Stability. A homogeneous system x’ = Ax with constant

Chapter 9, Problem 49E

(choose chapter or problem)

49. Stability. A homogeneous system \(x^{\prime}=A x\) with constant coefficients is stable if it has a fundamental matrix whose entries all remain bounded as \(t \rightarrow+\infty\). (It will follow from Lemma 1 in Section 9.8 that if one fundamental matrix of the system has this property, then all fundamental matrices for the system do.) Otherwise, the system is unstable. A stable system is asymptotically stable if all solutions approach the zero solution as

\(t \rightarrow+\infty\). Stability is discussed in more detail in Chapter \(12^{\dagger}\).

(a) Show that if A has all distinct real eigenvalues, then \(x^{\prime}(t)=A x(t)\) is stable if and only if all eigenvalues are nonpositive.

(b) Show that if A has all distinct real eigenvalues, then \(x^{\prime}(t)=A x(t)\) is asymptotically stable if and only if all eigenvalues are negative.

(c) Argue that in parts (a) and (b), we can replace “has distinct real eigenvalues” by “is symmetric” and the statements are still true.

Equation Transcription:

       

     

     

 

   

 

Text Transcription:

x'=Ax      

t \rightarrow+\infty

t \rightarrow+\infty

12  \dagger

x'(t)=Ax(t)    

x'(t)=Ax(t)