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I have this sets of equations \begin{align} x_1'[t] &= x_2[t] \\ x_2'[t] &= -p_2[t] \\ p_1'[t]&=-x_1[t] \\ p_2'[t]&=-p_1[t] - p_3 (-2 H_2 (3 - x_2[t]) + 2 H_1 (3 + x_2[t])) \\ p_3'[t]&=0 \end{align} where $H_i$ are Heaviside functions defined as

$ H_1= \begin{cases} 0,&for \qquad-(3 + x_2[t])\geq 0\\ 1,&for \qquad-(3 + x_2[t])<0 \end{cases} $

$ H_2= \begin{cases} 0,&for \qquad-(3 - x_2[t])\geq 0\\ 1,&for \qquad-(3 - x_2[t])<0 \end{cases} $

I know Mathematica can solve systems of equations, but I don't know how to integrate a Heaviside function.

How can I solve this system of equations. Actually I know initial values of $x_i$ but don't know $p_i$ since they are Lagrange multipliers of the Hamiltonian equation.

I know that;

$p_1[t_f]=p_2[t_f]=0$ where $t_f$ is not defined it is free.

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    Not actually. I don't know that much for Laplace transform for this type of equations. Btw, what is the transform of heaviside? – 2017-01-15
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    @Moo actually I have missed to replace u[t] it is equal to -p2[t] a am going to edit it. I need to find Laplace transofrm of all the 5 equations and than do the substitution to find them lastly I need to take inverse transform. Right? – 2017-01-15
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    I still coudn't compute the Laplace transform of Heaviside function. It does not depend explicitly to variable t and the function multiplied by. – 2017-01-15
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    I have already seen that page, but still couldn't make it similar to that in the forum. I have heavyside function like this $x_2 [t]-3$ which is a function of t or one of which $x_2$has negative sign. – 2017-01-15
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    You can see that $x_2 '[t]$ has terms like $H_1*(3-x_2[t])$ where $H_1$ is Heaviside defined as in the question. – 2017-01-15
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    I just imlimented that subscript so that I can indicate that those are two different Heavisides. Not for the shifting term. – 2017-01-15
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    One of my lecturer has asked as a homework question which is an example question from the book with a little change. And I red the section and the derivation of equations which defines those Heavisides but no solution of integration. – 2017-01-15

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