Register to reply

Linear control ODE - exponential convergence?

Share this thread:
Jul3-12, 07:27 PM
P: 1

I'm having hard times with the following simple linear ODE coming from a control problem:
$$u(t)' \leq \alpha(t) - u(t)\,,\quad u(0) = u_0 > 0$$
with a given smooth α(t) satisfying
$$0 \leq \alpha(t) \leq u(t) \quad\mbox{for all } t\geq 0.$$
My intuition is that $$\lim_{t\to\infty} u(t) - \alpha(t) = 0,$$
and that the convergence is exponential, i.e., $$|u(t) - \alpha(t)| = u(t) - \alpha(t) \leq c_1 e^{-c_2 t}.$$
For instance, if α was a constant, then the exponential convergence clearly holds (just solve the related ODE and use a "maximum principle").
Do you see a simple proof for time-dependent α (could not prove neither of the "statements" - probably I'm missing something very elementary); or is my intuition wrong?

Many thanks, Peter
Phys.Org News Partner Science news on
What lit up the universe?
Sheepdogs use just two simple rules to round up large herds of sheep
Animals first flex their muscles
Jul5-12, 12:17 AM
Sci Advisor
HW Helper
P: 9,839
For a u(t) to exist satisfying those constraints puts constraints on α(t). Not exactly that it is monotonic non-increasing, but something approaching that. Do you know of such a constraint (beyond that implied)?
Jul5-12, 12:36 AM
P: 2
For the fundamental solution set S={ex,e2x,e3x} can we construct a linear ODE with constant coefficients?

I have verified that the solution set is linearly independent via wronskian. I have got the annihilators as (D-1),(D-2),(D-3). However after that I'm not sure how to proceed. What do I do to get the ODE?


Jul5-12, 03:36 AM
Sci Advisor
HW Helper
P: 9,839
Linear control ODE - exponential convergence?

Consider α(t) as follows:
In the nth period of time B, α(t) = A > 0, except for the last e-n, where it is 0. If u'(t) = α(t) - u(t) and vn = u(Bn) - A,
vn+1 = vne-B - A(1-e-e-n)
> vne-B - Ae-n
So vn > wn where
wn+1 = wne-B - Ae-n
which I believe gives:
wn = Ce-Bn - Ae-n/(e-1-e-B)
wn tends to 0 as n goes to infinity, not going negative. Hence u(t) converges to A, and the difference between u and α exceeds A on occasions beyond any specified t.

Register to reply

Related Discussions
Linear system of exponential equations Calculus & Beyond Homework 18
How to calculate the convergence point for exponential function? Calculus & Beyond Homework 2
Linear independence of the set of exponential functions Calculus & Beyond Homework 1
Soving an eqn with exponential & linear parts Calculus & Beyond Homework 5
Radii of convergence for exponential functions Calculus & Beyond Homework 2