- #1
dmytro
- 7
- 0
I have a linear time-varying linearly perturbed ODE of the form:
<br /> \dot{x} = [A(t)+B(t)]x<br />
where A(t) is a bounded lower-triangular matrix with negative functions on the main diagonal, i.e. 0>a^0\ge a_{ii}(t). The matrix B(t) is bounded, so that ||B(t)|| \le \beta.
The question is whether there exists a sufficiently small bound \beta such that the origin is asymptotically stable (not necessarily exponentially! I don't care about that).
This paper gives a positive answer to the question, BUT the assumption is that \dot{x}=A(t)x is exponentially stable. In my case, while it is possible to show inductively stability of \dot{x}=A(t)x, it is not clear to me that it is exponentially stable, since, for example \dot{x}_2 = a_{22}(t)x_2 + a_{21}(t)x_1(t) such that x_1(t)\to0 is exponentially bounded. Is x_2(t) also exponentially bounded?
<br /> \dot{x} = [A(t)+B(t)]x<br />
where A(t) is a bounded lower-triangular matrix with negative functions on the main diagonal, i.e. 0>a^0\ge a_{ii}(t). The matrix B(t) is bounded, so that ||B(t)|| \le \beta.
The question is whether there exists a sufficiently small bound \beta such that the origin is asymptotically stable (not necessarily exponentially! I don't care about that).
This paper gives a positive answer to the question, BUT the assumption is that \dot{x}=A(t)x is exponentially stable. In my case, while it is possible to show inductively stability of \dot{x}=A(t)x, it is not clear to me that it is exponentially stable, since, for example \dot{x}_2 = a_{22}(t)x_2 + a_{21}(t)x_1(t) such that x_1(t)\to0 is exponentially bounded. Is x_2(t) also exponentially bounded?
