differential equations - Optimization of ODE with respect to the initial condition

One has a (system) of ODEs with a one-parameter family of initial conditions. For example,

f[kk_] := N[aa = kk; s = NDSolve[{y''[t] == y[t]^2, y[1] == aa, y'[1] == 1},
                                  y, {t, 2}]; Evaluate[(y[2] /. s)][[1]]]

where k is the parameter, and one sees $f(k)$ as a function of $k$ works fine, as one can try to plot

Plot[f[x], {x, -3, 0}]

and obtain

However, when one tries to find the min, using FindMinimum or NMinimize, it does not work as it seems Mathematica would first perform symbolic calculation with

NDSolve[{(y''[t] == -2y[t], y[1] == x, y'[1] == 1}, y, {t, 2}]

which results an error as the initial condition is not a number, and returns y[2.] before continuing evaluating NMinimize[y[2.], x].

Does anyone know how to get around it, or some better ways to do the optimization?

Answer

Use ParametricNDSolve or it's cousin ParametricNDSolveValue:

s = ParametricNDSolveValue[
    {y''[t] == y[t]^2, y[1] == aa, y'[1] == 1},
    y[2],
    {t, 2},

    aa
];

NMinimize[s[aa], aa]

{0.0964433, {aa -> -1.65429}}

Comments

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