Currently, I am solving for the coefficients of a quintic polynomial that passes through two states:
$f(x) = c_0 + c_1x + c_2x^2 + c_3x^3 + c_4x^4 + c_5x^5$
The states define $f(x)$, $f'(x)$, and $f''(x)$ at $x=0$ and $x=x_f$ so the problem has an exact solution. Also, $f(x_f) > f(x_0) >= 0$.
Now, however, I want to enforce the constraint that $f(x)$ be monotonically increasing, i.e. $f'(x) >= 0$, on $[0, x_f]$
I know that I can't necessarily achieve this and have the function pass exactly through both states, so I'll redefine the problem to be generating a quintic polynomial that:
- passes through the first state,
- is monotonically increasing on $[0, x_f]$, and
- comes as close to passing through the second state as possible.
I have two questions:
What are my options for solving this problem? I suppose this is some sort of optimisation problem.
Is it possible to solve the problem exactly (such that the polynomial is monotonically increasing and passes through both states) if I increase the order of the polynomial?