GATE ECE 2025 | Question: 43

Question

Consider a system where $x_{1}(t), x_{2}(t)$, and $x_{3}(t)$ are three internal state signals and $u(t)$ is the input signal. The differential equations governing the system are given by

$$\frac{d}{d t}\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \\ x_{3}(t) \end{array}\right]=\left[\begin{array}{rrr}  2 & 0 & 0 \\  0 & -2 & 0 \\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{l} x_{1}(t) \\ x_{2}(t) \\ x_{3}(t) \end{array}\right]+\left[\begin{array}{l}  1 \\ 1 \\ 1 \end{array}\right] u(t) $$

Which of the following statements is/are TRUE?

• A. The signals $x_{1}(t), x_{2}(t)$, and $x_{3}(t)$ are bounded for all bounded inputs
• B. There exists a bounded input such that at least one of the signals $x_{1}(t)$, $x_{2}(t)$, and $x_{3}(t)$ is unbounded
• C. There exists a bounded input such that the signals $x_{1}(t), x_{2}(t)$, and $x_{3}(t)$ are unbounded
• D. The signals $x_{1}(t), x_{2}(t)$, and $x_{3}(t)$ are unbounded for all bounded inputs

Answer 1

b,c