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Let the $\textbf{bold characters}$ represent the vectors/matrices.All the vectors/matrices/scalars only refer to real numbers.

In a continuous LTI system.

$\frac{d\mathbf{T}(t)}{dt}=\mathbf{A}\mathbf{T}(t)+\mathbf{B}$

in which $\mathbf{T}=[T_i]_{N\times 1}$, $\mathbf{B}=[B_i]_{N\times 1}$ and $\mathbf{A}=[A_{i,j}]_{N\times N}$. $\mathbf{A}$ is a given constant matrix. $\mathbf{T}(t)$ are real valued function of $t$. All $\mathbf{T}(t)$, $\mathbf{B}$ and $\mathbf{A}$ only contain real numbers.$t>0$.

$B_i$ is different square-pulse input, that $B_i(x_i)=u(x_i)-u(x_i-K_i)$, where $K_i$ is the given pulse width and $K_i>0$. $u(t)$ is a continuous unit-step function as $u(t)=1$ when $t\geq 0$; $u(t)=0$, otherwise.

How can we find a phase $\mathbf{X}=[x_i]_{N\times 1}$ such that the output $\max(\mathbf{T}(t))$ can be minimized?


So far, my idea is to use Sinusoidal to approximate the square inputs signal, then calculate their phase. Then, use superposition method. However, it still need to use numerical method to check the $t_i$ for different $B_i$. Is there any simpler way? Any paper talk about this? Thank you.

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