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I'm trying to solve an equation from the next form:

$$\begin{eqnarray*} x_1'(t)&=&x_2(t) \\ \\x_2'(t)&=&-0.25x_1(t)-u(t)x_2(t) \end{eqnarray*}$$

I'm not sure how to solve this because there is a multiple between the controller $u(t)$ and the state space $x_2(t)$.

My question is how can i proceed with solving this using Hamiltonian in this case.

Thank you.

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    You need some information about $u(t)$ – 2017-01-28
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    the information i have is: 0<=u(t)<=1. Thanks you. – 2017-01-28
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    You haven't stated what your optimization criteria and boundary conditions are. Is it a minimum time problem? – 2017-01-29
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    Find the minimum control which steers the system to the origin in minimum time. Thank you – 2017-01-29
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    Is $a$ positive or negative? This also affects the optimal trajectories. – 2017-01-29
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    I updated the question, a=0.25 – 2017-01-29
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    You could go straight, write the Hamiltonian $H=-1+\psi_1x_2-\psi_2(0.25x_1+ux_2)$, find the optimal control to be $u^*=0$ if $\psi_2x_2\ge 0$ and $u^*=1$ otherwise etc., but this will probably not bring you any further. Instead of this I'd suggest you to see the literature on time-optimal control of bilinear systems. Your system falls into this class and so, you may find some information specifically related to your problem. – 2017-01-30

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