Algebraic characterization of autonomy and controllability of behaviours of spatially invariant systems

We give algebraic characterizations of the properties of autonomy and of controllability of behaviours of spatially invariant dynamical systems, consisting of distributional solutions, that are periodic in the spatial variables, to a system pf partial differential equations corresponding to a polynomial matrix M in (C[\xi_1,...,\xi_d, \tau])^{m \times n}.


Introduction
Consider a homogeneous, linear, constant coefficient partial differential equation, in R d+1 described by a polynomial p ∈ C[ξ 1 , . . . , ξ d , τ ]: In the behavioural approach to control theory, the "behaviour" B W (p) associated with p in W (where W is an appropriate solution space, for example smooth functions C ∞ (R d+1 ) or distribution spaces like D ′ (R d+1 ) or S ′ (R d+1 ) and so on), is defined to be the set of all solutions w ∈ W that satisfy the above PDE (1.1), and one characterizes algebraically (in terms of algebraic properties of the polynomial p) certain analytical properties of B W (p) (for example, the control theoretic properties of autonomy, controllability, stability, and so on). See for example [8], [9], [1] for distinct takes on this in the context of systems described by partial differential equations. For example let us consider the property of "autonomy", which means the following.
Definition 1.1. Let W be a subspace of D ′ (R d+1 ) which is invariant under differentiation, that is, for all w ∈ W, ∂ ∂x k w ∈ W for k = 1, . . . , d, and ∂ ∂t w ∈ W.
If p ∈ C[ξ 1 , . . . , ξ d , τ ], then the behaviour (of p in W) is We call the behaviour B W (p) autonomous (with respect to W) if the only w ∈ B W (p) satisfying w| t<0 = 0 is w = 0.
corresponding to p is autonomous if and only if deg p = deg p(0, τ ).
Here by deg(·), we mean the total degree, which is the maximum (over the monomials occuring in the polynomial) of the sum of the degrees of the exponents of indeterminates in the monomial. Also, p(0, τ ) denotes the polynomial in C[τ ] obtained from p ∈ C[ξ 1 , . . . , ξ d , τ ] by making the substitutions ξ k → 0 for k = 1, . . . , d.
There has been recent interest in "spatially invariant systems", see for example [3], [4], where one considers solutions to PDEs that are periodic along the spatial direction. So it is a natural question to ask what the analogue of Proposition 1.2 is, when we replace the solution space D ′ (R d+1 ) with one that consists only of those solutions that are periodic in the spatial directions. In this article, our first main result is the following one, characterizing autonomy of spatially invariant systems.
is autonomous if and only if (V(C ξ (p))) (2πiA −1 Z d ) = ∅, where A is the matrix with its rows equal to the transposes of the column vectors a 1 , . . . , a d : Here D ′ A (R d+1 ) is, roughly speaking, the set of all distributions on R d+1 that are periodic in the spatial direction with a discrete set A of periods. The precise definition of D ′ A (R d+1 ) is given in Subsection 1.1. Also, in the above result, the condition is an algebraic-geometric condition, saying that the variety V(C ξ (p)) of the "ξ-content" of p does not meet the discrete set of points in 2πiA −1 Z d . The notion of the ξ-content of a polynomial is defined in Subsection 1.2.
From [5, §34], T is a tempered distribution, and from the above it follows by taking Fourier transforms that (1 − e 2πia k ·y ) T = 0 for k = 1, . . . , d. It can be seen that for some scalars α v ∈ C, and where A is the matrix with its rows equal to the transposes of the column vectors a 1 , . . . , a d : Also, in the above, δ v denotes the usual Dirac measure with support in v: By the Schwartz Kernel Theorem (see for instance [7, p. 128 , the space of all continuous linear maps from D(R) to D ′ (R d ), thought of as vector-valued distributions. For preliminaries on vectorvalued distributions, we refer the reader to [2]. We indicate this isomorphism by putting an arrow on top of elements of D ′ (R d+1 ). Thus for w ∈ D ′ (R d+1 ), we set w ∈ L(D(R), D ′ (R d )) to be the vector valued distribution defined by for ϕ ∈ D(R) and ψ ∈ D(R d ). That this specifies a well-defined distribution in D ′ (R d+1 ), can be seen using the fact that for every Φ ∈ D(R d+1 ), there exists a sequence of functions (Ψ n ) n that are finite sums of direct products of test functions, that is, Ψ n = k ψ k ⊗ ϕ k , where ψ k ∈ D(R d ) and ϕ k ∈ D(R), such that Ψ n converges to Φ in D(R d+1 ). We also have ∂ ∂x k w = 2πiy k w for k = 1, . . . , d, and ∂ ∂t w = ∂ ∂t w.

Proof of Theorem 1.3
Before we prove our main result, we illustrate the basic idea behind 'If' part: by taking Fourier transform, the partial derivatives with respect to the spatial variables are converted into the polynomial coefficients a k (2πiy), where y is the vector of Fourier transform variables y , . . . , y d . But the support of w is carried on a family of lines, indexed by n ∈ Z d , in R d+1 parallel to the time axis. So we obtain a family of ODEs, parameterized by n ∈ Z d , and by "freezing" an n ∈ Z d , we get an ODE, where for a solution we can indeed say that zero past implies a zero future, and so the proof can be completed easily.

Example 3.1 (Diffusion equation). Consider the diffusion equation
Here the ξ-content C ξ (p) is the ideal in C[ξ, . . . , ξ d ] generated by the two polynomials 1 and −ξ 2 1 − · · · − ξ 2 d , and so C ξ (p) is the full ring C[ξ, . . . , ξ d ]. Consequently, its variety in C d is empty. Hence the behaviour B D ′ A (R d+1 ) (p) is always autonomous, no matter what A is. Note that this is in striking contrast to what happens when we look at just distributional solutions: since deg(p(ξ 1 , . . . , ξ d , τ )) = deg(τ − (ξ 2 1 + · · · + ξ 2 d )) = 2 = deg(p(0, . . . , 0, τ )) = deg(τ ) = 1 we have from Proposition 1.2 that B D ′ (R d+1 ) (p) is not autonomous, and this outcome is physically unexpected. Indeed, if we imagine the case of diffusion of heat, in which case the w is the temperature, say along a metallic rod when d = 1, then zero temperature upto time t = 0 should mean that the temperature stays zero in the future as well (since the above PDE describes the situation when no external heat is supplied). However, when one considers distributional solutions, one can have pathological solutions with a zero past that are nonzero in the future! But if we choose the physically "correct" solution space in this context, namely functions which at each time instant have a spatial profile belonging to L ∞ (R), then it can be shown that solutions that are zero in the past are also zero in the future, as expected. So the real reason for the nonautonomy when one considers solutions in D ′ (R d+1 ) is that there is no restriction on the spatial profiles of the solutions at each time instant, and wild growth (such as something which grows faster than e |x| 2 ) is allowed. However, with a periodic profile in the spatial direction, namely when the spatial profile is in D ′ A (R d ), we know that the spatial profile is automatically tempered (see for example [5]), and as we have seen above, in this case the behaviour B D ′ A (R d+1 ) (p) is autonomous, in conformity with our physical expectation. Similarly, since c 0 = 1 = 0, it follows from Theorem 1.5 that B D ′ A (R d+1 ) (p) is never controllable. ♦ Remark 3.2. We end with an open question: Is there an algebraic-geometric characterization in terms of p of approximate controllability of B D ′ A (R d+1 ) (p)? Here, by approximate controllability, we mean the following.
It is not hard to show that for a nontrivial behaviour B D ′ A (R d+1 ) (p), one has the following heirarchy of properties: controllability ⇒ approximate controllability ⇒ ¬(autonomy).
Thus in our search for the appropriate algebraic-geometric condition characterizing approximate controllability of B D ′ A (R d+1 ) (p), we expect an algebraic-geometric condition that lies between the two characterizations of controllability and approximate controllability given in this article in Theorems 1.3 and 1.5.