Misc

[ocots]

Remarks in vrac.

@AnasXbouali

See the comments below.

[ocots]

• Here, add a space between to and (CS).
• In general, I would not add labels for the equations since there is no ref. There are others in other pages.

[AnasXbouali]

the change is made.

[ocots]

In the page here, we have:

• \texttt{ForwardDiff}
• \textit{pseudo-Hamiltonian}\footnote{the pseudo-Hamiltonian stands for the Hamiltonian flow associated with each arc.}

This should be fixed.

Still in the same place, in the paragraph:

We begin by defining the flow of the Hamiltonian associated with each arc. To accomplish this, we use the function Flow that can be found in the CTFlows.jl package. This latter allows to solve the Hamiltonian system over a given time interval from given initial values of the state and the adjoint vector. This function requires necessary libraries such as \texttt{ForwardDiff} for calculating gradients and Jacobians and OrdinaryDiffEq for solving ordinary differential equations.

The usage is the following:

• for the package do not use backquotes and write the suffix: CTFlows.jl instead of CTFlows.jl, OrdinaryDiffEq.jl instead of OrdinaryDiffEq.
• for code use backquotes: Flow is 👍

[AnasXbouali]

the change is made.

[ocots]

Zermelo problems: I would split in two different pages the two examples since it is quite long.

[AnasXbouali]

the change is made.

[ocots]

Erratum. It cannot be shortened as mentioned below since the dimension of the flows are different!

The following can be shortened:

ode_sol = fc((t0, tt1), x0, pp0, saveat=0.1)
ttt1 = ode_sol.t
xx1 = [ ode_sol[1:2, j] for j in 1:size(ttt1, 1) ]
pp1 = [ ode_sol[3:4, j] for j in 1:size(ttt1, 1) ]
uu1 = u11.(xx1, pp1)

pλ0 = 0.
py0 = 0.

ode_sol = fcl((tt1, tt2), [xx1[end] ; aa ; 0.0], [pp1[end] - [0. , jmp11]; pλ0 ; py0], saveat=0.1)
ttt2 = ode_sol.t
xx2 = [ ode_sol[1:2, j] for j in 1:size(ttt2, 1) ]
pp2 = [ ode_sol[5:6, j] for j in 1:size(ttt2, 1) ]
uu2 = a.*ones(length(ttt2)) ;

ode_sol = fc((tt2, tf), xx2[end], pp2[end] - [0. , jmp22], saveat=0.1)
ttt3 = ode_sol.t
xx3 = [ ode_sol[1:2, j] for j in 1:size(ttt3, 1) ]
pp3 = [ ode_sol[3:4, j] for j in 1:size(ttt3, 1) ]
uu3 = u11.(xx3, pp3)

tsol = [ ttt1 ; ttt2 ; ttt3 ]
xsol = [ xx1 ; xx2 ; xx3 ]
psol = [ pp1 ; pp2 ; pp3 ]
usol = [ uu1 ; uu2 ; uu3 ]

m = length(tsol)

x11 = [ xsol[i][1] for i=1:m ]
x22 = [ xsol[i][2] for i=1:m ]
p11 = [ psol[i][1] for i=1:m ]
p22 = [ psol[i][2] for i=1:m ]

You can use the concatenation of arcs. See here. But you need to add the jumps. I have added this possibility. See this example where I make a concatenation with a jump:

f = f0 * (t1, fc) * (t2, ν2, f0) # ν2 is a jump on the costate and has the same dimension as the state

Here f0 is followed by fc whenever t ≥ t1 which is followed by f0 again whenever t ≥ t2 adding a jump ν2 to the costate. Note that you can still call f as any flow from any initial time to any final time.

[AnasXbouali]

Consider the Zermelo problem with two loss control regions, let us recall the Flows:

# Hamiltonian: permanent region
H1(x, u, p)  = p' * F(x, u)                 # pseudo-Hamiltonian
u11(x, p)    = atan(p[2]/p[1])              # maximizing control
Hc(x, p)     = H1(x, u11(x, p) , p )        # Hamiltonian

# Flow
fc  = Flow(Hamiltonian(Hc))

# Hamiltonian: control loss region
H2(x, λ, y, p)   = p' * F(x, λ)   + y* p' *G(λ)    # pseudo-Hamiltonian
Hcl(X, P)        = H2(X[1:2], X[3], X[4], P[1:2])  # Hamiltonian

# Flow
fcl = Flow(Hamiltonian(Hcl))

If I understood correctly, the concatenation of the flows should look like this:

f        = fc * (s1, jump1, fcl) * (s2, jump2, fc) * (s3, jump3, fcl) * (s4, jump4, fc)

However, when I run the following command:

flow_sol = f((t0, tf), x0, p0)

I get an error.

[ocots]

As I said, I put an erratum at the beginning of the comment. You cannot concatenate flows with different state dimensions!

[ocots]

To concatenate, you would augment the flows with smaller dimension and add constant dynamics. Not sure it will be more readable.

[ocots]

We do not use NLSolve.jl but MINPACK.jl or NonlinearSolve.jl in our documentation: see here.

To see if we change here or not.

[AnasXbouali]

thanks! I used NonlinearSolve.jl.