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?)

One option is APM MATLAB that allows differential and algebraic equations with either continuous or mixed integer variables. You'll need to define the differential of the objective function and include it as a constraint. You can then include the integral of that variable in the objective function declaration.
