Maximizing the vertical distance from a repulsive time-varying barrier

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Continuing topic: https://mathematica.stackexchange.com/questions/246825/multidimensional-obstacle-avoidance-in-ode-part-ii

Given dynamical system:

$\boldsymbol{x}=f(\boldsymbol{x},u,t)$

where $\vec{\boldsymbol{x}}={x_1...x_n}$ - state-space variables.

Suppose I have chosen some variable from $\vec{\boldsymbol{x}}$ and want it not to exceed some repulsive trajectory that changes over time when moving towards a steady state.

How to maximize the vertical distance $S$ from the repulsive time-varying barrier?

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