On restrictions of n-D systems to 1-D subspaces

In this paper, we look into restrictions of the solution set of a system of PDEs to 1-d subspaces. We bring out its relation with certain intersection modules. We show that the restriction, which may not always be a solution set of differential equations, is always contained in a solution set of ODEs coming from the intersection module. Next, we focus our attention to restrictions of strongly autonomous systems. We first show that such a system always admits an equivalent first order representation given by an n-tuple of real square matrices called companion matrices. We then exploit this first order representation to show that the system corresponding to the intersection module has a state representation given by the restriction of a linear combination of the companion matrices to a certain invariant subspace. Using this result we bring out that the restriction of a strongly autonomous system is equal to the system corresponding to the intersection module.