Continuous-time and Discrete-time systems, Schur conditions, Routh-Hurwitz

[EconStu]

Hi guys,

I hope one of you can help me out with the following problem. I hope I posted the problems in the correct section. Problem:

Consider the following continuous-time and discrete-time systems with respect to
a common 2X2 matrix A:

x˙ = Ax (1) is a continuous- time system
xt+1 = xt + hAxt (2) is a discrete-time system

where 0 < h 1 and A is assumed to be asymptotically stable (AS). The intuition
is that for sufficiently small step sizes h, system (2) is a good approximation of
(1) and therefore AS, too. Your task is to find a positive number H such that
(2) is AS for all h < min{1,H}.
Hints:
• Write (2) in the form xt+1 = Bxt (where B depends on h, of course) and
use the Schur conditions.
• Denote the Routh-Hurwitz coefficients for the stability of (1) by a1, a2, and
the coefficients for the Schur conditions in (2) by b1, b2. Express the latter
in terms of a1, a2 and (of course) h, and use the information on the signs of
a1, a2 that you have from the stability assumption on A.
• To simplify matters, you are allowed to suppose that traceA + 2 > 0.

Here is what i have so far:

I applied Schur and Routh-Hurwitz conditions and expressed one in the form of the other as was advised in the hint.I got the following result:

My matrix A = [d e]
-------------[f g]

My matrix B = [1+dh eh]
------------[fh gh +1]

S1 (Schur Condition 1) dgh^2+ efh^2 >0 (B1)
S2 (Schur Condition 2) 4 + 2dh + 2gh + dgh^2 + efh^2 >0 (B2)
S3: -gh-dh-dgh^2+efh^2 (B3)

Routh-Hurwitz conditions:

RH1: -d-g < 0 (A1)
RH2: dg-ef > 0 (A2)

After expressing Shur Conditions in the form of Routh-Hurwritz Conditions, I got the following result:

-A2h^2 >0
4-2A1h-A2h^2 >0
A1h-A2h^2>0

I am not sure how to take it from there though.

I am a Master Student in Economics. My mathematics background is insufficient to solve this problem. I hope somebody can help.

Thanks in Advance

 

[jvicens]

Hi there,

Thank you for posting your problem in the correct section. I am a scientist with a background in mathematics and I will do my best to help you with your problem.

Firstly, I would like to commend you for successfully applying the Schur and Routh-Hurwitz conditions to express one in the form of the other. This is a good start and shows that you have a good understanding of the problem.

To continue, let's take a closer look at the result you have obtained so far:

-A2h^2 >0
4-2A1h-A2h^2 >0
A1h-A2h^2>0

From the Schur conditions, we know that B1 and B2 must be greater than 0 for the discrete-time system (2) to be asymptotically stable. This means that both A2h^2 and 4-2A1h-A2h^2 must be positive. This leads to two inequalities:

-A2h^2 >0 (from B1)
4-2A1h-A2h^2 >0 (from B2)

We also know from the Routh-Hurwitz conditions that A1h-A2h^2 must be greater than 0 for the continuous-time system (1) to be asymptotically stable. This leads to the inequality:

A1h-A2h^2>0 (from A1)

Combining these three inequalities, we can solve for h:

-A2h^2 >0
4-2A1h-A2h^2 >0
A1h-A2h^2>0

Solving for h, we get:

h < min{A1/A2, (4-A2)/2A1}

Therefore, the positive number H that we are looking for is:

H = min{A1/A2, (4-A2)/2A1}

This means that for all h < min{1,H}, the discrete-time system (2) will be asymptotically stable.

I hope this helps you to solve your problem. If you have any further questions, please do not hesitate to ask. Good luck with your research!
Scientist