Why a PDE is an infinite dimensional system

[zhidayat]

Hi,

I hope I posted in the right group. I read some papers about infinite dimensional systems and gave PDEs as examples of infinite dimensional systems. So far, I still cannot get why is that so.

Could everybody here help me giving relation between a PDE and an infinite dimensional system?

Thank you.

 

[HallsofIvy]

The solution to an ordinary differential equation, of order n, can be written as a linear combination of n independent solutions, with n undetermined constants- a vector space of dimension n.

The solution to a partial differential equation, of order n, can be written as a linear combination of n independent solutions but with n undetermined functions. The functions themselves constitute an infinite dimensional vector space.

 

[zhidayat]

Thank you ... :)

 

[Bacle]

Maybe if you told us the context--or the actual source--we may be better
able to help you.

 

[zhidayat]

the context is about control of distributed parameter systems which are modeled as PDEs. in papers i read (and try to understand), authors of the papers wrote the PDE and transform it into a state space representation using linear operator in Hilbert space and did the analysis using semigroups theory. since i am a beginner in the subject, i lost the link between the PDE and the semigroups analysis presented in the paper.

I would be grateful if i can get additional information.