What are the applications of the concept of various vector spaces like Banach space, Hilbert space, Eucledian space etc. in Control theory? What is the reason behind consideration of different spaces? Some simple examples in this context will be great.
Hi Anirudh,
Each of these vector spaces ensures certain useful mathematical properties that are useful in specific control problems. They make theorems and propositions easy to prove and established in those specific cases.
Hence their popularity. I am sure you can locate numerous examples in different contexts.
-Sanjay
Thank you Sir. Sir can you provide some simple examples for Banach and Hilbert spaces.
At the case of Hilbert space the main property is for example possibility of orthogonality of basis functions. Banach space theory is good for applications in the case of kinetic theory( transport theory ) without assumption of existence of scalar product . Pontrjagin's maximum principle does not always hold for the case of fractional calculus of variations.
Hilbert space is an inner product space (real or complex), that means, you can benefit from concepts of length, angle and distance. Also, it is a complete space for the norm induced by the inner product, that means you can benefit from topology ( open, closed sets, limit, ...). So look what Control theory requires and you will find the relation.
The fundamental idea is the transformation (of #s )from one constrained space to another , Eg: state space (independent of inputs ) and depends on the transfer function that characterizes the performance.
Transformation to another space can help in realizing the properties such as : direction (resultant vectors)/commutative /linearity/dependency or independence of variables (or points ) in the particular space.
Eg: Hilbert also show the ortho-normality
Design of optimal control, stability, stabilization etc. is specific for each case of these infinite dimensional systems.
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