What do you mean by “If the integral is surjective (as a linear operator), then we can find an input to achieve any desired state”?
Submitted by: asa Submitted on: December 6, 2004 Identifier: L5.1 Surjective is what is commonly known as “onto”. That is, if we have an operator f: X -> Y, then the image of f is all of Y. In this context, we have a linear operator (the operator, with integral, etc., that takes a system state plus input at time 0 to the system state at time T). This is an operator from the space of (initial conditions (cross) inputs u(tau)) to the space of (final system states). We really want to consider just the input term (the integral), which is an operator from the inputs u(tau) to system states. If this operator is surjective, that means we can reach any final state (accounting for the given initial condition), as long as we have complete control over the input.
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