This problem is asking to demonstrate that the solutions to the system

$\dot{x} = Ax + bw(t)$

where $x(t) \in \mathbb{R}^n$, $b \in \mathbb{R}^n$, $A \in \mathbb{R}^{n \times n}$ (with all eigenvalues having negative real parts), and $w(t)$ is bounded noise, are uniformly bounded and uniformly ultimately bounded.

Approach

1. Homogeneous system analysis (no noise term):
   The system can first be analyzed without the noise term ($w(t) = 0$). In this case, the equation reduces to: $\dot{x} = Ax$ Since $A$ is a matrix whose eigenvalues have negative real parts, the solutions of this homogeneous system will decay exponentially. Specifically, if $\lambda_1, \lambda_2, \dots, \lambda_n$ are the eigenvalues of $A$, then the general solution will behave like: $x(t) = e^{At} x(0)$ where $e^{At}$ is the matrix exponential. Since the eigenvalues have negative real parts, $\|e^{At}\|$ decays to zero as $t \to \infty$, which implies that the homogeneous system is stable and the solutions are bounded.

2. Non-homogeneous system (with noise term):
   Now, consider the non-homogeneous equation: $\dot{x} = Ax + bw(t)$ Using the method of variation of parameters, we can write the solution as: $x(t) = e^{At} x(0) + \int_0^t e^{A(t - \tau)} b w(\tau) \, d\tau$ The first term, $e^{At} x(0)$, decays to zero because of the stability of the homogeneous system, as established earlier. The second term involves the bounded noise $w(t)$.

   Since $w(t)$ is bounded, say $|w(t)| \leq M$ for some constant $M$, the integral term can be bounded by: $\left| \int_0^t e^{A(t - \tau)} b w(\tau) \, d\tau \right| \leq M \int_0^t \|e^{A(t - \tau)} b\| \, d\tau$ Given that $\|e^{A(t - \tau)}\|$ decays exponentially as $t - \tau$ increases, the integral converges and remains bounded for all $t$. This shows that the total solution $x(t)$ is uniformly bounded.

3. Uniform ultimate boundedness:
   The decay of $e^{At}$ ensures that the contribution of the initial condition decays over time. For large $t$, the solution is dominated by the integral term involving the noise. Since $w(t)$ is bounded and $e^{At}$ decays, there exists a bound $B$ such that: $\|x(t)\| \leq B \quad \text{for all} \ t \geq T$ for some sufficiently large $T$. This shows that the system is uniformly ultimately bounded.

Conclusion:

The system is both uniformly bounded and uniformly ultimately bounded due to the stability of the matrix $A$ (which has eigenvalues with negative real parts) and the bounded nature of the noise $w(t)$.

Would you like more details or have any specific questions?

Related questions:

1. What is the method of variation of parameters used in non-homogeneous systems?
2. How does the matrix exponential $e^{At}$ behave when the eigenvalues of $A$ have negative real parts?
3. What is the concept of uniform boundedness in dynamical systems?
4. How does bounded noise affect the stability of a linear system?
5. What conditions guarantee that a system is uniformly ultimately bounded?

Tip: In stability analysis, negative real eigenvalues of the system matrix ensure that perturbations decay over time, which is crucial for the system's boundedness.