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I have a function $f(x,t;k)$, a starting point $x_0$, a gradient $\operatorname{Grad}(f)$, and an equilibrium point $x^*$. I can adjust the parameter $k$ freely, and I know that for any $k$ the process will eventually reach $x^*$.

I want to know which value of $k$ will produce a trajectory that takes me to $x^*$ in the shortest time possible.

What kind of a differential equation problem is this? Is there a canonical solution? Is there a reference somewhere I can check to get more information?

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  • $\begingroup$ No, I have to set k at the beginning of the process and then keep it constant. Otherwise, yes, it's a standard control problem. $\endgroup$
    – David Pepper
    Commented Feb 11, 2020 at 20:08
  • $\begingroup$ What is 'the process' in "for any $k$ the process will eventually reach $x^*$"? $\endgroup$
    – LSpice
    Commented Feb 11, 2020 at 20:38
  • $\begingroup$ x starts at x_0 and then follows its gradient every period. $\endgroup$
    – David Pepper
    Commented Feb 11, 2020 at 20:55
  • $\begingroup$ Ok, I misunderstood your post, nevermind. What you are looking for is steepest descent, which is a heavily studied topic. Without more knowledge of how k appears, it is hard to give a reference. $\endgroup$
    – Piyush Grover
    Commented Feb 11, 2020 at 21:38

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