Question

For the state equation $\dot{x} = 3x$ , the system is _______.

• A. stable
• B. marginally stable
• C. unstable
• D. oscillatory

Hint:

For any system equation of the form $\dot{x} = ax$: - If $a < 0$, the system is stable (exponential decay). - If $a = 0$, the system is marginally stable (constant value). - If $a > 0$, the system is unstable (exponential growth). Here, $a = 3 > 0$, so it is unstable.

Answer 1

The Correct Option is C

Concept: A continuous-time linear system described by a state-space differential equation is stable if and only if all the eigenvalues of its system matrix (or the roots of its characteristic equation) have strictly negative real parts. If any eigenvalue has a positive real part, the system's time response grows exponentially, causing it to be unstable.

Step 1: Finding the explicit solution of the state equation.
We are given a first-order scalar state equation: \[ \dot{x}(t) = \frac{dx(t)}{dt} = 3x(t) \] We can solve this homogeneous linear differential equation by separating variables: \[ \frac{1}{x} \, dx = 3 \, dt \] Integrating both sides from an initial time $t = 0$ to a final time $t$: \[ \int_{x(0)}^{x(t)} \frac{1}{x} \, dx = \int_{0}^{t} 3 \, dt \] \[ \ln\left(\frac{x(t)}{x(0)}\right) = 3t \] Taking the exponential of both sides yields the final time-response expression: \[ x(t) = x(0) \cdot e^{3t} \]

Step 2: Evaluating the stability from the time response.
Let us analyze the behavior of the output $x(t)$ as time approaches infinity ($t \rightarrow \infty$): \[ \lim_{t \rightarrow \infty} x(t) = \lim_{t \rightarrow \infty} \left[ x(0) \cdot e^{3t} \right] \rightarrow \infty \quad (\text{for any initial state } x(0) \neq 0) \] Because the system contains an eigenvalue located at $+3$ (which lies in the right-half plane), the system exhibits unbounded exponential growth, making it unstable.