The Diffiety Institute Preprint Series

Preprint DIPS–11 November 14, 2001

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

The Diffiety Institute

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

Classical and higher symmetries of control systems

by

V. N. CHETVERIKOV, A. N. KANATNIKOV, A. P. KRISHCHENKO

Classical and higher symmetries of control systems

V. N. CHETVERIKOV, A. N. KANATNIKOV, A. P. KRISHCHENKO

Abstract. We study classical and higher infinitesimal symmetries of control systems. Defining equations for classical external symmetries are obtained in the general and affine cases. For computing higher symmetries we suggest a simple procedure involving algebraic operations and differentiation but not integration. Relations between classical symmetries and first integrals of control systems are established. An example is considered to illustrate our methods.

1. Introduction

Consider a nonlinear 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, f: E = ℝ^{1}×ℝ^{n}×ℝ^{m} → ℝ^{n} is a smooth (C^{∞}) function, ẋ ≡ dx/dt. Let rank(∂f/∂u) = m. Following (Aranda–Bricaire et al, 1995) and (Fliess et al, 1995) we regard system (1) as an underdetermined system of ordinary differential equations. This viewpoint allows to consider system (1) in the framework of geometrical theory of differential equations (see Krasil’shchik et al, 1999).

In this framework, system (1) is naturally related to two manifolds (E and E^{∞} below) with two distributions. The first manifold (E) is finite-dimensional, the second one (E^{∞}) is infinite-dimensional. Both distributions are called the Cartan ones. A map from one of these manifolds to itself is called a symmetry of system (1) if the map preserves the corresponding Cartan distribution. In the first case (E), a symmetry is called classical. In the case E^{∞}, it is called higher. Any classical symmetry generates a higher symmetry. Any symmetry of system (1) takes each solution of (1) to a solution again.

An infinitesimal version of the above construction leads to the concepts of infinitesimal classical and higher symmetries. If two control systems are static feedback equivalent, then the Lie algebras of their classical infinitesimal symmetries are isomorphic. Similarly, an equivalence by endogenous dynamic feedback generates an isomorphism of the corresponding Lie algebras of higher infinitesimal symmetries.

This paper is devoted to methods for calculation of classical and higher infinitesimal symmetries of control systems. To find classical infinitesimal symmetries, one needs to solve some system of partial differential equations. We give this system in the general (see Theorems 1 and 2) and affine (see Theorems 5 and 2) cases. Our method for calculation of higher infinitesimal symmetries is based on the infinitesimal Brunovsky form introduced in (Aranda–Bricaire et al, 1995). We assign a higher symmetry to an arbitrary collection of m functions and a classical symmetry of some system of ordinary differential equations (see Theorem 7 for details).

Key words and phrases. nonlinear control systems, symmetries, decomposition, flatness of control systems.

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

The results exposed in this paper were applied by the authors to the decomposition problem for control systems (Kanatnikov et al, 1994) and the flatness problem (Chetverikov, 2001).

The paper is organized as follows. Classical symmetries are studied in Sections 2—5. The method for calculation of higher infinitesimal symmetries is presented in Sections 6—8. In Sections 2 and 6 we give the two geometric interpretations of control systems (E and E^{∞} respectively). The conditions for classical infinitesimal symmetries are obtained in Section 3 in the general case and in Section 5 in the affine case. A relationship between classical symmetries and first integrals of control systems is discussed in Section 4. In Section 7 a construction introduced in (Aranda–Bricaire et al, 1995) is generalized to the nonautonomous case. This generalization is used in Section 8, where higher infinitesimal symmetries of control systems are described. Finally, Section 9 contains an example of calculation of classical and higher infinitesimal symmetries.

2. The first geometric interpretation

System (1) determines the trivial bundle π: E → ℝ^{1}, π(t, x, u) = t. Consider the 1-jet space J^{1}π of this bundle (see Krasil’shchik et al, 1999). Let (t, x, u, p, q) be local coordinates on J^{1}π with p = (p_{1}, …, p_{n}) corresponding to ẋ(t), q = (q_{1}, …, q_{m}) corresponding to u̇(t). System (1) can be written as p = f(t, x, u). Therefore it can be interpreted as the submanifold

E = {(t, x, u, p, q) ∈ J^{1}π| p − f(t, x, u) = 0}

of codimension n in J^{1}π. Each section (x(t), u(t)) of the bundle π has a prolongation onto J^{1}π as a curve l_{xu} of the form t ↦ (t, x(t), u(t), ẋ(t), u̇(t)). A section (x(t), u(t)) is a solution of (1) if and only if l_{xu} ⊂ E.

The Cartan distribution on J^{1}π is determined by the 1-forms ω_{i} = dx_{i} − p_{i}dt, τ_{j} = du_{j} − q_{j}dt, i = 1, 2, …, n, j = 1, 2, …, m. The curves l_{xu} are integral curves of the Cartan distribution. Let π_{1} : J^{1}π → ℝ^{1} be the natural projection, i. e., π_{1}(t, x, u, p, q) = t. It is known (see Krasil’shchik et al, 1999) that if a one–dimensional integral submanifold of the Cartan distribution is locally maximal, then it locally coincides with a curve l_{xu} (except for singular points of the map π_{1} : l_{xu} → ℝ^{1}). These manifolds are called Rmanifolds. We shall call R-manifolds contained in E generalized solutions of system (1). The restriction of the Cartan distribution on J^{1}π to E is called the Cartan distribution on E. Generalized solutions are locally maximal integral curves of this distribution.

A diffeomorphism from J^{1}π to itself is called a Lie transformation if it preserves the Cartan distribution. Lie transformations of J^{1}π send any R-manifold to an R-manifold again. If a Lie transformation translates the submanifold E into itself (and consequently any generalized solution to a generalized solution again), then it is called a (classical external) symmetry of (1). It can be proved (see Krasil’shchik et al, 1999) that in the case n + m > 1 each Lie transformation is lifted from J^{0}π = E.

One stated above is transferred on one–parameter groups of Lie transformations. These groups are connected with their infinitesimal generators — vector fields named Lie fields. In our case, when Lie transformations are lifted from the manifold E, Lie fields are also obtained as lifting of vector fields on E. Namely if a vector field X on E has the form

X = ξ(t, x, u)\frac{∂}{∂t} + ∑_{i=1}^{n} η_{i}(t, x, u)\frac{∂}{∂x_{i}} + ∑_{j=1}^{m} ϑ_{j}(t, x, u)\frac{∂}{∂u_{j}} , (2)

then its lifting X^{(1)} on J^{1}π is the vector field

X^{(1)} = X + ∑_{i=1}^{n} ζ_{i}(t, x, u, p, q)\frac{∂}{∂p_{i}} + ∑_{j=1}^{m} ε_{j}(t, x, u, p, q)\frac{∂}{∂q_{j}} , (3)

where ζ = (ζ_{1}, …, ζ_{n}) and ε = (ε_{1}, …, ε_{m}) are obtained by formulas

ζ = Dη − pDξ, ε = Dϑ − qDξ, (4)

with η = (η_{1}, …, η_{n}), ϑ = (ϑ_{1}, …, ϑ_{m}), and

D = \frac{∂}{∂t} + ∑_{i=1}^{n} p_{i} \frac{∂}{∂x_{i}} + ∑_{j=1}^{m} q_{j} \frac{∂}{∂u_{j}}

being the total derivative with respect to t on J^{1}π (see Krasil’shchik et al, 1999).

If Lie field (3) is tangent to the submanifold E, then Lie transformations of its one– parameter group translates E into itself. In this case the vector field is called an (infinitesimal classical external) symmetry of system (1). The condition necessary and sufficient to field (3) being tangent to E is the relation

X^{(1)}(p_{i} − f_{i})|_{E} = 0, i = 1, 2, …, n, (5)

where (f_{1}, …, f_{n}) = f.

3. Defining equations for classical symmetries

Using (2)–(4) relation (5) in coordinate terms reduces to

ξ \frac{∂f}{∂t} + \frac{∂f}{∂x}η + \frac{∂f}{∂u}ϑ − \frac{∂η}{∂t} − \frac{∂η}{∂x}f − \frac{∂η}{∂u}q + f (\frac{∂ξ}{∂t} + \frac{∂ξ}{∂x}f + \frac{∂ξ}{∂u}q) = 0, (6)

the latter being valid for all (t, x, u, q). System (6) is linear with respect to q and therefore decomposes into two subsystems

ξ \frac{∂f}{∂t} + \frac{∂f}{∂x}η + \frac{∂f}{∂u}ϑ − \frac{∂η}{∂t} − \frac{∂η}{∂x}f + f \frac{∂ξ}{∂t} + f \frac{∂ξ}{∂x}f = 0, (7)

\frac{∂η}{∂u} − f \frac{∂ξ}{∂u} = 0. (8)

We shall call equations (7)–(8) the defining equations for classical infinitesimal symmetries of system (1).

Any symmetry (2)–(3) of a control system is uniquely determined by its components ξ, η_{1}, …, η_{n} (see (4) and (10)). Let

H = ξ \frac{∂}{∂t} + ∑_{i=1}^{n} η_{i} \frac{∂}{∂x_{i}} , F = \frac{∂}{∂t} + ∑_{i=1}^{n} f_{i} \frac{∂}{∂x_{i}} .

Denote by F_{u} the distribution generated by fields F_{j} = [∂/∂u_{j}, F], j = 1, …, m. Note that dim F_{u} = rank(∂f/∂u) = m.

Theorem 1. (Kanatnikov et al, 1994) System (1) possesses a symmetry X of the form

X = H + ∑_{j=1}^{m} ϑ_{j}(t, x, u)\frac{∂}{∂u_{j}} (9)

if and only if the vector field [F, H] − F (ξ)F lies in the distribution F_{u} and system (8) holds. In this case, the components ϑ_{1}, …, ϑ_{m} of X are uniquely determined by the condition

[F, H] − F (ξ)F = ∑_{j=1}^{m} ϑ_{j}F_{j}. (10)

Theorem 2. (Kanatnikov et al, 1994) If a vector field X (2) is a symmetry of system (1) and rank(∂f/∂u) ≥ 2 everywhere on E, then the components ξ, η_{1}, …, η_{n} of X are independent of u and system (8) is trivial.

4. First integrals of control systems and symmetries

A first integral of system (1) is a function α(t, x, u) which is constant along any solution (x(t), u(t)) of the system. In other words, a first integral is a function α with its time–derivative α̇(t, x, u)|_{(1)} according to system (1) being equal to 0. Hence

\frac{∂α}{∂u_{j}} = 0, j = 1, …, m,

and

F (α) = \frac{∂α}{∂t} + ∑_{i=1}^{n} f_{i} \frac{∂α}{∂x_{i}} = 0.

First integrals of system (1) form a ring under the standard addition and multiplication.

Theorem 3. (Kanatnikov et al, 1994) If a vector field X is a symmetry of system (1) then for each first integral α the field αX is also a symmetry of system (1). The set of all symmetries of system (1) is a module over the ring of first integrals.

Theorem 4. (Kanatnikov et al, 1994) The family of all symmetries X of system (1) of the form (2) with ξ being a first integral, is involutive.

5. Affine systems

Affine control system

ẋ = a(t, x) + ∑_{j=1}^{m} b_{j}(t, x)u_{j}, (11)

where a, b_{1}, …, b_{m}: ℝ^{1}×ℝ^{n} → ℝ^{n} are smooth functions, corresponds uniquely to vector fields

A = \frac{∂}{∂t} + ∑_{i=1}^{n} a_{i} \frac{∂}{∂x_{i}} ,  B_{j} = ∑_{i=1}^{n} b_{ij} \frac{∂}{∂x_{i}} ,  j = 1, …, m,

with (a_{1}, …, a_{n}) = a(t, x), (b_{1j}, …, b_{nj}) = b_{j}(t, x). The field F has the form

F = A + ∑_{j=1}^{m} u_{j}B_{j}.

Denote by B the distribution generated by fields B_{1}, …, B_{m}.

Theorem 5. (Kanatnikov et al, 1994) Let the components ξ, η_{1}, …, η_{n} of a vector field H be independent of u. A vector field X of the form (9) is a symmetry of system (11) if and only if the vector fields

[A, H] − A(ξ)A, [B_{k}, H] − B_{k}(ξ)A (12)

for k = 1, …, m lie in the distribution B. The components ϑ_{1}, …, ϑ_{m} of X are uniquely determined by the conditions

ϑ_{j} = ϑ_{j}^{(0)}(t, x) + ∑_{k=1}^{m} ϑ_{jk}^{(1)}(t, x)u_{k} − F (ξ)u_{j}, j = 1, …, m,

[A, H] − A(ξ)A = ∑_{j=1}^{m} ϑ_{j}^{(0)}B_{j}, (13)

[B_{k}, H] − B_{k}(ξ)A = ∑_{j=1}^{m} ϑ_{jk}^{(1)}B_{j}, k = 1, …, m.

6. The second geometric interpretation

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)}, …), (14)

where u^{(l)} denotes the vector variable corresponding to the lth order derivative of u with respect to t. 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)}} + …, (15)

which is called the total derivative with respect to t on E^{∞}. The Lie derivative along D is simply the time–derivative according to system (1). We denote by Dω the Lie derivative of the form ω along D.

A smooth function on E^{∞} is a function smoothly depending on a finite (but arbitrary) number of coordinates (14). 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, (15)).

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

A motivation of the last definition is the following. A vector field on E^{∞} is called integrable if it possess a one–parameter group of diffeomorphisms (a flow). Since E^{∞} is an infinite-dimensional manifold, vector fields on E^{∞} are not usually integrable. Nevertheless let us first consider an integrable field Y. Suppose all diffeomorphisms of its flow take each solution of (1) to a solution again. Since solutions of (1) are integral curves of the field D, we have

[Y, D] = aD (16)

for some function a on E^{∞}. Consider now an arbitrary (may be nonintegrable) vector field Y on E^{∞} satisfying (16). It is uniquely represented as the sum of a vertical field X and a horizontal field hD for some function h on E^{∞}, i. e., Y = X + hD. Condition (16) means that [X, D] = 0 and h is an arbitrary function on E^{∞}. Thus the set of all fields satisfying (16) is split in equivalence classes and each class contains a higher symmetry.

7. Infinitesimal Brunovský form for nonautonomous systems

Here we remind some concepts from (Aranda–Bricaire et al, 1995) and simultaneously generalize them to the nonautonomous case.

Let C^{1}Λ(E) be the F(E)–module of differential 1–forms on E^{∞} belonging to the codistribution corresponding to 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 f ↦ df − D(f)dt. The operator d_{C} possesses many properties of the differential d. In particular,

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

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

Obviously, in coordinate system (14) 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 one has rank(∂f/∂u) = m and for any k > 0 the dimensions of H_{k} and H_{k} + D(H_{k}) are constant.

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}f = df, 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 (Aranda–Bricaire et al, 1995).

Theorem 6. 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) the functions χ_{1}, …, χ_{ρ} and their total derivatives with respect to t satisfy a system of the form

{χ̇_{i} = γ_{i}(t, χ_{1}, …, χ_{ρ}), i = 1, …, ρ; (17)

(3) {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).

The proof is similar to that of the corresponding theorem from (Aranda–Bricaire et al, 1995). Moreover, the infinitesimal Brunovský form given in the same work for the autonomous case can be generalized to the nonautonomous case.

8. Higher symmetries of control systems

By Theorem 6, it follows that in a neighborhood of a B–regular point there exist functions {g_{l,i}, h_{l,k,j}} on E^{∞} such that

d_{C}x_{l} = ∑_{i=1}^{ρ} g_{l,i}d_{C}χ_{i} + ∑_{k=1}^{m} ∑_{j=0}^{r_{k}} h_{l,k,j}D^{j}ω_{k} (18)

for any l = 1, …, n + m and some r_{1}, …, r_{m}, where x_{l} = u_{l−n} for l = n + 1, …, n + m.

Theorem 7. (Chetverikov, 1999) In a neighborhood of a B–regular point any higher symmetry of system (1) has the form

�_{φ} = ∑_{i=1}^{n} φ_{i} \frac{∂}{∂x_{i}} + ∑_{i=1}^{m} ∑_{j=0}^{∞} D^{j}φ_{i+n} \frac{∂}{∂u_{i}^{(j)}} , (19)

where

φ_{l} = ∑_{i=1}^{ρ} g_{l,i}a_{i} + ∑_{k=1}^{m} ∑_{j=0}^{r_{k}} h_{l,k,j}D^{j}ψ_{k}, (20)

for l = 1, …, n + m, ψ_{1}, …, ψ_{m} are arbitrary functions on E^{∞}, a_{1}, …, a_{ρ} are arbitrary functions of t, χ_{1}, …, χ_{ρ} such that

\frac{∂a_{i}}{∂t} + ∑_{α=1}^{ρ} γ_{α} \frac{∂a_{i}}{∂χ_{α}} = ∑_{α=1}^{ρ} a_{α} \frac{∂γ_{i}}{∂χ_{α}} , (21)

for i = 1, …, ρ.

The vector function φ = (φ_{1}, …, φ_{n+m}) is called the generating function of symmetry (19).

Remark 2. The vector field

∑_{α=1}^{ρ} a_{α} \frac{∂}{∂χ_{α}} (22)

is a symmetry of the system of ordinary differential equations (17). Condition (21) means that the commutator of the fields (22) and

\frac{∂}{∂t} + ∑_{α=1}^{ρ} γ_{α} \frac{∂}{∂χ_{α}}

vanishes.

9. Example

Find classical and higher symmetries of the control system

\{ \begin{matrix}ẋ_{1} & = & tx_{1}^{2} \\ ẋ_{2} & = & x_{4}u_{1} \\ ẋ_{3} & = & u_{1} \\ ẋ_{4} & = & u_{2}. \end{matrix} (23)

In the case of classical symmetries, we use results of Section 5. We have n = 4, m = 2, and

A = \frac{∂}{∂t} + tx_{1}^{2} \frac{∂}{∂x_{1}} .

The distribution B is generated by fields

B_{1} = x_{4} \frac{∂}{∂x_{2}} + \frac{∂}{∂x_{3}} , B_{2} = \frac{∂}{∂x_{4}} .

The corresponding codistribution is generated by forms

α_{1} = dt,  α_{2} = dx_{1},  α_{3} = dx_{2} − x_{4}dx_{3}.

By Theorem 2, it follows that the components ξ, η_{1}, …, η_{4} of a desired symmetry X are independent of u_{1} and u_{2}. By Theorem 5, the defining equations for symmetries can be expressed as

α_{i}(Y) = 0 (24)

for any field Y of the form (12) and i = 1, 2, 3. Since equalities (24) are trivial in the case i = 1, we obtain 6 differential equations for the components ξ, η_{i}. Introducing the functions

z = η_{1} − tx_{1}^{2}ξ, v = η_{2} − x_{4}η_{3}, (25)

these equations can be written as

A(z) = 2tx_{1}z,  B_{1}(z) = 0,  B_{2}(z) = 0, (26)
A(v) = 0,  B_{1}(v) = η_{4},  B_{2}(v) = η_{3}. (27)

Solving the system of equations (26), we get

z = a(t^{2} + \frac{2}{x_{1}} )x_{1}^{2},

where a is an arbitrary smooth function of one variable. From the first equation in (27) it follows that

v = b(t^{2} + \frac{2}{x_{1}} , x_{2}, x_{3}, x_{4}),

where b is an arbitrary function of four variable. Using the second and the third equations in (27), we get expressions for η_{3} and η_{4}. Finally, using (25), we find η_{1} and η_{2}.

Thus any infinitesimal classical symmetry of system (23) has the form

X = ξ \frac{∂}{∂t} + ∑_{i=1}^{4} η_{i}(t, x, u)\frac{∂}{∂x_{i}} + ∑_{j=1}^{2} ϑ_{j}(t, x, u)\frac{∂}{∂u_{j}}

and is uniquely determined by functions a, b, and ξ. The component ξ is an arbitrary function of t, x_{1}, …, x_{4}. Besides,

η_{1} = tx_{1}^{2}ξ + a(t^{2} + \frac{2}{x_{1}} )x_{1}^{2}, η_{2} = v + x_{4} \frac{∂v}{∂x_{4}} , η_{3} = \frac{∂v}{∂x_{4}} , η_{4} = x_{4} \frac{∂v}{∂x_{2}} + \frac{∂v}{∂x_{3}} .

From (10) it follows that

ϑ_{1} = F (η_{3}) − F (ξ)u_{1}, ϑ_{2} = F (η_{4}) − F (ξ)u_{2},

where F = A + u_{1}B_{1} + u_{2}B_{2}.

To obtain higher symmetries of system (23), we use results of Section 7 and 8. In our case, any element of H_{0} has the form

ω = ∑_{i=1}^{4} f_{i}d_{C}x_{i}, f_{i} ∈ F(E).

We get

Dω = ∑_{i=1}^{4} Df_{i}d_{C}x_{i} + f_{1}2tx_{1}d_{C}x_{1} + f_{2}(x_{4}d_{C}u_{1} + u_{1}d_{C}x_{4}) + f_{3}d_{C}u_{1} + f_{4}d_{C}u_{2}. (28)

If Dω ∈ H_{0}, then the coefficients of d_{C}u_{1} and d_{C}u_{2} in (28) vanish. Whence

f_{2}x_{4} + f_{3} = 0, f_{4} = 0.

Therefore the module H_{1} is generated by d_{C}x_{1} and d_{C}x_{2} − x_{4}d_{C}x_{3}.

In the same way, the condition

D(f_{1}d_{C}x_{1} + f_{2} (d_{C}x_{2} − x_{4}d_{C}x_{3})) ∈ H_{1},

means that f_{2} = 0 and d_{C}x_{1} ∈ H_{2}. We see that k^{∗} = 2 and d_{C}x_{1} ∈ H_{k^{∗}}. Thus ρ = 1, χ_{1} = x_{1}, system (17) consists of the first equation of system (23), and ω_{1} = d_{C}x_{2} − x_{4}d_{C}x_{3} ∈ H_{1}. The 1–form ω_{2} should be chosen such that

{d_{C}χ_{1}, 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 {u_{1} ≠ 0} and we obtain

d_{C}x_{1} = d_{C}χ_{1},  d_{C}x_{2} = ω_{1} + x_{4}ω_{2},  d_{C}x_{3} = ω_{2},
d_{C}x_{4} = \frac{1}{u_{1}} Dω_{1} + \frac{u_{2}}{u_{1}} ω_{2},  d_{C}u_{1} = Dω_{2},  d_{C}u_{2} = D (d_{C}x_{4}) .

Condition (21) has the form

\frac{∂a}{∂t} + tx_{1}^{2} \frac{∂a}{∂x_{1}} = 2tx_{1}a.

Solving the last equation and using Theorem 7, we get generating functions of all higher symmetries:

φ_{1} = x_{1}^{2}a(t^{2} + \frac{2}{x_{1}} ),  φ_{2} = ψ_{1} + x_{4}ψ_{2},  φ_{3} = ψ_{2},

φ_{4} = \frac{1}{u_{1}} Dψ_{1} + \frac{u_{2}}{u_{1}} ψ_{2},  φ_{5} = Dψ_{2},  φ_{6} = Dφ_{4},

where a is an arbitrary function of one variable, ψ_{1}, ψ_{2} are arbitrary functions on E^{∞}.

References

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

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

[3] Chetverikov, V.N. (2001). New flatness conditions for control systems, in Proc. of NOLCOS’01, 168–173.

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

[5] Kanatnikov, A.N. and A.P. Krishchenko. (1994). Symmetries and Decomposition of Nonlinear Systems. Differential Equations. 30 (11), 1735–1745.

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

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