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:
onsider 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 from there though.
I am a Master Student in Economics. My mathematics background is insufficient to solve this problem. I hope somebody can help. The problem is due tomorrow. Thank you in advance for your help.