the sequence of states, inputs (via `Control_trajectory`) and costs of a trajectory.
+"""
 struct Cost_control_trajectory{T1, T2, T3}
     control_trajectory::Control_trajectory{T1, T2}
     costs::Trajectory{T3}
diff --git a/utils/example_SF_abst_1.jl b/utils/example_SF_abst_1.jl
index 495c38eaf..523f3c771 100644
--- a/utils/example_SF_abst_1.jl
+++ b/utils/example_SF_abst_1.jl
@@ -8,8 +8,12 @@ const DI = Dionysos
 const UT = DI.Utils
 const SY = DI.Symbolic
 
-# Example to illustrate the cost of a transition between two ellispoids as a function of meta-parameters
-# such as the volume of the initial ellispoid and the contraction factor
+# The aim of this file is to generate Figure 2 of the paper "State-feedback Abstractions for Optimal 
+# Control of Piecewise-affine Systems" from L.N. EGIDIO, T.A. LIMA and R.M. JUNGERS (CDC 2022).
+# This paper investigates symbolic abstractions that capture the behavior of piecewise-affine systems
+# under input constraints and bounded external noise. This file illustrates the cost of a transition 
+# between two ellispoids as a function of meta-parameters such as the volume of the initial ellispoid 
+# and the contraction factor.
 
 lib = CDDLib.Library() #polyhedron lib
 # aux functions
diff --git a/utils/example_ellipsoid_inclusion.jl b/utils/example_ellipsoid_inclusion.jl
index a80c736a3..6fd9697ad 100644
--- a/utils/example_ellipsoid_inclusion.jl
+++ b/utils/example_ellipsoid_inclusion.jl
@@ -2,6 +2,12 @@ using Dionysos
 const UT = Dionysos.Utils
 using Plots, Colors, LinearAlgebra, LaTeXStrings
 
+# The aim of this file is to generate Figure 1 of the paper "An Efficient Method to Verify the Inclusion
+# of Ellipsoids" from J. CALBERT, L.N. EGIDIO and R.M. JUNGERS (arXiv 2022).
+# The objective is to illustrate Theorem 1 which provides a criterion to check the inclusion of two
+# ellipsoids. They consider 3 different cases : when there is an intersection but no inclusion, 
+# when there is inclusion with a contact point and finally, when there is a strict inclusion.
+
 myblue = RGB(108 ./ 256, 142 ./ 256, 191 ./ 256)
 myblueN = RGB(32 ./ 256, 103 ./ 256, 205 ./ 256)
 myorange = RGB(255 ./ 280, 158 ./ 280, 56 ./ 280)
@@ -147,7 +153,7 @@ function plotSecularFunction(El0, El, intervalx, intervalfx, ϵ, h; eps = 1e-10)
     return display(p)
 end
 
-# intersection but non inclusion
+# CASE 1: Intersection but no inclusion
 function fig1()
     a = 1.0
     c0 = [1.6 + a; 1.4 + a]
@@ -191,7 +197,7 @@ function fig1()
     return plotSecularFunction(El0, El, intervalx, intervalfx, ϵ, h)
 end
 
-# inclusion with contact point
+# CASE 2: Inclusion with contact point
 function fig2()
     a = 0.89
     c0 = [1.6 + a; 1.4 + a]
@@ -234,7 +240,7 @@ function fig2()
     return plotSecularFunction(El0, El, intervalx, intervalfx, ϵ, h)
 end
 
-# strict inclusion
+# CASE 3: Strict inclusion
 function fig3()
     a = 0.6
     c0 = [1.6 + a; 1.4 + a]
@@ -599,6 +605,5 @@ end
 
 # tansformations()
 # fig3()
-test6()
 # particularCase()
 fig1()
diff --git a/utils/example_hierarchical_abstraction.jl b/utils/example_hierarchical_abstraction.jl
index 7d310edfb..8c0dd38e0 100644
--- a/utils/example_hierarchical_abstraction.jl
+++ b/utils/example_hierarchical_abstraction.jl
@@ -1,5 +1,9 @@
 using StaticArrays, JuMP, Plots, Test
 
+# The objective of this file is to provide an example of the application of the hierarchical abstraction-based technique 
+# developed in Dionysos for a reach-avoid optimal control problem. The whole procedure is detailed and commented, and 
+# the results are displayed at the end.
+
 # At this point, we import Dionysos.
 using Dionysos
 const DI = Dionysos
@@ -11,9 +15,10 @@ const PR = DI.Problem
 const OP = DI.Optim
 const AB = OP.Abstraction
 
+# We export the system from the problems folder.
 include("../problems/simple_problem.jl")
 
-## specific functions
+# Specific functions
 function post_image(abstract_system, concrete_system, xpos, u)
     Xdom = abstract_system.Xdom
     x = DO.get_coord_by_pos(Xdom.grid, xpos)
@@ -79,6 +84,13 @@ end
 
 minimum_transition_cost(symmodel, contsys, source, target) = 1.0
 
+# We define the concrete problem that we want to solve. For instance:
+# - the whole domain for the state and the obstacles/constraints 
+# - the domain for the input and the obstacles/constraints 
+# - the initial and targets sets
+# - the state and transition costs 
+# - the time step
+# - etc.
 concrete_problem = SimpleProblem.problem(;
     rectX = UT.HyperRectangle(SVector(0.0, 0.0), SVector(60.0, 60.0)),
     obstacles = [UT.HyperRectangle(SVector(22.0, 21.0), SVector(25.0, 32.0))],
@@ -95,6 +107,7 @@ concrete_problem = SimpleProblem.problem(;
     measnoise = SVector(0.0, 0.0),
 )
 
+# The concrete system is the real system
 concrete_system = concrete_problem.system
 
 # Local optimizer parameters
@@ -149,7 +162,7 @@ end
 # Get the results
 abstract_system = MOI.get(optimizer, MOI.RawOptimizerAttribute("abstract_system"))
 
-# ## Simulation
+# Simulation
 println("Solved : ", optimizer.solved)
 if optimizer.solved
     x0 = UT.get_center(concrete_problem.initial_set)
@@ -162,24 +175,24 @@ if optimizer.solved
     println("Cost:\t $(cost)")
 end
 
-# ## Display the results
-# # Display the specifications, domains and trajectory
+# Display the results
+# First, display the specifications, domains and trajectory
 fig1 = plot(; aspect_ratio = :equal)
 
-#We display the concrete domain
+# Then, we display the concrete domain
 plot!(fig1, concrete_system.X; color = :yellow, opacity = 0.5)
 
-#We display the abstract domain
+# We display the abstract domain
 plot!(fig1, abstract_system.symmodel.Xdom; color = :blue, opacity = 0.5)
 
-#We display the concrete specifications
+# We display the concrete specifications
 plot!(fig1, concrete_problem.initial_set; color = :green, opacity = 0.8)
 plot!(fig1, concrete_problem.target_set; dims = [1, 2], color = :red, opacity = 0.8)
 
 #We display the concrete trajectory
 plot!(fig1, cost_control_trajectory; ms = 0.5)
 
-# # Display the lazy abstraction 
+# Finally, we display the lazy abstraction 
 fig2 = plot(; aspect_ratio = :equal)
 
 plot!(