# Some type of Mixed Integer Nonlinear Programming Problem

This is a minimisation problem, to minimise the integral over possible $0\leq t \leq T$, $T$ is free,

$$J = \text{min} \int_0^T (\alpha + \beta_1\cdot v \cdot R_T \cdot q+ \beta_2 \cdot \frac{M}{1000} \cdot v \cdot a^2 \cdot p \cdot q)\,dt$$

where,

$$R_T = b_1 + b_2 \cdot v^2 + \frac{M}{1000}\cdot a$$

$$p = \text{sgn}(a) = \begin{cases} 1, & \text{if a > 0} \\ 0, & \text{otherwise} \end{cases}$$

$$q = \text{sgn}(R_T) = \begin{cases} 1, & \text{if R_T > 0} \\ 0, & \text{otherwise} \end{cases}$$

$s$ is travel distance, $v$ is velocity, $a$ is acceleration and subject to dynamics, $$\begin{cases} \dot s(t)= v(t)\text{;}\ v(0) = 0\text{,}\ s(T) = D \\ \dot v(t)= a(t)\text{;}\ a(0) = 0 \end{cases}$$

In the above expressions, $\alpha$, $\beta_1$, $\beta_2$, $b_1$, $b_2$, $M$, $D$ are constants.

Any suggestions about computing the minimum $J$ are greatly appreciated. (Is there any routine in matlab that solves this problem?)

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