The Diffiety Institute Preprint Series

Preprint DIPS–01 January 24, 2002

Available via INTERNET: http://diffiety.ac.ru/

The Diffiety Institute

Polevaya 6-45, Pereslavl-Zalessky, 152140 Russia.

Flatness conditions for control systems

by

V. N. CHETVERIKOV

Flatness conditions for control systems

V. N. CHETVERIKOV

Abstract. We study the problem of looking for flat control systems and flat outputs. A control system is flat if there exists a change of variables transforming this system to the trivial system: ẋ = u. The change of variables can be of most general type. In this work, a necessary and sufficient condition for flatness is obtained. We give also a flatness test and a method for constructing the change of variables. An example is considered to illustrate the test and the method.

1. Introduction

Flat control systems were introduced by Fliess, Lévine, Martin, and Rouchon. Later on, numerous interesting examples of flat control systems was found (see, for example, [3, 4]). Since flatness is equivalent to endogenous dynamic feedback linearization [1, 3], problems of control theory are solvable for flat systems (see [4]). This facts explain the interest in looking for flatness conditions [1].

Using geometric methods, Aranda-Bricaire, Moog, and Pomet introduced an infinitesimal Brunovsky form for nonlinear systems and obtained a necessary and sufficient condition for flatness [1]. Namely flatness of a control system means existence of an invertible differential operator of some type satisfying some conditions (see Theorem 2 below). In geometry of differential equations [6, 7] the operators of this type are investigated and are called C-differential.

In this work, we regard a dynamical control system as an underdetermined system of ordinary differential equations and use a geometric approach to nonlinear differential equations developed by Vinogradov and his school [6, 7]. Main objects of this approach are diffieties. In the case of control systems, the diffieties are infinite dimensional manifolds endowed with one-dimensional distributions, which are called the Cartan distributions.

Besides, we extend the deformation theory of pseudogroup structures defined on finite dimensional manifolds to our infinite dimensional case. This theory was developed by Spencer [9], Guillemin and Sternberg [5] and solves the following problem. Recall that a pseudogroup is a transformation group whose elements are solutions of some partial differential equation. Let some transitive continuous pseudogroup act on a manifold M. Consider some geometric structure on M. This structure is transformated under the action of the pseudogroup. The deformations (transformations) of the structure under the action of the pseudogroup are investigated in this theory. In our case, the manifold M is the diffiety for the control system under consideration. The geometric structure is a vector version of the complex of Cartan forms (9). The pseudogroup is the group of changes of variables on the diffiety.

Key words and phrases. nonlinear control systems, flatness, C-transformation, dynamic feedback linearization.

This work was supported in part by RFBR grant 99-01-00863.

Using properties of C-differential operators and ideas of the deformation theory, we rewrite the condition from [1] and get new flatness conditions (see Theorem 4 below). These conditions are equations on some form-valued differential operator R. To find a flat output it is necessary and sufficient to solve these equations.

The paper is organized as follows. In section 2, we present a geometric framework for control systems, like [4, 8]. In section 3, results from [1] are generalized to the nonautonomous case. Section 4 contains a result about higher symmetries of control systems. In geometry of differential equations higher symmetries of an equation are interpreted as vector fields on a space of solutions of the equation. We use higher symmetries in section 5, where we give our main result (Theorem 4) and a method for search of flat outputs. In section 6, we solve our flatness conditions by means of a small parameter method exposed in [5]. An example of application of our methods is also discussed there.

2. A geometric framework

Consider a smooth control system

ẋ = f(t, x, u), (1)

where x = (x_{1}, …, x_{n}) ∈ ℝ^{n} is the state, u = (u_{1}, …, u_{m}) ∈ ℝ^{m} is the control input. For system (1), the diffiety (or infinitely prolonged system) is an infinite dimensional manifold E^{∞} with coordinates

(t, x, u^{(0)}, u^{(1)}, …, u^{(l)}, …), (2)

where u^{(l)} denotes the vector variable corresponding to the lth order derivative of u with respect to t. Every solution s = (s_{x}(t), s_{u}(t)) of system (1) defined in a neighborhood of a point t_{0} determines a point of E^{∞} with the coordinates

t = t_{0},  x = s_{x}(t_{0}),  u^{(l)} = \frac{∂^{l}s_{u}(t_{0})}{∂t^{l}},  l ≥ 0.

This point is called the infinite jet of the solution s at the point t_{0}. From the existence theorem for systems of ordinary differential equations it follows that any point of the diffiety E^{∞} is an infinite jet of some solution of (1).

The Cartan distribution on E^{∞} is one-dimensional and is determined by the vector field

D = \frac{∂}{∂t} + f(t, x, u^{(0)})\frac{∂}{∂x} + u^{(1)} \frac{∂}{∂u^{(0)}} + ⋯ + u^{(s+1)} \frac{∂}{∂u^{(s)}} + …, (3)

which is called the total derivative with respect to t. The Lie derivative along D is simply the time-derivative according to system (1). We denote by Dω the Lie derivative of a differential form (or a function) ω along D.

A smooth function on E^{∞} is a function smoothly depending on a finite (but arbitrary) number of coordinates (2). By F(E) denote the ℝ-algebra of smooth functions on E^{∞}. Differential forms on E^{∞} are finite sums, whereas vector fields may be given by infinite sums with coefficients in F(E) (see, for example, (3)).

Let E^{∞} and S^{∞} be two diffieties, F : E^{∞} ⟶ S^{∞} a diffeomorphism preserving the independent variable, i. e., F^{∗}(t) = t. The diffeomorphism F is called a C-transformation [6], a Lie–Bäcklund isomorphism [3] or an equivalence by endogenous dynamic feedback [1, 3] if it preserves the Cartan distribution, i. e., F_{∗}(D) = D. In this definition the diffieties E^{∞} and S^{∞} can be changed to their open subsets.

Let E^{∞} be the diffiety for system (1), θ ∈ E^{∞}. System (1) is said to be flat [3] or endogenous dynamic feedback linearizable [1, 3] around θ if there exists a C-transformation F of a neighborhood of θ to an open subset of the diffiety for the trivial system

ẋ = u, x ∈ ℝ^{m}, u ∈ ℝ^{m}.

In what follows, E^{∞} denotes the diffiety for system (1).

3. B-basises and B-regular points

Here we remind some concepts from [1] and simultaneously generalize them to the nonautonomous case.

Let C^{1}Λ(E) be the F(E)-module of differential 1-forms annihilating the Cartan distribution, i. e.,

ω ∈ C^{1}Λ(E) ⟺ ω(D) = 0.

Define the operator d_{C} : F(E) ⟶ C^{1}Λ(E) by the rule g ↦ dg − D(g)dt. The operator d_{C} possesses many properties of the de Rham differential d. In particular,

d_{C}g(x, u^{(0)}, …) = ∑_{i} \frac{∂g}{∂x_{i}} d_{C}x_{i} + ∑_{j} \frac{∂g}{∂u_{j}^{(0)}} d_{C}u_{j}^{(0)} + ….

However d_{C}g = 0 iff g is a function of t.

Obviously, in coordinate system (2) the module C^{1}Λ(E) is generated by forms

d_{C}x_{1}, …, d_{C}x_{n}, d_{C}u_{1}^{(0)}, …, d_{C}u_{m}^{(0)}, …, d_{C}u_{1}^{(l)}, …, d_{C}u_{m}^{(l)}, ….

Denote by H_{0} the F(E)-submodule of C^{1}Λ(E) generated by forms d_{C}x_{1}, …, d_{C}x_{n}. By definition, put

H_{k+1} = {ω ∈ H_{k} | Dω ∈ H_{k}}, k ≥ 0.

The dimension of some submodule H ⊂ C^{1}Λ(E) at a point θ ∈ E^{∞} is the dimension of the space of covectors {ω_{θ} | ω ∈ H}. A point θ ∈ E^{∞} is called Brunovský-regular (or shortly B-regular) if in a neighborhood of θ the rank of the matrix ∂f/∂u is equal to m and the dimensions of H_{k} and H_{k} + D(H_{k}) are constant for any k > 0.

Note that the dimension of H_{k} at any point is finite and H_{k+1} ⊂ H_{k}. It follows that in a neighborhood of a B-regular point there exists an integer k^{∗} such that H_{k+1} = H_{k} = H_{k^{∗}} for k ≥ k^{∗}. By ρ denote the dimension of H_{k^{∗}} in a neighborhood of a B-regular point under consideration.

Remark 1. In the autonomous case we can consider only functions, differential forms, and vector fields that are independent of t. In this case, d_{C}g = dg, E^{∞} is a manifold with coordinates (x, u^{(0)}, …, u^{(l)}, …) (without t), C^{1}Λ(E) is identified with Λ^{1}(E^{∞}). Also, all concepts and facts from this section are transformed to concepts and facts from [1].

Theorem 1. [1] In a neighborhood of a B-regular point for system (1) there exist ρ functions χ_{1}, …, χ_{ρ} of t, x_{1}, …, x_{n} and m forms ω_{1}, …, ω_{m} ∈ H_{0} such that

(1) {d_{C}χ_{1}, …, d_{C}χ_{ρ}} is a basis of the module H_{k^{∗}};

(2) {d_{C}χ_{1}, …, d_{C}χ_{ρ}} ∪ {D^{j}(ω_{k}) | k = 1, …, m, j ≥ 0} is a basis of the module C^{1}Λ(E).

It is easily shown that nonautonomous system (1) has an infinitesimal Brunovský form similar to that given in [1] for the autonomous case.

Further, if system (1) is flat around a B-regular point θ, then ρ = 0. In this case, the set {ω_{1}, …, ω_{m}} is called a B-basis or a linearizing Pfaffian system [1] at the point θ. The proof of Theorem 1 is similar to the proof of the corresponding theorem from [1] and gives a procedure for finding a B-basis. Namely for ρ = 0 we have H_{k^{∗}} = 0 and H_{k^{∗}−1} ≠ 0. Choose a basis ω_{1}, …, ω_{m_{1}} (m_{1} ≤ m) of the module H_{k^{∗}−1}. Then the 1-forms ω_{1}, …, ω_{m_{1}}, D(ω_{1}), …, D(ω_{m_{1}}) are F(E)-linear independent and lie in H_{k^{∗}−2}. To obtain a basis of the module H_{k^{∗}−2} we add some 1-forms ω_{m_{1}+1}, …, ω_{m_{2}} (m_{1} ≤ m_{2} ≤ m) to this collection. Repeating this construction for H_{k^{∗}−3}, …, H_{0}, we get a B-basis {ω_{1}, …, ω_{m}}.

Denote

C^{s}Λ(E) = \underbrace{C^{1}Λ(E) ∧ ⋯ ∧ C^{1}Λ(E)}_{s times}, s > 1,

C^{0}Λ(E) = F(E), C^{∗}Λ(E) = ⨁_{s=0}^{∞} C^{s}Λ(E).

The elements of C^{s}Λ(E) are called Cartan s-forms on E^{∞}. A differential operator of the type

Δ = g_{0} + g_{1}D + g_{2}D^{2} + ⋯ + g_{k}D^{k}, g_{0}, g_{1}, …, g_{k} ∈ F(E), (4)

is called a C-differential operator of order k. A C-differential operator acts on Λ^{∗}(E^{∞}) and takes C^{∗}Λ(E) to itself.

Remark 2. The composition of two C-differential operators is a C-differential operator. The set of all C-differential operators on E^{∞} forms a noncommutative ℝ-algebra. The set C^{1}Λ(E) is a left module over this algebra. From Theorem 1 it follows that in the case ρ = 0, this module is free and a B-basis is its basis.

Let s = s(t) be a (local) solution of (1). Denote by g|_{s} the restriction of the function g ∈ F(E) to the set of infinite jets of the section s. Clearly, g|_{s} is a function of t. The differential operator

g_{0}|_{s} + g_{1}|_{s} \frac{d}{dt} + g_{2}|_{s} \frac{d^{2}}{dt^{2}} + ⋯ + g_{k}|_{s} \frac{d^{k}}{dt^{k}}

is called the restriction of the C-differential operator (4) to s and is denoted by Δ|_{s}.

Remark 3. It is easily shown that Δ(g)|_{s} = Δ|_{s}(g|_{s}) for any function g ∈ F(E) and for any solution s. Moreover, if for some differential operator Δ and for any solution s there exists a differential operator Δ|_{s} satisfying this equality, then the operator Δ is a C-differential one [7].

A matrix whose entries are C-differential operators determines in the standard way the operator on the set of columns of differential forms. This operator is called a matrix C-differential operator. The order of a matrix C-differential operator is the maximal order of its entries.

Theorem 2. [1] Let θ be a B-regular point for system (1), ρ = 0, and {ω_{1}, …, ω_{m}} be a B-basis at θ. System (1) is flat at θ if and only if in a neighborhood of θ there exist m functions h_{1}, …, h_{m} on E^{∞} and an invertible matrix C-differential operator Δ such that

( d_{C}h_{1} ; ⋮ ; d_{C}h_{m} ) = Δ ( ω_{1} ; ⋮ ; ω_{m} ) . (5)

The functions h_{1}, …, h_{m} from Theorem 2 are called linearizing or flat output [1, 3].

4. Higher symmetries of control systems

A vector field on E^{∞} without the term ∂/∂t is called vertical. A vertical field X on E^{∞} is called a higher symmetry of system (1) if [X, D] = 0.

Let ρ = 0. From Theorem 1 it follows that in a neighborhood of a B-regular point there exists a matrix C-differential operator P of order k^{∗} such that

( d_{C}x_{1} ; ⋮ ; d_{C}u_{m} ) = P ( ω_{1} ; ⋮ ; ω_{m} ) , (6)

where {ω_{1}, …, ω_{m}} is a B-basis. By P_{i} denote ith row of the matrix P, i. e.,

d_{C}x_{i} = P_{i}(ω), d_{C}u_{j} = P_{n+j}(ω),

where ω is the m-column consisting of the forms ω_{1}, …, ω_{m}.

Theorem 3. [2] Suppose θ is a B-regular point for system (1) and ρ = 0. Then in a neighborhood of θ any higher symmetry of system (1) has the form

S_{φ} = ∑_{i=1}^{n} P_{i}(φ)\frac{∂}{∂x_{i}} + ∑_{k=0}^{∞} ∑_{j=1}^{m} D^{k}(P_{n+j}(φ))\frac{∂}{∂u_{j}^{(k)}} ,

where φ is a m-column of arbitrary functions φ_{1}, …, φ_{m} ∈ F(E).

The vector function φ is called the generating function of the symmetry S_{φ}. Clearly, the Lie derivative along any symmetry S_{φ} takes C^{i}Λ(E) to C^{i}Λ(E). Below the Lie derivative along S_{φ} is also denoted by S_{φ}.

5. Main results

A map A of C^{∗}Λ(E) to C^{∗}Λ(E) such that

A(Ω) = α_{0} ∧ Ω + α_{1} ∧ DΩ + ⋯ + α_{k} ∧ D^{k}Ω, α_{0}, α_{1}, …, α_{k} ∈ C^{r}Λ(E), (7)

is called a CΛ-operator of order k and of grading r. The CΛ-operator (7) is denoted by

A = α_{0} ∧ 1 + α_{1} ∧ D + ⋯ + α_{k} ∧ D^{k}. (8)

A matrix whose entries are CΛ-operator of order ≤ k and of grading r is called matrix CΛ-operator of order ≤ k and of grading r. Obviously, matrix C-differential operators of order ≤ k are matrix CΛ-operators of order ≤ k and of grading 0. Matrix CΛ-operators are main objects of our approach.

The operator d_{C} : C^{0}Λ(E) = F(E) ⟶ C^{1}Λ(E) can be extended to Λ^{∗}(E^{∞}) [7]. We use only the extensions to C^{∗}Λ(E) given by

d_{C}(fd_{C}g_{1} ∧ ⋯ ∧ d_{C}g_{i}) = d_{C}f ∧ d_{C}g_{1} ∧ ⋯ ∧ d_{C}g_{i},

where i > 0 and f, g_{1}, …, g_{i} ∈ F(E). Since the module C^{1}Λ(E) is generated by the image of the operator d_{C}, the map d_{C} from C^{i}Λ(E) to C^{i+1}Λ(E) is uniquely defined by the last equality for any i > 0. It can be proved that the sequence

0 ⟶ F(E) ⟶^{d_{C}} C^{1}Λ(E) ⟶ ⋯ ⟶ C^{i}Λ(E) ⟶^{d_{C}} C^{i+1}Λ(E) ⟶ ⋯ (9)

is exact in all terms except for the term F(E). The homology in the term F(E) is equal to the algebra of all functions of t. The map d_{C} is called the Cartan differential on E^{∞}.

Further, extend d_{C} (and S_{φ}) to columns of forms by the rule (ω_{i}) ↦ (d_{C}ω_{i}). Denote by [,] and [,]_{+} the commutator and the anticommutator of maps, i. e.,

[A, B] = A ∘ B − B ∘ A, [A, B]_{+} = A ∘ B + B ∘ A.

Note that d_{C} and S_{φ} are not CΛ-operators. But their commutators or their anticommutators with CΛ-operators can be CΛ-operators. Indeed, consider the CΛ-operator (8) of grading r. Since d_{C} and D commute, the CΛ-operator

d_{C}α_{0} ∧ 1 + d_{C}α_{1} ∧ D + ⋯ + d_{C}α_{k} ∧ D^{k}

is [d_{C}, A] for even r and [d_{C}, A]_{+} for odd r. Hence [d_{C}, A] for even r and [d_{C}, A]_{+} for odd r are CΛ-operators of order ≤ k and of grading r + 1. Similarly, [S_{φ}, A] is a CΛ-operator of order ≤ k and of grading r for any r ≥ 0. Obviously, the same is valid for a matrix CΛ-operator A.

The internal product X⌋A of a vector field X on E^{∞} and a CΛ-operator A of grading r of the form (8) is the following CΛ-operator of grading r − 1:

X⌋A = (X⌋α_{0}) ∧ 1 + (X⌋α_{1}) ∧ D + ⋯ + (X⌋α_{k}) ∧ D^{k},

where X⌋α denotes the internal product of X and a differential form α. For a matrix CΛ-operator R with entries R_{ij}, the matrix CΛ-operator X⌋R with entries X⌋R_{ij} is called the internal product of X and R.

Consider a matrix CΛ-operator R of grading 1 (see Theorem 4). For any matrix CΛ-operator A and any symmetry S_{φ}, denote

S_{φ}{A} = [S_{φ}, A] + (S_{φ}⌋R) ∘ A,
S_{φ}⟨A⟩ = [S_{φ}, A] − A ∘ (S_{φ}⌋R).

Theorem 4. Let θ be a B-regular point for system (1), ρ = 0, and let ω be the mcolumn of 1-forms ω_{1}, …, ω_{m} forming a B-basis at θ. System (1) is flat at θ if and only if in a neighborhood of θ there exists a matrix CΛ-operator R of grading 1 such that (A) (d_{C} + R)(ω) = 0; (B) [d_{C}, R]_{+} + R ∘ R = 0; (C) the orders of the operators

S_{φ_{k}}{… S_{φ_{2}}{S_{φ_{1}}⌋R} … } and S_{φ_{k}}⟨… S_{φ_{2}}⟨S_{φ_{1}}⌋R⟩ … ⟩

are limited for any positive integer k and any collection of higher symmetries S_{φ_{1}}, …, S_{φ_{k}}.

The condition (B) from the theorem is a well-known equation from the deformation theory (see [9] and [5]). The condition (C) is a typical one for the infinite-dimensional case and allows us to pass to the finite dimensional case. Any nilpotent operator R satisfies the condition (C).

Proof of necessity. By Theorem 2, there exist an invertible matrix C-differential operator Δ and a m-column h of the functions h_{1}, …, h_{m} satisfying (5). Set

R = Δ^{−1} ∘ [d_{C}, Δ]. (10)

Obviously,

d_{C} + R = Δ^{−1} ∘ d_{C} ∘ Δ. (11)

If we combine this with d_{C}^{2} = 0 and Δ(ω) = d_{C}h, we get (A) and (B):

(d_{C} + R)(ω) = (Δ^{−1} ∘ d_{C} ∘ Δ)(ω) = (Δ^{−1} ∘ d_{C} ∘ d_{C})(h) = 0,
[d_{C}, R]_{+} + R ∘ R = (d_{C} + R)^{2} = Δ^{−1} ∘ d_{C} ∘ Δ ∘ Δ^{−1} ∘ d_{C} ∘ Δ = 0.

Since S_{φ} and D commute, we see that

S_{φ}⌋R = S_{φ}⌋(Δ^{−1} ∘ [d_{C}, Δ]) = Δ^{−1} ∘ [S_{φ}, Δ]. (12)

It now follows that the order of the operator S_{φ}⌋R is less than or equal to the sum of the orders of Δ^{−1} and of Δ. Using (12), we get also

[S_{φ_{2}}, [S_{φ_{1}}, Δ]] = [S_{φ_{2}}, Δ ∘ (S_{φ_{1}}⌋R)] = [S_{φ_{2}}, Δ] ∘ (S_{φ_{1}}⌋R) + Δ ∘ [S_{φ_{2}}, S_{φ_{1}}⌋R]

= Δ ∘ ((S_{φ_{2}}⌋R) ∘ (S_{φ_{1}}⌋R) + [S_{φ_{2}}, S_{φ_{1}}⌋R]) = Δ ∘ S_{φ_{2}}{S_{φ_{1}}⌋R}.

As above, the order of the operator S_{φ_{2}}{S_{φ_{1}}⌋R} is less than or equal to the same sum of the orders.

On the other hand,

0 = [d_{C}, Δ^{−1} ∘ Δ] = [d_{C}, Δ^{−1}] ∘ Δ + Δ^{−1} ∘ [d_{C}, Δ].

Using this equality, we get consecutively

[d_{C}, Δ^{−1}] = −R ∘ Δ^{−1},
[S_{φ}, Δ^{−1}] = S_{φ}⌋[d_{C}, Δ^{−1}] = −(S_{φ}⌋R) ∘ Δ^{−1},
[S_{φ_{2}}, [S_{φ_{1}}, Δ^{−1}]] = −[S_{φ_{2}}, (S_{φ_{1}}⌋R) ∘ Δ^{−1}] = −[S_{φ_{2}}, S_{φ_{1}}⌋R] ∘ Δ^{−1}
− (S_{φ_{1}}⌋R) ∘ [S_{φ_{2}}, Δ^{−1}] = −S_{φ_{2}}⟨S_{φ_{1}}⌋R⟩ ∘ Δ^{−1}.

Arguing as above, we see that the order of the operator S_{φ_{2}}⟨S_{φ_{1}}⌋R⟩ is also less than or equal to the sum of the orders of Δ^{−1} and of Δ. Continuing this line of reasoning, we get (C). ▶

To prove the sufficiency, we need two lemmas.

Lemma 5. Let an infinite jet θ ∈ E^{∞} of a local solution s of (1) be a B-regular point. Suppose ρ = 0, ω is a B-basis at θ, and a matrix CΛ-operator R of grading 1 satisfies the conditions (A), (B), and (C) in a neighborhood of θ. Then in a neighborhood of θ there exist unique matrix CΛ-operators Δ and ∇ of grading 0 satisfying

(13)  [d_{C}, Δ] = Δ ∘ R,  Δ|_{s} = id,
(14)  [d_{C}, ∇] = −R ∘ ∇,  ∇|_{s} = id,

where id denotes the identical operator.

Lemma 6. A left-invertible C-differential operator with a square matrix is two-sided invertible.

The proof of these lemmas are given after the proof of the second part of Theorem 4. Proof of sufficiency. From the first equalities in (13) and (14) it follows that

[d_{C}, Δ ∘ ∇] = [d_{C}, Δ] ∘ ∇ + Δ ∘ [d_{C}, ∇] = 0.

Using the second equalities in (13) and (14), we get

(Δ ∘ ∇)|_{s} = Δ|_{s} ∘ ∇|_{s} = id . (15)

On the other hand, the homology of the complex (9) in the term F(E) is the ℝ-algebra of smooth functions of t. Therefore coefficients of the operator Δ ∘ ∇ at D^{l}, l ≥ 0, are functions of t only. Combining this with (15), we get Δ∘∇ = (Δ∘∇)|_{s} = id. By Lemma 6, the operator Δ is two-sided invertible.

Thus we have (10) and thereby (11). Combining (11) with the condition (A), we get d_{C}(Δ(ω)) = 0. Since the complex (9) is exact in the term C^{1}Λ(E), we have Δ(ω) = d_{C}h for some vector function h. By Theorem 2, system (1) is flat. ▶

Proof of Lemma 5. Let the orders of the operators from the condition (C) be limited by an integer L. Consider the set C_{R} of matrix C-differential operators Δ of dimension m×m and of order ≤ L such that for any positive integer k and any collection of higher symmetries S_{φ_{1}}, …, S_{φ_{k}} the order of the operator Δ ∘ S_{φ_{k}}{… S_{φ_{2}}{S_{φ_{1}}⌋R} … } is less than or equal to L. The set C_{R} is nonempty, since it contains the identical operator id.

We shall prove that in a neighborhood of the point θ there exists an operator Δ from C_{R} satisfying (13). Clearly, C_{R} is a module over F(E). Besides, the orders of operators from C_{R} are limited. Therefore the module C_{R} has a finite basis {Δ_{1}, …, Δ_{l}}. Suppose

Δ = ∑_{i=1}^{l} f_{i}Δ_{i}, f_{i} ∈ F(E); (16)

then equation (13) is written as

∑ d_{C}f_{i} ∧ Δ_{i} + ∑ f_{i}([d_{C}, Δ_{i}] − Δ_{i} ∘ R) = 0. (17)

Now we show that S_{φ}⟨Δ⟩ ∈ C_{R} if Δ ∈ C_{R}. For any symmetry S_{φ} and any matrix C-differential operators Δ and □ one has

[S_{φ}, Δ ∘ □] = S_{φ}⟨Δ⟩ ∘ □ + Δ ∘ S_{φ}{□}. (18)

Put □ = S_{φ_{k}}⟨… S_{φ_{2}}⟨S_{φ_{1}}⌋R⟩ … ⟩. Since Δ ∈ C_{R}, the orders of the operators Δ ∘ □ and Δ ∘ S_{φ}{□} are less than or equal to L. Commutation with a symmetry S_{φ} does not increase order of a C-differential operator. Therefore, the order of [S_{φ}, Δ∘□] is less than or equal to L. Due to (18), the same is valid for S_{φ}⟨Δ⟩ ∘ □. That is S_{φ}⟨Δ⟩ ∈ C_{R}.

On the other hand, for any symmetry S_{φ} one has

S_{φ}⌋([d_{C}, Δ_{i}] − Δ_{i} ∘ R) = S_{φ}⟨Δ_{i}⟩ ∈ C_{R}.

Since higher symmetries generate F(E)-module of vertical fields, the last inclusion means that

[d_{C}, Δ_{i}] − Δ_{i} ∘ R = ∑_{j=1}^{l} ω_{ij} ∧ Δ_{j}, i = 1, …, l, (19)

where ω_{ij} ∈ C^{1}Λ(E) for any i, j.

Combining (17) and (19), we get

∑_{i=1}^{l} (d_{C}f_{i} + ∑_{j=1}^{l} f_{j}ω_{ji}) ∧ Δ_{i} = 0.

Since {Δ_{1}, …, Δ_{l}} is a basis, we see that equation (13) is equivalent to the system

d_{C}f_{i} + ∑_{j=1}^{l} f_{j}ω_{ji} = 0, i = 1, …, l. (20)

Note that the forms ω_{ji}, Dω_{ji}, j, i = 1, …, l, depend on a finite number of coordinates (2). Denote by z_{1}, …, z_{s} the coordinates from the collection (2) without t, on which these forms depend. To prove solvability of system (20), we introduce auxiliary variables v_{1}, …, v_{l} and consider a manifold M with the coordinates t, z_{1}, …, z_{s}, v_{1}, …, v_{l}. Let P be the distribution determined by the forms

dt, dv_{i} + ∑_{j=1}^{l} v_{j}ω_{ji}, i = 1, …, l.

Our aim now is to prove that the distribution P is integrable. Since dω_{ji} = d_{C}ω_{ji} + dt ∧ Dω_{ji}, we have

d(dv_{i} + ∑_{j=1}^{l} v_{j}ω_{ji}) = ∑_{j=1}^{l}(dv_{j} ∧ ω_{ji} + v_{j}d_{C}ω_{ji} + v_{j}dt ∧ Dω_{ji}).

Whence,

d(dv_{i} + ∑_{j=1}^{l} v_{j}ω_{ji}) = ∑_{j=1}^{l}(dv_{j} + ∑_{p=1}^{l} v_{p}ω_{pj}) ∧ ω_{ji}

+ ∑_{j=1}^{l}(v_{j}dt ∧ Dω_{ji}) + ∑_{j=1}^{l} v_{j}(d_{C}ω_{ji} − ∑_{p=1}^{l} ω_{jp} ∧ ω_{pi}).

Since the forms d_{C}ω_{ji} − ∑ ω_{jp} ∧ ω_{pi} contain d_{C}z_{α} only and do not contain dv_{α} and dt, the distribution P satisfies the Frobenius condition iff

∑_{j=1}^{l} v_{j}(d_{C}ω_{ji} − ∑_{p=1}^{l} ω_{jp} ∧ ω_{pi}) = 0, i = 1, …, l.

Further, the forms d_{C}ω_{ji} − ∑ ω_{jp} ∧ ω_{pi} are independent of v_{j}. Therefore the last system is equivalent to

d_{C}ω_{ji} = ∑_{p=1}^{l} ω_{jp} ∧ ω_{pi}, j, i = 1, …, l. (21)

Now let us prove (21). Using the condition (B) and properties of the commutator and the anticommutator, we get

[d_{C}, [d_{C}, Δ_{i}] − Δ_{i} ∘ R]_{+} = −[d_{C}, Δ_{i}] ∘ R − Δ_{i} ∘ [d_{C}, R]_{+} = −([d_{C}, Δ_{i}] − Δ_{i} ∘ R) ∘ R.

Combining this with (19), we get

[d_{C}, ∑_{j} ω_{ij} ∧ Δ_{j}]_{+} + (∑_{j} ω_{ij} ∧ Δ_{j}) ∘ R = 0.

It now follows that

0 = ∑_{j}(d_{C}ω_{ij} ∧ Δ_{j} − ω_{ij} ∧ ([d_{C}, Δ_{j}] − Δ_{j} ∘ R)) = ∑_{j}(d_{C}ω_{ij} ∧ Δ_{j} − ∑_{p} ω_{ij} ∧ ω_{jp} ∧ Δ_{p}).

Since {Δ_{1}, …, Δ_{l}} is a basis, we obtain (21).

Thus the distribution P is integrable. Every maximal integral manifold of P is given by a system of equations of the form

t = const, v_{1} = g_{1}, …, v_{l} = g_{l},

where g_{1}, …, g_{l} are some functions on E^{∞} depending on the coordinates z_{1}, …, z_{s}. This integral manifold is uniquely determined by a value of t and values of the functions g_{1}, …, g_{l} at a point of E^{∞}.

Let the given point θ ∈ E^{∞} be the infinite jet of the solution s at a point t_{0}. Any solution f_{1}, …, f_{l} ∈ F(E) of system (20) corresponds to the collection of maximal integral manifold of P of the form

t = t_{1}, v_{1} = f_{1}|_{t=t_{1}}, …, v_{l} = f_{l}|_{t=t_{1}}

for t_{1} from some neighborhood of the point t_{0}. So there exists a unique solution of system (20) satisfying the initial conditions

f_{i}|_{s} = e_{i}|_{s}, i = 1, …, l,

where ∑_{i} e_{i}Δ_{i} is the identical operator id. This means that there exists a unique matrix C-differential operators Δ such that conditions (13) hold.

In the same way, it can be proved existence of the operator ∇ satisfying conditions (14). In this case, it should be replaced the module C_{R} by the module of matrix Cdifferential operators ∇ of order ≤ L such that for any positive integer k and any collection of higher symmetries S_{φ_{1}}, …, S_{φ_{k}} the order of the operator S_{φ_{k}}⟨… S_{φ_{2}}⟨S_{φ_{1}}⌋R⟩ … ⟩∘ ∇ is less than or equal to L. ▶

Proof of Lemma 6. Let Δ and ∇ be C-differential operators with square matrixes and

Δ ∘ ∇ = id . (22)

Denote by Q the matrix C-differential operator ∇ ∘ Δ − id. To prove that the operator Q vanishes, we introduce some concepts.

Any matrix C-differential operator P can be presented in the form

P = ∑_{i=0}^{s} P_{i}D^{i},

where s is the order of P and P_{0}, P_{1}, …, P_{s} are matrixes of functions on E^{∞}. The matrix P_{s} ≠ 0 is called the symbol of the operator P and is denoted by smbl P. Clearly, the composition of the operators possesses the following property:

(23) P ∘ q = (smbl P smbl q) D^{s+l} + …,

where q is another matrix C-differential operator of the corresponding dimension, l is its order, the symbols of the operators P and q are multiplied as matrixes, and the dots denote the terms with D^{i}, i < s + l.

Further note that m-columns of C-differential operators determine in the standard way operators from F(E) to the set of m-columns of functions from F(E). Denote by CDiff_{l}^{m} the set of the m-columns whose entries are C-differential operators of order ≤ l. This set is a module over F(E) under the following multiplication. The product of an operator p ∈ CDiff_{l}^{m} and a function f ∈ F(E) is the operator taking a function g ∈ F(E) to the m-column p(fg). Let k be the order of the operator ∇ from (22). Denote by h_{∇}^{l} the F(E)-module homomorphism from CDiff_{l}^{m} to CDiff_{l+k}^{m} taking q to ∇ ∘ q. From (22) it follows that ker h_{∇}^{l} = 0 for any l ≥ 0.

Assume now that Q ≠ 0. Let smbl Q ≠ 0 at a point θ_{0} and r > 0 be the rang of the matrix smbl Q at this point. Using (22), we get

Q ∘ ∇ = ∇ ∘ Δ ∘ ∇ − ∇ = 0. (24)

This together with formula (23) yields (smbl Q smbl q) = 0 for any operator q ∈ im h_{∇}^{l} and for any l ≥ 0. Hence values of symbols of the operators from im h_{∇}^{l} at the point θ_{0} belong to a space of dimension n − r for any l ≥ 0. Therefore,

dim im h_{∇}^{l}|_{θ_{0}} ≤ (n − r)(l + k + 1), l ≥ 0.

On the other hand,

dim im h_{∇}^{l}|_{θ_{0}} = n(l + 1)

because ker h_{∇}^{l}|_{θ_{0}} = 0. However the inequality n(l + 1) ≤ (n − r)(l + k + 1) cannot hold for sufficiently large l. This contradiction concludes the proof. ▶

Theorems 2 and 4 and Lemma 5 give the following algorithm for search of flat outputs.

Step 1: For the control system under consideration, calculate a B-basis (see p. 3), B-regular points, and the operator P (see (6)).

Step 2: Find a matrix CΛ-operator R of grading 1 satisfying the conditions (A), (B), and (C).

Step 3: Solving equation (13), one obtains Δ. Here s = (x(t), u(t)) is some solution of the control system. To calculate Δ one has no need to know the functions x(t) and u(t) (see the example below).

Step 4: From equation (5) find a flat output.

6. Small parameter method

In this section we discuss step 2 of the algorithm given above. Suppose there exists a parameter ε such that the desired CΛ-operator R can be expanded in powers of ε. The parameter ε can be introduced in the following way. Consider a coordinate system on E^{∞}

(t, z_{1}, …, z_{k}, y_{1}, …, y_{i}, …) (25)

such that the total derivative has the form

D = \frac{∂}{∂t} + ∑_{i=1}^{k} f_{i}(t, z, y)\frac{∂}{∂z_{i}} + ∑_{j≥1} y_{m+j} \frac{∂}{∂y_{j}} , (26)

where f_{1}(t, z, y), …, f_{k}(t, z, y) are analytic functions in the variables y_{1}, y_{2}, . . .. Assume also that a B-basis ω and the desired CΛ-operator R are analytic in y.

Let us replace y_{j} by εy_{j} in D, ω, R for any j ≥ 1, where ε is a constant. By construction, D, ω, R are analytic in ε. Since d_{C} = d − dt ∧ D, the differential d_{C} is also analytic in ε. Let

ω = ∑_{i≥0} ε^{i}ω_{(i)},  D = ∑_{i≥0} ε^{i}D_{(i)},  d_{C} = ∑_{i≥0} ε^{i}d_{(i)}.

Expanding the operator R in powers of D and ε, we obtain

R = ∑_{i≥0} ε^{i}R_{(i)}, R_{(i)} = ∑_{j≥0} R_{ij}D^{j}, (27)

where the matrix R_{ij} is independent of ε for any i, j. Notice that R_{(i)} depends on ε because D depends on ε. But the operator R_{(i)} is uniquely determined by the operator

R_{(i)}^{0} = ∑_{j≥0} R_{ij}D_{(0)}^{j},

which is independent of ε.

It follows easily that

D_{(0)} = \frac{∂}{∂t} + ∑_{i=1}^{k} f_{i}(t, z, 0)\frac{∂}{∂z_{i}} + ∑_{j≥1} y_{m+j} \frac{∂}{∂y_{j}} . (28)

Besides, the matrixes R_{0j}, j ≥ 0, are independent of y_{1}, y_{2}, … and do not contain d_{C}y_{1}, d_{C}y_{2}, . . .. More precisely, R_{ij}, j ≥ 0, are homogeneous polynomials of degree i in y_{1}, y_{2}, …, d_{C}y_{1}, d_{C}y_{2}, . . ..

Consider the dynamical system

{ż_{i} = f_{i}(t, z, 0), i = 1, …, k, (29)

understood as a control system with the control input (y_{1}, …, y_{m}). Denote by E_{0}^{∞} the corresponding diffiety. A coordinate system on E_{0}^{∞} is (25). The total derivative on E_{0}^{∞} is vector field (28). Thus the diffieties E^{∞} and E_{0}^{∞} have the same infinite dimensional manifold, but different Cartan distributions.

Further, since d_{(0)} = d − dt ∧ D_{(0)} is the Cartan differential on E_{0}^{∞}, the sequence

0 ⟶ F(E_{0}) ⟶^{d_{(0)}} C^{1}Λ(E_{0}) ⟶ … ⟶ C^{i}Λ(E_{0}) ⟶^{d_{(0)}} C^{i+1}Λ(E_{0}) ⟶ … (30)

coincides with sequence (9) for system (29) and consequently is exact in all terms except for the term F(E).

Finally note that the equalities (A) and (B) are analytic in ε. Expanding differential forms from (A) and operators from (B) in powers of ε, we obtain the collection of equations on operators R_{(i)}^{0}, i = 0, 1, . . .. These equations can be solved inductively. Namely in the case i = 0, we obtain

(d_{(0)} + R_{(0)}^{0})(ω_{(0)}) = 0, [d_{(0)}, R_{(0)}^{0}]_{+} + R_{(0)}^{0} ∘ R_{(0)}^{0} = 0. (31)

Assume that we know R_{j} = R_{(0)} + εR_{(1)} + ⋯ + ε^{j}R_{(j)}. From (A) and (B) it follows that

(d_{C} + R_{j})(ω) = ε^{j+1}Ω_{(j+1)} + …, (32) [d_{C}, R_{j}]_{+} + R_{j} ∘ R_{j} = ε^{j+1}Q_{(j+1)} + …, (33)

where the 2-form Ω_{(j+1)} and the operator Q_{(j+1)} are independent of ε, the dots denote the terms with ε^{s}, s > j + 1. Taking into account coefficients at ε^{j+1} in (A) and (B), we obtain

R_{(j+1)}^{0}(ω_{(0)}) = −Ω_{(j+1)}, [d_{(0)} + R_{(0)}^{0}, R_{(j+1)}^{0}]_{+} = −Q_{(j+1)}. (34)

Theorem 7. Let the total derivative D for system (1) be of the form (26).

(1) If system (1) is flat at a B-regular point θ and all the functions y_{1}, y_{2}, … vanish at θ, then in a neighborhood of θ there exists an invertible matrix C-differential operator Δ_{0} on E_{0}^{∞} such that the CΛ-operator on E_{0}^{∞}

R_{(0)}^{0} = Δ_{0}^{−1} ∘ [d_{(0)}, Δ_{0}] (35)

satisfies (31).

(2) If the CΛ-operator R_{(0)}^{0} of the form (35) satisfies (31), then for any j ≥ 0 system (34) is solvable with respect to R_{(j+1)}^{0}.

Proof. Let Δ be the invertible matrix C-differential operator from Theorem 2. As above, we introduce the parameter ε. From the well-known Hadamard lemma it follows that any smooth function g depending on ε can be represented in the form g = g_{0} +εg_{1}, where g_{0} and g_{1} are smooth functions, g_{0} is independent of ε. This yields that

Δ = Δ_{0} + εΔ_{f}, Δ^{−1} = ∇_{0} + ε∇_{f}, (36)

where Δ_{0}, Δ_{f}, ∇_{0}, and ∇_{f} are smooth differential operators, Δ_{0}, ∇_{0} are independent of ε. Obviously, Δ_{0} and ∇_{0} are matrix C-differential operators on E_{0}^{∞}. Since

Δ ∘ Δ^{−1} = Δ_{0} ∘ ∇_{0} + ε(Δ_{f} ∘ ∇_{0} + Δ_{0} ∘ ∇_{f} + εΔ_{f} ∘ ∇_{f}) = id,

we see that Δ_{0} ∘ ∇_{0} = id. Similarly, ∇_{0} ∘ Δ_{0} = id. Therefore, ∇_{0} = Δ_{0}^{−1}.

Consider now the CΛ-operator (10) on E^{∞}. The operator R_{(0)}^{0} defined on page 12 satisfies R = R_{(0)}^{0} + εR_{f} for some differential operator R_{f}. Setting ε = 0 in (10), we get (35). In the same way, from the conditions (A) and (B) we get (31). This proves the first statement of the theorem.

To prove the second one, we first show that Ω_{(j+1)} in (32) is a column of Cartan 2-forms on E_{0}^{∞} and Q_{(j+1)} in (33) is a CΛ-operator of grading 2 on E_{0}^{∞}. It is readily seen that

α ∈ C^{i}Λ(E) ⟺ D⌋α = 0,

where D⌋α denotes the internal product of D and α. Since the left-hand side of equality (32) is a column of Cartan 2-forms on E^{∞}, we have

ε^{j+1}D⌋Ω_{(j+1)} + ⋯ = 0.

Taking into account coefficients at ε^{j+1}, we obtain D_{(0)}⌋Ω_{(j+1)} = 0. This yields that Ω_{(j+1)} in (32) is a column of Cartan forms on E_{0}^{∞}.

Similarly, the left-hand side of equality (33) is a CΛ-operator of grading 2 on E^{∞}. Therefore its coefficients at D^{l}, l ≥ 0, are matrixes of Cartan 2-forms on E^{∞}. Let us expand these forms in powers of ε. As above, the coefficients at ε^{j+1} are Cartan forms on E_{0}^{∞}. To obtain Q_{(j+1)}, one needs also to expand the vector field D in powers of ε. Evidently, the operator Q_{(j+1)} contains D_{(0)} and the mentioned Cartan forms on E_{0}^{∞} only. Therefore it is a CΛ-operator on E_{0}^{∞}.

Further, we rewrite (35) as

d_{(0)} + R_{(0)}^{0} = Δ_{0}^{−1} ∘ d_{(0)} ∘ Δ_{0}. (37)

Using this equality, we can represent (34) as

R̃(ω̃) = −Ω̃, (38) [d_{(0)}, R̃]_{+} = −Q̃, (39)

where

R̃ = Δ_{0} ∘ R_{(j+1)}^{0} ∘ Δ_{0}^{−1}, Q̃ = Δ_{0} ∘ Q_{(j+1)}^{0} ∘ Δ_{0}^{−1}

are CΛ-operators on E_{0}^{∞},

ω̃ = Δ_{0}(ω_{(0)}),  Ω̃ = Δ_{0}(Ω_{(j+1)})

are columns of Cartan forms on E_{0}^{∞}. Besides, from the first equation in (31) it follows that d_{(0)}(ω̃) = 0.

Let us expand CΛ-operators Q̃ and R̃ in powers of D_{(0)}, i. e.,

Q̃ = ∑_{α≥0} B_{α}D_{(0)}^{α},  R̃ = ∑_{α≥0} A_{α}D_{(0)}^{α},

where entries of matrixes B_{α} and A_{α} are Cartan forms on E_{0}^{∞}. Since d_{(0)} and D_{(0)} commute, equation (39) can be written as the system

d_{(0)}(A_{α}) = −B_{α} ∀α, (40)

where entries of the matrix d_{(0)}(A_{α}) are the images of the corresponding entries of the matrix A_{α} under the differential d_{(0)}. Formula (40) means that the matrix B_{α} consists of d_{(0)}-exact forms for any α. Since complex (30) is exact in the term C^{2}Λ(E_{0}), system (40) is solvable if and only if

d_{(0)}(B_{α}) = 0 ∀α.

In other notation,

[d_{(0)}, Q̃] = 0. (41)

To prove (41), note that

[d_{C} + R_{j}, (d_{C} + R_{j})^{2}] = 0. (42)

Since d_{C}^{2} = 0, the operator (d_{C} +R_{j})^{2} coincides with the left-hand side of equality (33). Combining this with (33) and (42), we get

ε^{j+1}[d_{C} + R_{j}, Q_{(j+1)}] + ⋯ = 0.

Taking into account coefficients at ε^{j+1}, we obtain [d_{(0)} + R_{(0)}^{0}, Q_{(j+1)}] = 0 and consequently (41).

Thus system (40) is solvable. Let {Ā_{α}} be some solution of (40). Then another solution {A_{α}} satisfies

d_{(0)}(A_{α} − Ā_{α}) = 0 ∀α.

Since complex (30) is also exact in the term C^{1}Λ(E_{0}), we get

A_{α} = Ā_{α} + d_{(0)}(C_{α})

for any α and some matrixes C_{α} of functions on E_{0}^{∞}. Consequently equation (39) is also solvable and any solution of (39) can be represented as

R̃ = R̄ + [d_{C}, □], (43)

where R̄ = ∑_{α} Ā_{α}D_{(0)}^{α} is some fixed solution of (39) and □ = ∑_{α} C_{α}D_{(0)}^{α} is an arbitrary matrix CΛ-operator of grading 0 on E_{0}^{∞}.

Now we show that there exists an operator □ such that the operator (43) satisfies (38), i. e.,

(R̄ + [d_{(0)}, □])(ω̃) = −Ω̃.

Since d_{(0)}(ω̃) = 0, the last equality is equivalent to

d_{(0)}(□(ω̃)) = −R̄(ω̃) − Ω̃. (44)

From (33) and (34) it follows that the operator (d_{C} + R_{j+1})^{2} has the terms only with ε^{s}, s ≥ j + 2. On the other hand, using the definition of R_{j+1} and (32), we get

(d_{C} + R_{j+1})^{2}(ω) = (d_{C} + R_{j+1})(ε^{j+1}Ω_{(j+1)} + ⋯ + ε^{j+1}R_{(j+1)}(ω)).

Taking into account coefficients at ε^{j+1}, we obtain

(d_{(0)} + R_{(0)}^{0})(Ω_{(j+1)} + R_{(j+1)}^{0}(ω_{(0)})) = 0.

Combining this with (37), we get

d_{(0)}(R̄(ω̃) + Ω̃) = 0. (45)

Since complex (30) is exact in the term C^{2}Λ(E_{0}), we see that

R̄(ω̃) + Ω̃ = d_{(0)}β_{0}

for some column β_{0} of Cartan 1-forms on E_{0}^{∞}.

Substituting d_{C} for d_{(0)} in β_{0}, we obtain the column β of Cartan 1-forms on E^{∞} such that β|_{ε=0} = β_{0}. Since ω is a B-basis on E^{∞}, there exists matrix C-differential operator ∇ on E^{∞} such that ∇(ω) = −β. If we put ε = 0, we get ∇|_{ε=0}(ω_{(0)}) = −β_{0}. Arguing as above, we see that ∇|_{ε=0} is a matrix C-differential operator on E_{0}^{∞}. To conclude the proof, it remains to note that the matrix C-differential operator □ = ∇|_{ε=0} ∘ Δ_{0}^{−1} satisfies (44). ▶

Using the theorem, we obtain series (27). However this series can diverge and is not unique. If an operator R_{(j+1)}^{0} satisfies (34) and □ is an arbitrary matrix C-differential operator such that

(d_{(0)} + R_{(0)}^{0})(□(ω_{(0)})) = 0,

then the operator

R_{(j+1)}^{0} + [d_{(0)} + R_{(0)}^{0}, □]

also satisfies (34).

When D and ω are polynomials in y, the operator R can be polynomial in y. In this case, applying theorem 7, we obtain an operator R satisfying the conditions (A) and (B). Then the condition (C) is checked.

Example. Consider the control system

ẋ_{1} = u_{1},
(46)  ẋ_{2} = u_{2},
ẋ_{3} = sin \frac{u_{1}}{u_{2}} .

To find a B–basis, note that any element of H_{0} has the form

ω = ∑_{i=1}^{3} a_{i}d_{C}x_{i}, a_{i} ∈ F(E).

We have

Dω = ∑_{i=1}^{3} Da_{i}d_{C}x_{i} + a_{1}d_{C}u_{1} + a_{2}d_{C}u_{2} + a_{3} cos(u_{1}/u_{2}) (\frac{1}{u_{2}} d_{C}u_{1} − \frac{u_{1}}{u_{2}^{2}} d_{C}u_{2}) . (47)

If Dω ∈ H_{0}, then the coefficients at d_{C}u_{1} and d_{C}u_{2} in (47) vanish. Whence,

a_{1} + a_{3} cos(u_{1}/u_{2})\frac{1}{u_{2}} = 0, a_{2} − a_{3} cos(u_{1}/u_{2})\frac{u_{1}}{u_{2}^{2}} = 0.

Therefore the module H_{1} is generated by

ω_{1} = d_{C}x_{1} − \frac{u_{1}}{u_{2}} d_{C}x_{2} − \frac{u_{2}}{cos(u_{1}/u_{2})}d_{C}x_{3}.

In the same way, the condition D(fω_{1}) ∈ H_{1} means that f = 0. We see that k^{∗} = 2 and ρ = 0. The 1-form ω_{2} should be chosen such that {ω_{1}, Dω_{1}, ω_{2}} is a basis of the module H_{0}. We put ω_{2} = d_{C}x_{3}. In this case, the set of B-regular points is {D(u_{1}/u_{2}) ≠ 0} and ω = {ω_{1}, ω_{2}} is a B–basis.

To obtain a polynomial form for D and ω, we introduce the following control variables:

v_{1} = \frac{u_{1}}{u_{2}} , v_{2} = u_{2}.

It is readily seen that D and ω are polynomials in the variables x_{2}, v_{2}, D^{i}v_{j}, i > 0, j = 1, 2. The corresponding coordinates on the diffiety are x_{1}, x_{2}, x_{3}, v_{1}, v_{2}, D^{i}v_{j}, i > 0, j = 1, 2. We have

ω_{(0)} = ( \begin{matrix}d_{C}x_{1} \\ d_{C}x_{3} \end{matrix} ) , ω_{(1)} = ( \begin{matrix}−v_{1}d_{C}x_{2} \\ 0 \end{matrix} ) .

Since d_{C}ω_{(0)} = 0, we can put R_{(0)} = 0. Equations (34) for j = 0 have the form

R_{(1)}^{0}(ω_{(0)}) = −d_{C}ω_{(1)}, [d_{C}, R_{(1)}^{0}]_{+} = 0.

Whence,

R_{(1)} = ( \begin{matrix}0 & −d_{C}(x_{2} cos v_{1}) ∧ D \\ 0 & 0 \end{matrix} ) .

It is clear that the operator R = R_{(1)} satisfies the conditions (A), (B), and (C). To find the operator Δ, note that the operators from (C) have the form

( \begin{matrix}0 & gD \\ 0 & 0 \end{matrix} ) , g ∈ F(E).

Consequently the module C_{R} consists of all the operators of the form

Δ = ( \begin{matrix}f_{1} & f_{3}D + f_{5} \\ f_{2} & f_{4}D + f_{6} \end{matrix} ) , f_{i} ∈ F(E).

Therefore equation (13) has the form

( \begin{matrix}d_{C}f_{1} & d_{C}f_{3} ∧ D + d_{C}f_{5} \\ d_{C}f_{2} & d_{C}f_{4} ∧ D + d_{C}f_{6} \end{matrix} ) = ( \begin{matrix}0 & −f_{1}d_{C}(x_{2} cos v_{1}) ∧ D \\ 0 & −f_{2}d_{C}(x_{2} cos v_{1}) ∧ D \end{matrix} ) .

So we get

(48)  d_{C}f_{3} = −f_{1}d_{C}(x_{2} cos v_{1}),  d_{C}f_{4} = −f_{2}d_{C}(x_{2} cos v_{1}),
d_{C}f_{1} = d_{C}f_{2} = d_{C}f_{5} = d_{C}f_{6} = 0.

Whence f_{i} are functions of t, x_{2}, and v_{1}. Let s be a solution (x_{i} = q_{i}(t), v_{j} = p_{j}(t)) of (46). From the initial condition Δ|_{s} = id and (48) it follows that

f_{1} = 1, f_{2} = 0, f_{3} = −x_{2} cos v_{1} + q_{2}(t) cos p_{1}(t), f_{4} = f_{5} = 0, f_{6} = 1.

Thus

Δ = ( \begin{matrix}1 & (−x_{2} cos v_{1} + q_{2}(t) cos p_{1}(t))D \\ 0 & 1 \end{matrix} ) .

It is easy to prove that for any functions q_{2}(t) and p_{1}(t) the operator Δ satisfies condition (13). Take q_{2}(t) = p_{1}(t) = 0. From Theorem 2, we obtain flat outputs h_{1} = x_{1} − x_{2}u_{1}/u_{2}, h_{2} = x_{3}.

References

[1] Aranda–Bricaire, E., Moog, C. H., Pomet J.-B.: An infinitesimal Brunovsky form for nonlinear systems with applications to dynamic linearization. In: Geometry in Nonlinear Control and Differential Inclusions (ed. by B. Jakubczyk, W. Respondek, and T. Rzezuchowski), pp. 19-33. Banach Center Publications, 32, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland 1995

[2] Chetverikov, V. N.: Higher symmetries and infinitesimal Brunovský form of control systems. Nauchniy Vestnik MGTU GA 16 (1999), 77-83 (in Russian).

[3] Fliess, M., Lévine, J., Martin, Ph., Rouchon, P.: Flatness and defect of non-linear systems: Introductory theory and examples. Int. J. Contr. 61, n. 6 (1995), 1327-1361.

[4] Fliess, M., Lévine, J., Martin, Ph., Rouchon, P.: A Lie–Bäcklund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Automat. Control 44, n. 5 (1999), 922-937.

[5] Guillemin, V., Sternberg, S.: Deformation theory of pseudogroup structures. Mem. Amer. Math. Soc. 64 (1966).

[6] Krasil’shchik, I. S., Lychagin, V. V., Vinogradov, A. M.: Geometry of Jet Spaces and Nonlinear Partial Differential Equations. Gordon and Breach, New York 1986

[7] Krasil’shchik, I. S., Vinogradov, A. M., Eds.: Symmetries and Conservation Laws for Differential Equations of Mathematical Physics. Amer. Math. Soc., Providence, RI 1999

[8] Pomet, J.-B.: A differential geometric setting for dynamic equivalence and dynamic linearization. In: Geometry in Nonlinear Control and Differential Inclusions (ed. by B. Jakubczyk, W. Respondek, and T. Rzezuchowski), pp. 319-339. Banach Center Publications, 32, Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland 1995

[9] Spencer, D. C.: Deformation of structures on manifolds defined by transitive continuous pseudogroups. I: Annals of Math. 76 (1962), p. 306-445, II, III: Annals of Math. 81 (1965), p. 389-450.

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