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Kalman's fundamental notion of a controllable state space system, firstdescribed in Moscow in 1960, has been generalised to higher order systems byJ.C.Willems, and further to distributed systems defined by partial differentialequations. It turns out, that for systems defined in several important spacesof distributions, controllability is now identical to the notion of vectorpotential in physics, or of vanishing homology in mathematics.  These lectures will explain this relationship, and a few of its consequences.It will also pose an important question: does a controllable system, in anyspace of distributions, always admit a vector potential? In other words, isKalman's notion of a controllable system, suitably generalised, nothing more -nor less - than the possibility of describing the dynamics of the system bymeans of a vector potential?

Controllability and Vector Potential: Six Lectures at Steklov