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

Abstract

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. Then we look into restrictions of a general autonomous system, not necessarily strongly autonomous. We first define the notion of a free subspace of the domain—a 1-d subspace where every possible 1-d trajectory can be obtained by restricting the trajectories of the autonomous system. Then we give an algebraic characterization of free-ness of a 1-d subspace for a scalar autonomous system. Using this algebraic criterion we then give a full geometric characterization of free (and non-free) subspaces. As a consequence of this we show that the set of non-free 1-d subspaces is a closed linear set in the projective (n−1)-space. Finally, we show that restriction to a non-free subspace always equals the solution set of the ODEs coming from the intersection ideal. As a corollary to this we give a necessary and sufficient condition for a system to be stable in a given direction.