Problem 21 - Advanced Engineering Mathematics

Question

Show that the nonhomogeneous linear system \(\mathbf{X}^{\prime}=\mathbf{A} \mathbf{X}+\mathbf{F}\) has a unique critical point \(\mathbf{X}_{1}\) when \(\Delta=\operatorname{det} \mathbf{A} \neq 0\). Conclude that if \(\mathbf{X}=\mathbf{X}(t)\) is a solution to the nonhomogeneous system, \(\tau<0\) and \(\Delta>0\), then \(\lim _{t \rightarrow \infty} \mathbf{X}(t)=\mathbf{X}_{1} .[\) Hint \(:\) \(\left.\mathbf{X}(t)=\mathbf{X}_{c}(t)+\mathbf{X}_{1} \cdot\right]\)

Step-by-Step Solution

Answer

The unique critical point is \( \mathbf{X}_1 = -\mathbf{A}^{-1} \mathbf{F} \), and \( \lim_{t \to \infty} \mathbf{X}(t) = \mathbf{X}_1 \) for \( \tau < 0 \) and \( \Delta > 0 \).

Step 1: Understanding the System Dynamics

We consider the nonhomogeneous linear system given by \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} + \mathbf{F} \). A critical point \( \mathbf{X}_1 \) for this system is the point where the derivative is zero, \( \mathbf{X}^{\prime} = 0 \). This implies the equation \( \mathbf{A} \mathbf{X}_1 + \mathbf{F} = 0 \).

Step 2: Solving for the Critical Point

Set \( \mathbf{A} \mathbf{X}_1 + \mathbf{F} = 0 \). Rearranging gives \( \mathbf{A} \mathbf{X}_1 = -\mathbf{F} \). Since \( \Delta = \operatorname{det} \mathbf{A} eq 0 \), \( \mathbf{A} \) is invertible. Thus, the unique solution for \( \mathbf{X}_1 \) is \( \mathbf{X}_1 = -\mathbf{A}^{-1} \mathbf{F} \).

Step 3: Expressing the General Solution

Given the hint that \( \mathbf{X}(t) = \mathbf{X}_c(t) + \mathbf{X}_1 \), we acknowledge that \( \mathbf{X}_c(t) \) is a solution to the corresponding homogeneous system \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} \), without the nonhomogeneous component \( \mathbf{F} \).

Step 4: Analyzing Stability and Convergence

The homogeneous solution \( \mathbf{X}_c(t) \) depends on the eigenvalues of \( \mathbf{A} \). Given \( \tau < 0 \), the real part of all eigenvalues of \( \mathbf{A} \) is negative, indicating stability and forcing the homogeneous solution to decay to zero over time. Hence, \( \mathbf{X}(t) = \mathbf{X}_c(t) + \mathbf{X}_1 \) converges to \( \mathbf{X}_1 \) as \( t \to \infty \).

Step 5: Conclusion on Long-Term Behavior

Under the conditions \( \tau < 0 \) and \( \Delta > 0 \), the system stabilizes to the critical point \( \mathbf{X}_1 \). The convergence follows from the decay of \( \mathbf{X}_c(t) \) due to negative eigenvalue real parts.

Key Concepts

Critical Points

Critical points in the context of nonhomogeneous linear systems are specific solutions where the system's rate of change becomes stationary. For the system \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} + \mathbf{F} \), a critical point occurs when \( \mathbf{X}^{\prime} = 0 \). At such a point, the forces changing the system cancel out, leading to a stable or unstable state depending on other factors.In this specific case, to find the critical point \( \mathbf{X}_1 \), we solve the equation \( \mathbf{A} \mathbf{X}_1 + \mathbf{F} = 0 \). Rearranging gives \( \mathbf{A} \mathbf{X}_1 = -\mathbf{F} \). If the determinant of \( \mathbf{A} \) is non-zero \( (\Delta = \text{det} \mathbf{A} eq 0) \), \( \mathbf{A} \) is invertible, and thus the critical point is unique.This critical point represents where the system will naturally settle if no other forces act upon it. It's the foundation for analyzing the behavior of the system over time.

System Stability

The stability of a system determines whether, over time, the system will tend to return to a critical point or diverge. In linear systems, stability is often linked to the nature of a system’s eigenvalues.In the nonhomogeneous system \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} + \mathbf{F} \), the stability is primarily influenced by the homogeneous part \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} \). This part of the system has solutions that can be expressed using the eigenvalues of matrix \( \mathbf{A} \).If the real parts of these eigenvalues are negative (as in the condition \( \tau < 0 \)), the solutions of the homogeneous system decay towards zero over time. This indicates that the system is stable, as solutions tend to return to the critical point \( \mathbf{X}_1 \). Stability in this context means that perturbations or deviations from the critical point dampen out, aligning the system back to its steady state.

Eigenvalues

Eigenvalues are pivotal in understanding the behavior of differential equations, especially in linear systems. They are values that, if associated with a square matrix \( \mathbf{A} \), provide insight into the transformation characteristics of that matrix.Within the nonhomogeneous system \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} + \mathbf{F} \), the eigenvalues of \( \mathbf{A} \) reveal how solutions to the homogeneous system \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} \) will behave over time.

• If all eigenvalues have negative real parts, the system is stable. This means solutions will decay to zero, ensuring the complete system \( \mathbf{X}(t) \) converges to the critical point \( \mathbf{X}_1 \).
• Conversely, positive eigenvalue real parts suggest instability; solutions would grow unbounded.

Thus, eigenvalues are essential for predicting whether a system will stabilize over time or not.

Invertible Matrices

An invertible matrix, often called a non-singular matrix, possesses an inverse. This property is crucial in linear algebra, especially when solving systems of equations.For the nonhomogeneous system \( \mathbf{X}^{\prime} = \mathbf{A} \mathbf{X} + \mathbf{F} \), the matrix \( \mathbf{A} \) must be invertible to solve for the unique critical point. The invertibility is confirmed by a non-zero determinant (\( \Delta = \text{det} \mathbf{A} eq 0 \)).When \( \mathbf{A} \) is invertible, it guarantees that we can find a unique solution \( \mathbf{X}_1 \) to \( \mathbf{A} \mathbf{X}_1 = -\mathbf{F} \). This shows that every input vector corresponds exactly to one output vector, solidifying that the system can uniquely resolve its critical point.Without an invertible matrix, many linear systems would be challenging or impossible to solve analytically, underlining the significance of this concept in mathematics.