To determine the category of the continuous-time system described by the equation \(y(t) = x(t^2)\), we need to analyze its properties: causality, linearity, and time-invariance.

Analyzing System Causality

A system is considered causal if its output at any time \(t\) depends only on the input at time \(t\) and past times (\(\tau \le t\)). In the given system, the output \(y(t)\) at time \(t\) is equal to the input \(x(t^2)\) at time \(t^2\).

• If \(t \gt 1\), then \(t^2 \gt t\). For example, at \(t=2\), the output \(y(2)\) depends on the input \(x(2^2) = x(4)\). Here, the output at \(t=2\) depends on the input at a future time \(t=4\).
• If \(0 \le t \le 1\), then \(t^2 \le t\). For example, at \(t=0.5\), the output \(y(0.5)\) depends on the input \(x((0.5)^2) = x(0.25)\). Here, the output at \(t=0.5\) depends on the input at a past time \(t=0.25\).
• If \(t \lt 0\), then \(t^2 \gt 0\). For example, at \(t=-2\), the output \(y(-2)\) depends on the input \(x((-2)^2) = x(4)\). Here, the output at \(t=-2\) depends on the input at a future time \(t=4\).

Since there are instances where the output \(y(t)\) depends on the input \(x(\tau)\) where \(\tau \gt t\) (future input), the system is **non-causal**.

Analyzing System Linearity

A system is linear if it satisfies the superposition principle, which includes additivity and homogeneity. Let's consider two inputs \(x_1(t)\) and \(x_2(t)\) and their respective outputs \(y_1(t)\) and \(y_2(t)\).

• If \(x_1(t) \rightarrow y_1(t) = x_1(t^2)\)
• If \(x_2(t) \rightarrow y_2(t) = x_2(t^2)\)

Now, consider a combined input \(x(t) = a x_1(t) + b x_2(t)\), where \(a\) and \(b\) are constants. The output of the system for this input would be:

\[ y(t) = x(t^2) = (a x_1 + b x_2)(t^2) = a x_1(t^2) + b x_2(t^2) \]

We know that \(y_1(t) = x_1(t^2)\) and \(y_2(t) = x_2(t^2)\). Substituting these into the equation:

\[ y(t) = a y_1(t) + b y_2(t) \]

The output for the combined input is the same linear combination of the individual outputs. Therefore, the system satisfies the superposition principle and is **linear**.

Analyzing System Time-Invariance

A system is time-invariant if a time shift in the input signal results in an identical time shift in the output signal. Let the output for an input \(x(t)\) be \(y(t) = x(t^2)\).

Now, consider a time-shifted input \(x_{shifted}(t) = x(t - t_0)\), where \(t_0\) is the time shift. The output of the system for this shifted input is:

\[ y_{new}(t) = x_{shifted}(t^2) = x(t^2 - t_0) \]

Next, let's consider the original output \(y(t)\) shifted by the same amount \(t_0\):

\[ y(t - t_0) = x((t - t_0)^2) = x(t^2 - 2tt_0 + t_0^2) \]

For the system to be time-invariant, \(y_{new}(t)\) must be equal to \(y(t - t_0)\) for all \(t\) and \(t_0\). Comparing the two expressions:

\[ x(t^2 - t_0) \text{ vs } x(t^2 - 2tt_0 + t_0^2) \]

These expressions are generally not equal. For example, if \(t_0 = 1\), then \(y_{new}(t) = x(t^2 - 1)\) and \(y(t - 1) = x(t^2 - 2t + 1)\). These are different functions of \(t\). Therefore, the system is **time-variant**.

Summary of System Properties

Based on the analysis, the continuous-time system \(y(t) = x(t^2)\) has the following properties:

Thus, the system is non-causal, linear, and time-variant.