Minimizing a function by using large coefficients. we have two energies ${||u(t)||}^2$ and ${||z(t)||}^2$. They are inversely related i.e if we decrease $u$, $z$ increases and vice versa. Now we need to minimize a function given as
$J = \int {||z(t)||}^2 + p{||u(t)||}^2 dt$
My text says that if z(t) is large we can minimize J, by  using a large p. How can increasing a value decrease an integral that sums up to infinity? 
P.S: To me using a large p will increase the value of the product $p{||u(t)||}^2$ and intuitively it should increase the value of the integral, not minimize it.
You can read further at page 4 of this text.
 A: I think you've misread your text. Quoting (I think) the relevant part:

In LQR one seeks a controller that minimizes both
  energies. However, decreasing the energy of the controlled output will require a large control signal and a small control signal will lead to large controlled outputs. The role of the constant $\rho$ is to establish a trade-off between these conflicting goals:
  
  
*
  
*When we chose ρ very large, the most effective way to decrease $J_\text{LQR}$ is to use little control, at the expense of a large controlled output.
  
*When we chose ρ very small, the most effective way to decrease $J_\text{LQR}$ is to obtain a very small controlled output, even if this is achieved at the expense of a large controlled output.

Most importantly, $\rho$ isn't something chosen to solve the optimization problem. Instead, it's chosen to describe what we want our solution to look like. The constant $\rho$ represents how much we'd prefer to have a small controlled output versus a small control signal. If we choose $\rho>1$, changes in the size of the control signal, $u(t)$, will exert more influence over the size of 
$$J=\int \lvert\lvert z(t)\rvert\rvert^2+\rho\lvert\lvert u(t)\rvert\rvert^2 \text{d}t$$
than $z(t)$. Conversely, if we choose $\rho<1$, we state that we care more about minimizing controlled output, and $J$ is, correspondingly, more sensitive to changes in $z(t)$ than it is $u(t)$. 

Here is an example that might help spell it out further.  
Suppose that out of personal whim, we are much happier with small control signals than small controlled outputs. Consequently, we choose $\rho=4$ to say that a decrease in $\lvert\lvert u(t)\rvert\rvert^2$ is four times more valuable to us than a decrease in $\lvert\lvert z(t)\rvert\rvert$, and set about solving the problem $$J_\text{LQR}=\int_0^\infty\lvert\lvert z(t)\rvert\rvert+4\lvert\lvert u(t)\rvert\rvert\text{d}t$$.
Now, $\rho$ has already been set before we set about trying to find a solution to the LQR problem. What balance of $\lvert\lvert z(t)\rvert\rvert^2$ and $\lvert\lvert u(t)\rvert\rvert^2$ will give us a minimum? It's hard to know off hand, but it is easy to see that a decrease in $\lvert\lvert u(t)\rvert\rvert$ will count for much more than $z(t)$.
