Dynamic Equivalence of Control Systems and Infinite Permutation Matrices

To each dynamic equivalence of two control systems is associated an infinite permutation matrix. We investigate how such matrices are related to the existence of dynamic equivalences.


Introduction
A control system is an underdetermined ODE system of the form 9 x " f pt, x, uq, where x " px i q are called the state variables, u " pu α q the control variables. The meaning of "control" is clear: under suitable regularity conditions, specifying a control function uptq and an initial value xpt 0 q uniquely determines a local 'trajectory' xptq that satisfies the ODE system and the initial value. When f does not explicitly depend on t, the control system is said to be autonomous. In the current paper, all control systems considered are autonomous.
Let 9 x " f px, uq and 9 y " gpy, vq be two control systems. Suppose that there exist mappings φ " φpx, u, 9 u, : u, ..., u ppq q and ψ " ψpy, v, 9 v, : v, ..., v pqq q such that, ‚ for any solution pxptq, uptqq of the first system, the function py, vq " φpx, u, 9 u, : u, ..., u ppq q is a solution of the second; ‚ for any solution pyptq, vptqq of the second system, the function px, uq " ψpy, v, 9 v, : v, ..., v pqq q is a solution of the first; ‚ moreover, applying φ and ψ successively to a solution pxptq, uptqq of the first system yields the same solution pxptq, uptqq, and, similarly, applying ψ and φ successively to a solution pyptq, vptqq of the second system yields the same solution pyptq, vptqq. If all these conditions are satisfied, we say that the pair of maps pφ, ψq defines a dynamic equivalence between the two control systems. Intuitively, a dynamic equivalence provides a one-to-one correspondence between the spaces of solutions of the two control systems.
Fixing a dynamic equivalence defined by a pair of maps pφ, ψq, one can always find the smallest p, q ě 0 so that φ " φpx, u, 9 u, : u, ..., u ppq q and ψ " ψpy, v, 9 v, : v, ..., v pqq q. We call such a pair pp, qq the height of the corresponding dynamic equivalence. A dynamic equivalence with height p0, 0q is known as a static (feedback) equivalence, in which case φ, ψ are inverses of each other as diffeomorphisms.
An immediate question is: How much more general is the notion of dynamic equivalence than that of static equivalence. Classical results (see [Pom95]) suggest that the answer depends on the number of control variables. In particular, a dynamic equivalence between two control systems with a single control variable is necessarily static. It is also well known that the number of control variables is invariant under a dynamic equivalence. However, in the cases of 2 or more controls, a precise answer to the question above remains largely unknown.
In [Sta13], the author considered all control-affine systems with 3 states and 2 controls, proving that three statically non-equivalent systems are pairwise dynamically equivalent at height p1, 1q. In addition, he introduced a new method of studying dynamic equivalences of two control systems. He proved that to each dynamic equivalence is associated an infinite permutation matrix. Intuitively, such a matrix tells us how the 'generating 1-forms' of certain prolongations of the two control systems (viewed as Pfaffian systems), when chosen appropriately, relate under a dynamic equivalence.
In the current work, we present further properties of dynamic equivalences that can be derived using the associated infinite permutation matrices. First we prove that there is a rank matrix (Definition 3.1) associated to a dynamic equivalence, which has a more 'invariant' nature than an associated infinite permutation matrix (Proposition 3.2). Then we prove several inequalities and equalities (Propositions 4.1 and 4.2 ) satisfied by the rank matrix. Using these results, we prove an inequality satisfied by the height pp, qq of a dynamic equivalence (Theorem 4.1). In particular, this inequality implies the Theorem 4.3. Two control systems with the same number of state variables and 2 control variables can only be dynamically equivalent at height pp, qq with p " q.

Control Systems and Dynamic Equivalence
2.1. Control Systems of Type pn, sq.
Definition 2.1. A control system of type pn, sq ps ă nq is an underdetermined ODE system (1) 9 x " f px, uq, where x " px i q P R n , u " pu α q P R s , and f " pf i q : R n`s Ñ R n is a smooth function satisfying rankˆB f i Bu α˙" s on some open domain 1 in R n`s . Here, x i are called the state variables, u α the control variables.
For a control system, there is an equivalent geometric characterization. Let D Ă R n be the domain of the state variables x " px i q. The admissible tderivatives of x i , as imposed by the control system, are given by specifying a submanifold Σ Ă T D that submerses onto D with rank-s fibers. Each fiber is precisely parametrized by the control variables u " pu α q. The submanifold Σ induces an embedding which is the identity in the R-factor (with coordinate t) and t-independent in the Σ-factor. In coordinates, this embedding may be written as In other words, the system (1) corresponds to the Pfaffian system pM, C M q, where M :" RˆΣ, and C M is the restriction to M of the standard contact system C " xdx i´9 x i dty n i"1 on the jet bundle J 1 pR, Dq -RˆT D. Conversely, let Σ Ă T D be a submanifold that submerses onto a domain D Ă R n with rank-s fibers. The Pfaffian system pM, C M q corresponds to a control system of type pn, sq.

Prolongations of a Control System.
Definition 2.2. Let pM, C M q be a control system. For any p P M , an integral element at p is a 1-dimensional vector subspace L " Ru Ă T p M satisfying θpuq " 0 for all 1-forms θ P C M and dtpuq ‰ 0.
In coordinates, at each point p P M , the 1-forms in C M are linear combinations of dx i´fi px, uqdt pi " 1, ..., nq. It follows that, once we specify values of s auxiliary constants u p1q α , an integral element at p is uniquely determined by the vanishing of s extra 1-forms du α´u p1q α dt pα " 1, ..., sq. In other words, the space of integral elements at p P M is parametrized by the s parameters u p1q α . Let u p1q :" pu p1q α q.
Definition 2.3. The first total prolongation of a control system pM, C M q is the Pfaffian system pM p1q , C p1q q, where M p1q is the space of integral elements of pM, C M q, with the standard coordinates pt, x, u, u p1q q; C p1q is the Pfaffian system generated by dx i´fi px, uqdt, du α´u p1q α dt, pi " 1, ..., n; α " 1, ..., sq.
1 Since our study is local, we henceforth assume that such a domain is the entire R n`s .
Let k be a positive integer. One can start from a control system pM, C M q of type pn, sq and generate total prolongations successively for k times. The result will be denoted as pM pkq , C pkq q, where M pkq has the coordinates pt, x, u, u p1q , ..., u pkq q and C pkq is generated by the 1-forms pi " 1, ..., n; α " 1, ..., s; ℓ " 1, ..., k´1q.
When k " 0, we simply let pM p0q , C p0q q denote pM, C M q.
It is clear that pM pkq , C pkq q is a control system of type pn`ks, sq.

Dynamic Equivalence.
Given two control systems, it is natural to regard them as equivalent if one can establish a one-to-one correspondence between their solutions.
Of course, two control systems 9 x " f px, uq and 9 y " gpy, vq are equivalent in the sense above when they can be transformed into each other by a change of variables of the form y " φpxq, v " ψpx, uq and x " φ´1pyq, u " ρpy, vq. This notion of equivalence is called static equivalence, which, in particular, requires that the two equivalent control systems are of the same type. However, it is possible for two systems of different types to have a one-to-one correspondence between their solutions, as is indicated by the following standard property of jet bundles.
Proposition 2.1. Let pM, C M q be a control system. Let π : M pkq Ñ M be the canonical projection from its k-th total prolongation. Any integral curve τ : R Ñ M of pM, C M q has a unique lifting τ pkq : R Ñ M pkq (i.e., satisfying π˝τ pkq " τ ) to an integral curve of pM pkq , C pkq q. In addition, for each integral curve σ : R Ñ M pkq of pM pkq , C pkq q, its projection π˝σ is an integral curve of pM, C M q.
In other words, given two control systems, a one-to-one correspondence between their solutions may involve differentiation. This motivates the following notion of equivalence.
Definition 2.4. Two control systems pM, C M q and pN, C N q are said to be dynamically equivalent if there exist positive integers p and q and submersions Φ : M ppq Ñ N and Ψ : N pqq Ñ M that satisfy 2 i. Φ, Ψ preserve the t-variable and are t-independent in the state and control components; ii. Φ cannot factor through any M pkq for k ă p; Ψ cannot factor through any N pℓq for ℓ ă q; iii. for each integral curve τ : R Ñ M of pM, C M q, Φ˝τ ppq is an integral curve of pN, C N q; for each integral curve σ : R Ñ N of pN, C N q, Ψ˝σ pqq is an integral curve of pM, C M q; iv. letting τ and σ be as in iii, we have: For the convenience of the reader, we present the commutative diagram: (1) It is easy to verify that Definition 2.4 defines an equivalence relation.
(2) By this definition, a control system pM, C M q is dynamically equivalent to each of its total prolongations pM pkq , C pkq q.
(3) A dynamic equivalence with p " q " 0 is a static equivalence. To see this, let pt, x, uq and pt, y, vq be coordinates on M and N , respectively. Represent Φ in local coordinates as pt, y, vq " pt, φpx, uq, ψpx, uqq. Since Φ maps integral curves of pM, C M q to integral curves of pN, C N q, it is necessary that, for each dy i´gi py, vqdt P C N , its pull-back is contained in C M . It follows that φpx, uq is independent of u. A similar argument applies to Ψ. Finally, Condition iv in Definition 2.4 implies that Φ and Ψ are inverses of each other.
Definition 2.5. We call the pair of integers pp, qq in Definition 2.4 the height of the corresponding dynamic equivalence.
Given two dynamically equivalent control systems pM, C M q and pN, C N q, if needed, one could always apply a partial prolongation (for details, see [Sta13]) to one of them such that the resulting systems have the same number of states and are still dynamically equivalent. This perspective suggests understanding all dynamic equivalences between control systems with the same number of states.
Proposition 2.2. Let pM, C M q and pN, C N q be control systems with the same number of states. The height pp, qq of a dynamic equivalence between them must satisfy either p " q " 0 or p, q ą 0.
Proof. Suppose that the following commutative diagram represents a dynamic equivalence of height pp, 0q pp ą 0q between pM, C M q and pN, C N q: By the assumption, rankpC M q " rankpC N q. Since and since π, Φ, Ψ are all submersions, it is necessary that xπ˚C M y " xΦ˚C N y. Now, Φ is constant along the Cauchy characteristics of Φ˚C N , since the Cartan system (see [BCG`13]) of C N generates the entire cotangent bundle of N . On the other hand, the Cauchy characteristics of π˚C M are precisely the fibres of π. This proves that Φ factors through M , a contradiction to the choice of p. The case when p " 0, q ą 0 is similar.

Infinite Permutation Matrices Associated to a Dynamic Equivalence
In this section, we assume p, q ą 0 unless otherwise noted. Given a control system pM, C M q, one automatically obtains a system of projections π k,j : M pkq Ñ M pjq , pk ě jq.
The inverse limit of this projective system is denoted as Let π k : M p8q Ñ M pkq be the canonical projections. Since πk ,j C pjq Ď C pkq for all k ě j ě 0, one can define C p8q to be the differential system generated by ď kě0 πk C pkq .
The pair pM p8q , C p8q q is called the infinite prolongation of pM, C M q. Now suppose that pM, C M q p 9 x " f px, uqq and pN, C N q p 9 y " gpy, vqq are two control systems of types pn 1 , s 1 q and pn 2 , s 2 q, respectively, between which a dynamic equivalence of height pp, qq is given by It is shown in [Sta13] that Φ and Ψ induce, respectively, maps Moreover, if we let ω 0 " dx´f px, uqdt, ω k " du pk´1q´upkq dt pk ě 1q, η 0 " dy´gpy, vqdt, η k " dv pk´1q´vpkq dt pk ě 1q, then there exist matrices A i j , B i j pi, j ě 0q satisfying 3 : P1. for k ě 0, we have p`k are equal for k ě 1; similarly for B k q`k . Hence, we denote A 8 :" A k p`k , B 8 :" B k q`k , pk ě 1q. P3. rankpA 0 p q " rankpA 8 q ą 0, rankpB 0 q q " rankpB 8 q ą 0. For proofs of Properties P1 and P2, see [Sta13]. To see why P3 holds, note that there exists an n 1ˆs1 matrix F such that

Consequently,
F A 1 p`1 "´A 0 p . By Definition 2.1, F has full rank. Therefore, rankpA 1 p`1 q " rankpA 0 p q. The case for B i j is similar. Property P3 follows. 3.1. Two Classical Theorems.
Concerning dynamic equivalences between two control systems, the following two classical theorems are of fundamental importance.   Car14]) A dynamic equivalence between two control systems of the same type pn, 1q can only have height pp, qq " p0, 0q (i.e., the equivalence is static).
It turns out that these two theorems are consequences of the fact that the following matrices are inverses of each other, after taking into account the maps Φ p8q and Ψ p8q : (2) A "¨A

‚.
In fact, suppose that pM, C M q and pN, C N q are of types pn 1 , s 1 q and pn 2 , s 2 q, respectively. We can assume n 1 " n 2 ": n, because partial prolongations preserve the number of controls. Consequently, there are two possible cases: p " q " 0 or p, q ą 0. In the former case, we have static equivalence.
In the latter case, the form of matrix A implies that the n`rs 1 linearly independent components of ω 0 , ..., ω r are all linear combinations of the np r`pqs 2 components of η 0 , ..., η r`p . When s 1 ą s 2 , this is impossible because n`pr`pqs 2 ă n`rs 1 as long as r ą ps 2 s 1´s2 .
Theorem 3.1 follows. Furthermore, when p, q ą 0, A and B being inverses of each other requires that either A 8 B 8 " 0 or B 8 A 8 " 0. This is impossible when s 1 " s 2 " 1, in which case A 8 and B 8 are just nonvanishing functions. Theorem 3.2 then follows from Proposition 2.2. More generally, we have the Lemma 3.3. Suppose that E is a dynamic equivalence of height pp, qq pp, q ą 0q between two control systems with s controls. It is necessary that the associated matrices A 8 and B 8 satisfy (3) 2 ď rankpA 8 q`rankpB 8 q ď s.

The Infinite Permutation Matrix S.
Let pM, C M q, pN, C N q, ω i , η i , A, B be as above. In [Sta13], it is proved that there exist transformations where g i i " g i`1 i`1 , h i i " h i`1 i`1 for all i ě 1, such that, pointwise,
In addition, we have Proposition 3.1. Assume that p, q ą 0. The matricesĀ andB satisfȳ A "B T . In particular, the infinite permutation matrixĀ must be of the form in other words,Ā q`k k "B T 8 pk ě 1q, andĀ q`ℓ k " 0 for all ℓ ą k.
Definition 3.1. Let S, taking the form of (5), be an infinite permutation matrix obtained from a dynamic equivalence with height pp, qq pp, q ą 0q between two control systems. Let r i j " rankpĀ i j q. We define the rank matrix associated to S to be RpSq :" pr i j q. Given a dynamic equivalence, an associated matrix S may depend on the choice of the transformationsω " Gω andη " Hη. However, we have Proposition 3.2. If S 1 and S 2 are two infinite permutation matrices obtained from the same dynamic equivalence, then their rank matrices satisfy RpS 1 q " RpS 2 q.
Proof. Suppose that the underlying dynamic equivalence has height pp, qq pp, q ą 0q. One can write S 1 and S 2 in block forms: Let u i j :" rankpU i j q and v i j :" rankpV i j q. Since S 1 , S 2 arise from the same dynamic equivalence, there exist invertible block lower triangular matrices 4 K "¨k where k i i " k i`1 i`1 , ℓ i i " ℓ i`1 i`1 for all i ě 1, such that As results of the forms of K and L, we have To see why this is true, consider the submatrix pW 0 p , which must be equal to its column rank u 0 To see this, consider the submatrix p`1 q must be equal to its column rank pu 0 p`1 q. Let i decrease from p and use piiq. The desired result follows. (iv) u j i " v j i for j ě 2, 0 ď i ă p`j. This can be verified by a similar comparison between the column and row ranks of the submatrices This completes the proof.
As a consequence of Proposition 3.2, we have Corollary 3.4. If S 1 and S 2 are two infinite permutation matrices obtained from a dynamic equivalence with height pp, qq pp, q ą 0q, then there exist block diagonal matrices , where k i i and ℓ i i are usual permutation matrices of appropriate sizes, 5 such that S 1 " LS 2 K.
Proof. This is because S 1 and S 2 (i) are permutation matrices; and (ii) have the same rank in each pair of corresponding blocks.
Corollary 3.5. Let S be an infinite permutation matrix obtained from a dynamic equivalence of height pp, qq pp, q ą 0q. Using the notations in (2), the associated rank matrix RpSq " pr i j q satisfies r k p`k " rankpA 8 q, r q`k k " rankpB 8 q, pk ě 0q.
Proof. To see why this is true, first notice that a transformation (4) preserves the ranks of A 8 and B 8 ; then use Property P3.

The Height of a Dynamic Equivalence
Given two control systems pM, C M q (of type pn 1 , sq) and pN, C N q (of type pn 2 , sq) that are dynamically equivalent, it is interesting to ask: What are the possible heights of a dynamic equivalence? A particular instance is Theorem 3.2, which tells us that the height suggests how control systems with s " 1 and s ą 1 are qualitatively different. The current section will present some new results in this direction.

Some Rank Equalities and Inequalities.
Throughout this section, let pM, C M q and pN, C N q be as above. Suppose that a dynamic equivalence between them has height pp, qq with p, q ą 0. Let S be an associated infinite permutation matrix (Equation (5)), obtained from a choice of coframes pω 0 ,ω 1 , ...q and pη 0 ,η 1 , ...q. Let RpSq " pr i j q be the corresponding rank matrix (Definition 3.1).
Proof. This is because the matrix S is an infinite permutation matrix.
The proofs of ii and iii are similar; we leave them to the reader.
Proof. The rank matrix associated to E takes the form where r k p`k " rankpA 8 q " r 1 , r q`k k " rankpB 8 q " r 2 for all k ě 0. Let By Proposition 4.1, we have (15) 0 ď C ď pq´1qδ, 0 ď D ď pp´1qδ.

The conclusion follows.
Corollary 4.2. Let E be a dynamic equivalence between two control systems of types pn 1 , sq and pn 2 , sq, respectively. If rankpA 8 q`rankpB 8 q " s, then the height pp, qq of E must satisfy: when p, q ą 0, we have (16) n 1`r ankpA 8 q¨p " n 2`r ankpB 8 q¨q.
Proof. This is an immediate consequence of setting δ " 0 in (14).
Theorem 4.3. The height pp, qq of a dynamic equivalence between two systems of the same type pn, 2q must satisfy p " q.
For any p ą 1, the following pair of submersions Φ : M ppq Ñ N and Ψ : N ppq Ñ M define a dynamic equivalence with height pp, pq between pM, C M q and pN, C N q: py, vq " Φpx, u, ..., u ppq q " pu pp´1q 2´x 1 , x 2 , x 3 ; u ppq 2´u 1 , u 2 q; px, uq " Ψpy, v, ..., v ppq q " pv pp´1q 2´y 1 , y 2 , y 3 ; v ppq 2´v 1 , v 2 q. Of course, the dynamic equivalences given in Example 2 are somewhat artificial, since the underlying control systems are already statically equivalent. It is therefore more meaningful to ask: Fixing two control systems that are dynamically equivalent, is there any method of finding the minimum height pp, qq of a dynamic equivalence between them? This question remains open.