# Optimization with differential inequality constraint

Consider the closed set $[t_1,t_2]⊂R_{>0}$ and $V(t):[t1,t2]→R_{>0}$ being a continuous and piecewise continuously differentiable function. We want to find a continuously differentiable function $x(t):[t1,t2]→R_{>0}$ such that it minimizes:

$Q=\int_{t_1}^{t_2} (V(t)-x(t)) \, dt$

Under the following inequality constraints:

1) $V(t)-x(t) \geq 0 \,\, \forall \, t\in[t_1,t_2]$

2) $a_{max} \geq \dot x(t) \geq-a_{max} \,\, \forall \, t\in[t_1,t_2]$

In simple words, I am looking for a signal $x(t)$ that follows a given signal $V(t)$ as closely as possible while always stays below $V(t)$ and keeps its rate limited under a pre-specified bound($a_{max}$).

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