On the Willems closure with respect to Ws

The behavioural theory of Willems exploits the correspondence between the algebraic properties of the module describing the behaviour and the properties of the behaviour. For excellent introductions to the behavioural theory in the 1 D case and the n D case, we refer the reader to Polderman & Willems (1998) and Pillai & Shankar (1998), respectively. As opposed to the case of 1 D linear dynamical systems corresponding to a set of linear ODEs with constant coefficients, in the n D case there is a greater variety of possible solution spaces and the correspondence between modules and the associated behaviours may not be bijective: indeed, it depends on the solution space considered. There exists a bijective correspondence between modules and behaviours if one considers the space of smooth functions or distributions, and this was established in Oberst (1990). (In the 1 D case this was known, and it is the content of Theorem 3.6.2 on page 100 of Polderman & Willems, 1998.) However, this bijective correspondence does not go through for several classical spaces, such as the space of tempered distributions, S ′ (Rn). This naturally brings one to the notions of a Willems module and the Willems closure of a module with respect to a given solution space, which were first introduced in the works of Pillai & Shankar (1998) and Shankar (1999, 2001). This is analogous to the definition of the radical of an ideal in a polynomial ring and the correspondence between affine varieties and radical ideals. Roughly speaking, the notion of a Willems module can be explained as follows. Start with a given set of equations and find the corresponding behaviour in a certain solution space, say W . Now find all the equations that this behaviour satisfies. If this set of equations turns out to be the same set one started off with, then the original set is said to be Willems with respect to the solution space under consideration. The Willems submodules play an important role in the behavioural theory and furthermore, from a purely mathematical point of view, the determination of Willems submodules is the Nullstellensatz for systems of PDEs, the analogue of Hilbert Nullstellensatz, where as opposed to looking at the zeros in Cn of a set of polynomial equations, one now looks at the solutions of a set of linear PDEs with constant coefficients. In Shankar (1999), it is determined when a module is Willems with respect to the Schwartz space of tempered distributions. In this paper, following Shankar (1999), we perform a similar calculation for another space, which we call ‘the space of distributions