How to know if a system is dissipative?
Jul12-12, 01:58 AM
Hi guys. I got stuck in this problem and I am wondering anyone can help.
1. The problem statement, all variables and given/known data
For the the linear system describe by d^2/dt^2 y(t) + (5)d/dt y(t) + (6)y(t) = f(t),
where f(t) is the input to the linear system, and the solution to the differential equation is the response of the system. Determine if the system is dissipative.
2. Relevant equations
None I guess.
3. The attempt at a solution
I solve for the 2nd order differential equation by looking for the roots of its characteristic equation, y^2+5y+6=0, and got (y+2)(y+3)=0, so the roots are -2 and -3 and the general solution to the differential equation is y(t)=-2(C1)e^-2t-3(C2)e^-3t, and this express is defined as the response of the linear system, according to the problem statement. Then I am not sure how to proceed.
I assume a dissipative system means that the input is always bigger than the output? How am I suppose to compare "-2(C1)e^-2t-3(C2)e^-3t" against "d^2/dt^2 y(t) + (5)d/dt y(t) + (6)y(t)"? I can't find any similar example online. Any input will be appreciated!
Or do I take the limit of -2(C1)e^-2t-3(C2)e^-3t as t approaches infinity? Then I will get zero so I say the system is dissipative? Thanks!
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