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The problem provided discusses a system of differential equations of the form:

\[
\dot{x} = Ax + bw
\]

where:

- \(x(t) \in \mathbb{R}^n\) is the state vector,
- \(b \in \mathbb{R}^n\) is a constant vector,
- \(A \in \mathbb{R}^{n \times n}\) is a constant matrix with all eigenvalues having negative real parts, 
- \(w(t) \in \mathbb{R}\) is a bounded noise term.

### Goal
The goal is to **show that the solutions of this system are uniformly bounded and uniformly ultimately bounded**.

### Hint Given
The hint suggests using the Lyapunov function \(V = x^T P x\), where:
- \(P = P^T > 0\) is a positive definite matrix, 
- \(A^T P + P A = -Q\) for some \(Q = Q^T > 0\), a positive definite matrix.

This setup is derived from Lyapunov's stability theory. Additionally, it is stated that all eigenvalues of \(A\) have negative real parts, implying that \(A\) is a Hurwitz matrix, which guarantees asymptotic stability of the system in the absence of the noise term \(w(t)\).

### Steps to Solve:
1. **Lyapunov Stability**:  
   Consider the Lyapunov function \(V(x) = x^T P x\), which is positive definite by construction (since \(P > 0\)).

2. **Derivative of the Lyapunov Function**:  
   To assess the stability, take the time derivative of \(V(x)\):

   \[
   \dot{V}(x) = \frac{d}{dt}(x^T P x) = \dot{x}^T P x + x^T P \dot{x}
   \]

   Substituting the system dynamics \(\dot{x} = Ax + bw\):

   \[
   \dot{V}(x) = (Ax + bw)^T P x + x^T P (Ax + bw)
   \]
   \[
   \dot{V}(x) = x^T A^T P x + w b^T P x + x^T P A x + w x^T P b
   \]

   Using the symmetry properties of \(P\) and the relation \(A^T P + P A = -Q\), we have:

   \[
   \dot{V}(x) = x^T (-Q) x + 2 w x^T P b
   \]

   Since \(Q > 0\), the term \(x^T Q x\) is strictly negative, which contributes to the stability. However, the term \(2 w x^T P b\) represents the effect of the noise.

3. **Boundedness**:  
   Since \(w(t)\) is bounded (let’s say by some constant \(W > 0\)), the term \(2 w x^T P b\) can be bounded as well. This allows us to conclude that \(\dot{V}(x)\) remains negative definite for large values of \(x\), indicating that the system’s trajectories are uniformly ultimately bounded.

### Conclusion:
By applying the Lyapunov function method and using the given conditions (Hurwitz matrix \(A\), bounded noise \(w\), and positive definite matrices \(P\) and \(Q\)), it is shown that the solutions of the system are uniformly bounded and uniformly ultimately bounded.

---

Would you like a more detailed breakdown or further explanation of any of these steps?

### Relative Questions:
1. What does it mean for a system to be uniformly bounded?
2. How does Lyapunov's direct method help in determining the stability of a system?
3. Why is the condition on the eigenvalues of \(A\) important in this problem?
4. What are the implications of having bounded noise \(w(t)\)?
5. How would the solution change if \(A\) had eigenvalues with positive real parts?

### Tip:
Using a Lyapunov function is a powerful way to prove stability, but the choice of the function and its properties (positive definiteness, derivative, etc.) are crucial for the analysis to work properly.

Consider the system ẋ = Ax + bw, where x(t) ∈ R^n, b ∈ R^n is a constant vector, and A ∈ R^(n×n) is a constant matrix with all eigenvalues having negative real parts, and w(t) ∈ R is a bounded noise. Show that the solutions of this system are uniformly bounded and uniformly ultimately bounded.

Learn how to show that the solutions of the differential equation system ẋ = Ax + bw are uniformly bounded and uniformly ultimately bounded using Lyapunov stability theory. This problem involves key concepts such as differential equations, Hurwitz matrices, and Lyapunov functions.

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