Mathematics – Optimization and Control
Scientific paper
2008-05-06
SIAM J. Control Optim. 48 (2009), no. 2, 925-940
Mathematics
Optimization and Control
SIAM Journal on Control and Optimization To appear (2008)
Scientific paper
10.1137/080723351
If two control systems on manifolds of the same dimension are dynamic equivalent, we prove that either they are static equivalent --i.e. equivalent via a classical diffeomorphism-- or they are both ruled; for systems of different dimensions, the one of higher dimension must ruled. A ruled system is one whose equations define at each point in the state manifold, a ruled submanifold of the tangent space. Dynamic equivalence is also known as equivalence by endogenous dynamic feedback, or by a Lie-B\"acklund transformation when control systems are viewed as underdetermined systems of ordinary differential equations; it is very close to absolute equivalence for Pfaffian systems. It was already known that a differentially flat system must be ruled; this is a particular case of the present result, in which one of the systems is assumed to be "trivial" (or linear controllable).
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