The Diffiety Institute Preprint Series

Preprint DIPS–2/2000 October 10, 2000

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Full symmetry algebra for ODEs and control systems

by

A. Samokhin

Full symmetry algebra for ODEs and control systems

A. Samokhin

Abstract. A description of the full symmetry algebra (i.e., including higher symmetries) for a general nonlinear system of ordinary differential equations is given in terms of its general solution and differential constants. More precisely, the full symmetry algebra of a system is a module over the ring of its differential constants; the module is generated by partial derivatives of the general solution by the independent constants. Given a general solution, this description is both effective and explicit. Special solutions, such as an envelope of a family of solutions is described naturally in this context. These results are extended to control systems; in this case the differential constants become operators on controls. Examples are provided.

1. Introduction

The study of symmetries of ordinary differential equations was initiated by Sophus Lie himself and has a long history which is described briefly in [1]. The latest results were obtained in [2] and [3].

To find symmetries for an individual equation still remains a hard task. The present paper deals, however, with another problem. We give a full description of a symmetry algebra of a system of ODE in a nondegenerate situation using the general solution whose (local) existence is guaranteed by classical theorems. For a linear system of ODEs this result was obtained in [1]. It was generalized to the normal form scalar ODEs in [3].

Given a general solution, this description is both effective and explicit. Special solutions, such as an envelope of a family of solutions is described naturally in this context.

Of course, these results are of little practical importance since there is no need in symmetries when a general solution is known. Symmetries are used to obtain new solutions, not the other way round. Yet the interconnection between differential invariants, symmetries and a general solution are quite transparent in the case of ODEs and may be used as a model aplicable in other situations.

In this paper, we give two such applications. First, we describe the symmetries of a boundary / initial value problem for a one-dimensional wave equation. The second application deals with symmetries of control systems. In both cases differential invariants become nonlocal ones.

1991 Mathematics Subject Classification. 58G37, 34H05 .

Key words and phrases. Symmetry, control system, general solution.

This work was partially supported by INTAS grant 96-0793 .

The paper is organized as follows. Section 2 describes the full symmetry algebra for a general nonlinear system of ordinary differential equations. It also contains the description of special solutions as invariants of basic symmetries for a given general solution, (subsection 2.3) and examples (subsection 2.4). Section 3 is an application of this approach to control systems; examples are also provided.

2. Full symmetry algebra for a general nonlinear ordinary differential equation and a system of equations

2.1. General solution and differential constants. We begin with trivialities to introduce notation.

Let E denote a general scalar ordinary differential equation of nth order

y^{(n)} − F (x, y, y′, …, y^{(n−1)}) = 0. (1)

The equation’s general solution (or a general integral) is of the form

Φ(x, y, c_{1}, c_{2}, …, c_{n}) = 0. (2)

When (2) is solved with respect to y, we get

y = f(x, c_{1}, …, c_{n}); (3)

almost any solution of (1) is obtained from (3) by a proper choice of the constants c_{i}. (The solution that is not produced by the general one is called a special solution. Such solutions are discussed below.)

Differentiating (3) by x we obtain the following system of n independent equations

\{ \begin{matrix}y & = & f(x, c_{1}, …, c_{n}) \\ y′ & = & f′(x, c_{1}, …, c_{n}) \\ & … \\ y^{(n−1)} & = & f^{(n−1)}(x, c_{1}, …, c_{n}) \end{matrix} (4)

(further differentiating produce dependent equations since y^{(k)}, k ≥ n are expressed in y^{(i)}, i < n via (1)).

One can obtain an expression (not necessary explicit) for c_{i} solving (4). Thus

c_{i} = c_{i}(x, y, y′, …, y^{(n−1)}), i = 1, …, n. (5)

In this way all c_{i} are differential constants of order less than n. In other words, they are differential operators of order n − 1 or functions on the jet space J^{n−1}(ℝ).

In the case of a system of m differential equations,

y^{(n)} − F(x, y, y′, …, y^{(n−1)}) = 0, (6)

where y = (y_{1}, …, y_{m}), F = (F_{1}, …, F_{m}), the general solution is of the form

Φ_{k}(x, y, c_{1}, c_{2}, …, c_{mn}) = 0, k = 1, …, m (7)

or

y = f(x, c_{1}, …, c_{mn}). (8)

Almost any solution of (6) is obtained from (8) by a proper choice of the constants c_{i}.

2.2. Full symmetry algebra. By definition of a solution, if righthand side of (3), f(x, y, c_{1}, …, c_{n}) is substituted for y in (1), we obtain an identity

f^{(n)} − F (x, f, f′, …, f^{(n−1)}) ≡ 0. (9)

Hence

∀i : \frac{∂}{∂c_{i}} (f^{(n)} − F (x, f, f′, …, f^{(n−1)})) = 0 (10)

or

∀i : (D_{x}^{n} − ∑_{j=1}^{n} \frac{∂F (x, y, y′, …, y^{(n−1)})}{∂y_{j}}D_{x}^{j})|_{y=f(x,y,c_{1},...,c_{n})} f_{c_{i}} = 0, (11)

where D = \frac{d}{dx} is the total derivative with respect to x.

Recall that

L_{y^{(n)}−F} =^{def} D_{x}^{n} − ∑_{j=1}^{n} \frac{∂F (x, y, y′, …, y^{(n−1)})}{∂y_{j}}D_{x}^{j} (12)

is called an universal linearization of the operator y^{(n)} − F and that a solution ϕ of the equation

(L_{y^{(n)}−F} ) ϕ|_{E} = 0 (13)

is a symmetry of E.

Theorem 1. Partial derivatives f_{c_{i}}, i = 1, …, n form a full functionally independent basis of symmetries for equation (1).

Proof. The difference between (11) and (13) is that the same operator is restricted to formally different objects. However, note that the set

{y = f(x, c_{1}, …, c_{n}), y′ = f′(x, y, c_{1}, …, c_{n}) … |∀c_{i} ∈ ℝ} ⊂ J^{n}(ℝ)

essentially coincides with E. Indeed, dim J^{n}(ℝ) = n + 2, codim E ⊂ J^{n}(ℝ) = 1, so dim E = n + 1. It follows from the existence theorem for an ordinary differential equation that there is a solution containing any initial value point x_{0}, y_{0}, y′_{0}, …, y_{0}^{n−1} ∈ E. Now, since (3) produces almost all solutions and

dim{y = f(x, y, c_{1}, …, c_{n}), y′ = f′(x, y, c_{1}, …, c_{n}) … |∀c_{i} ∈ ℝ} = n + 1

we conclude that (11) coincides with the symmetry equation (13) almost everywhere on E.

Therefore, f_{c_{i}}, i = 1, …, n are symmetries of equation (1). Moreover, they form a basis of the symmetry algebra.

Indeed, let φ be a symmetry. Then it defines a flow on a set of solutions by the formula :

\frac{∂y}{∂τ} = φ|_{y}, (14)

where y = f(x, y, c_{1}, …, c_{n}). It can be solved (see [4]) and a solution of this equation is a one-parameter family of solutions of (1). By (3), it has a form

y = f(x, c_{1}(τ), …, c_{n}(τ)) (15)

On the other hand, differentiating (15) by τ, we obtain (via (14)) that

φ|_{y} = (∑_{i=1}^{n} \frac{∂c_{i}}{∂τ} f_{c_{i}})|_{y} (16)

on any solution y of equation (1). Therefore,

φ = ∑_{i=1}^{n} \frac{∂c_{i}}{∂τ} f_{c_{i}} (17)

holds everywhere on E.

Note that the derivatives \frac{∂c_{i}}{∂τ} |_{y} depend on y, that is, on c_{1}, …, c_{1}, which are functions on J^{n−1}(ℝ) by virtue of (5). Since any choice of arbitrary functions c_{i}(τ) define some symmetry by (15), the functions \frac{∂c_{i}}{∂τ} |_{y} are also arbitrary.

Thus, we got the general form of a symmetry for equation (1)

φ = ∑_{i=1}^{n} A_{i}(c_{1}, …, c_{n})\frac{∂}{∂c_{i}} f(x, y, c_{1}, …, c_{n}); (18)

here f is a general solution, A_{i} are arbitrary functions and c_{i} are functions on J^{n−1}(ℝ) given by system (4).

Formula (18) also completes the proof of the theorem.

Remark 1. A full symmetry algebra is a module over the ring of the equation’s differential constants. The module is generated by partial derivatives of a general solution by the independent constants.

Remark 2. Formula (18) gives a representation of the algebra of vector fields on ℝ^{n} in the full symmetry algebra of (6) by the isomorphism

∑_{i=1}^{n} A_{i}(c_{1}, …, c_{n})\frac{∂}{∂c_{i}} ⟷ ∑_{i=1}^{n} A_{i}(c_{1}, …, c_{n})\frac{∂}{∂c_{i}} f(x, c_{1}, …, c_{n})

(On the left-hand side, c_{i} are coordinates in ℝ^{n}; on the right-hand side they denote differential invariants (5) of (1) or special functions on J^{n−1}(ℝ)).

Remark 3. Theorem 1 generalizes easily to the case of a system of differential equations (6). Its full symmetry algebra is isomorphic to the algebra of vector fields on ℝ^{mn}: the representation is given by

∑_{i=1}^{mn} A_{i}(c_{1}, …, c_{mn})\frac{∂}{∂c_{i}} ⟷ ∂f × A,

where ∂f, A are respectively m × mn and mn × 1 matrices with matrix elements given by the formulas

(∂f)_{j,i} = \frac{∂f_{j}}{∂c_{i}} , (A)_{i} = A_{i}

A variant of Theorem (1) is also valid in the case of even more general system of ordinary differential equations,

y_{j}^{(n_{j})} − F_{j}(x, y_{1}, y′_{1}, …, y_{1}^{(n_{1}−1)}, …, y_{m}, y′_{m}, …, y_{1}^{(n_{m}−1)}) = 0.

It is not hard to right down the correspondent isomorphism between vector fields on the solution space and symmetries in this case too. Yet the formula is awkward to read and therefore it is omitted here. See [2] for relevant technicalities.

Let us call f_{c_{i}}, i = 1, …, n basic symmetries. They correspond to the flows y(τ) = f(x, c_{1}, …, c_{i} + τ, …, c_{n}). Thus, in the case of an explicit general solution (3) basic symmetries are f_{c_{i}} = y_{c_{i}}.

Remark 4. If general solution of (1) is given in an implicit form (2) then

\frac{dΦ}{dc_{i}} = \frac{∂Φ}{∂c_{i}} + \frac{∂Φ}{∂y} \frac{∂y}{∂c_{i}} = 0.

It follows immediately that basic symmetries are given by

y_{c_{i}} = − (\frac{∂Φ}{∂c_{i}}) ⧸ (\frac{∂Φ}{∂y}) . (19)

This formula generalizes in a straightforward way in the case of a system of equations.

2.3. Special and invariant solutions. Invariant or self-similar solution y of (1) is the solution that satisfy the condition φ(y) = 0 for some symmetry φ of the form (18). Hence an invariant solution satisfy a system of equations

\{ \begin{matrix}E(f) = y^{(n)} − F (x, y, y′, …, y^{(n−1)}) = 0 \\ ϕ(y) = ∑_{i=1}^{n} A_{i}(c_{1}(y), …, c_{n}(y))\frac{∂}{∂c_{i}} f(x, y, c_{1}(y), …, c_{n}(y)) = 0 \end{matrix} (20)

Since c_{i} are constants on solutions of (1) so are A_{i}(c_{1}(y), …, c_{n}(y)). Thus (20) is simply

\{ \begin{matrix}E(f) & = & y^{(n)} − F (x, y, y′, …, y^{(n−1)}) = 0 \\ ϕ(y) & = & ∑_{i=1}^{n} A_{i}f_{c_{i}}(x, y, c_{1}, …, c_{n}) = 0 \end{matrix} (21)

with constant A_{i} and c_{i}. The second condition in (21) means that basic symmetries are linearly dependent on an invariant solution. If rank{f_{c_{1}}, …, f_{c_{n}}}|_{y} = n − k, it is natural to introduce a notion of a k-invariant solution.

Remark 5. Recall that f_{c_{i}} represent independent vector fields on ℝ^{n}. In this way the structure of invariant solutions of ordinary differential equation is connected with the structure of degenerate points of a system of n independent vector fields on ℝ^{n}.

Consider a simple case of (21),

\{ \begin{matrix}y^{(n)} − F (x, y, y′, …, y^{(n−1)}) = 0 \\ f_{c_{i}} = 0 \end{matrix} (22)

Its solution is a fixed point of the flow c_{i} → c_{i} + τ. Geometrically, such a solution is an envelope for the family of solution generated by this flow, see section (2.4).

2.4. Examples.

Example 1.

y′′ + \frac{9}{8}(y′)^{4} = 0

This equation is invariant with respect to the translations in both x and y, hence its symmetry algebra is obvious. Its general solution is as follows

Φ(x, y, c_{1}, c_{2}) = (y + c_{1})^{3} − (x + c_{2})^{2} = 0,

or

y = f(x, c_{1}, c_{2}) = (x + c_{2})^{\frac{2}{3}} − c_{1}

Therefore, its basic symmetries are f_{c_{1}} = −1, f_{c_{2}} = \frac{2}{3}(x + c_{2})^{− \frac{1}{3}} . They depend on the differential constants c_{1}, c_{2} that may be found from the system (4),

\begin{aligned}(y + c_{1})^{3} & = & (x + c_{2})^{2}, \\ 3y′(y + c_{1})^{2} & = & 2(x + c_{2}). \end{aligned}

It follows that

\begin{aligned}c_{1} & = & (\frac{2}{3y′})^{2} − y, \\ c_{2} & = & (\frac{2}{3y′})^{3} − x. \end{aligned}

Now, basic symmetries come to

\begin{aligned}f_{c_{1}} & = & −1, \\ f_{c_{2}} & = & y′, \end{aligned}

which are (not surprisingly) translation in y and x respectively. So the general symmetry for this equation is of the form (18)

φ = A_{1}(c_{1}, c_{2})f_{c_{1}} + A_{2}(c_{1}, c_{2})f_{c_{2}} =

− A_{1} ((\frac{2}{3y′})^{2} − y, (\frac{2}{3y′})^{3} − x) + A_{2} ((\frac{2}{3y′})^{2} − y, (\frac{2}{3y′})^{3} − x) y′,

where A_{1}, A_{2} are arbitrary functions of two variables.

Invariant solutions have to satisfy the system (21)

\begin{aligned}A + y′B & = & 0, \\ y′′ + \frac{9}{8}(y′)^{4} & = & 0, \end{aligned}

for some constants A, B. It follows that y′ = 0, so y = const. This is a family of special solutions (in the sense they are not obtained from the general integral). Each special solution is an envelope for the family

(y − const)^{3} − (x + c_{2})^{2} = 0

for all c_{2}, see figure 1.

Example 2.

yy′′ + 2(y′^{2} + 1) = 0

The general integral in this case is as follows

Φ = ∫ \frac{y^{2} dy}{\sqrt{c_{1} − y^{4}}} ± x + c_{2}.

8 A. Samokhin

Figure 1. Enveloping invariant solution y = 2 Basic symmetries are obtained here by the formula (19):

\begin{matrix}φ_{1} = −\frac{Φ_{c_{1}}}{Φ_{y}} = \frac{1}{2} ^{\frac{\sqrt{c_{1}−y^{4}}}{y^{2}}} ∫ \frac{y^{2} dy}{(\sqrt{c_{1}−y^{4}})^{3}} , \\ φ_{2} = −\frac{Φ_{c_{2}}}{Φ_{y}} = −^{\frac{\sqrt{c_{1}−y^{4}}}{y^{2}}} . \end{matrix}

To obtain a final form for these symmetries it remains to express differential constants as functions on J^{1}(ℝ) using (4):

\begin{matrix}∫ \frac{y^{a} dy}{\sqrt{c_{1}−y^{2a}}} ± x + c_{2} = 0 \\ y′^{\frac{\sqrt{c_{1}−y^{4}}}{y^{2}}} ± 1 = 0. \end{matrix}

It follows immediately that

\begin{matrix}c_{1} = y^{4}(y′^{2} + 1), \\ c_{2} = ± ∫ dx ∓ x = c_{2}. \end{matrix}

Substituting these expressions into basic symmetries we obtain

\begin{matrix}φ_{1} = \frac{y′}{2} ∫ \frac{dy}{y′^{3}y^{4}} , \\ φ_{2} = y′. \end{matrix}

Note that ϕ_{1} is a nonlocal symmetry. Example 3. Linear equations (cf.[2])

y^{(n)} + ∑_{i=0}^{n−1} a_{i}(x)y^{(i)} = 0

Here the general integral is if the form

y = ∑_{i=1}^{n} c_{i}f_{i}(x),

where f_{i}(x) are independent solutions, i.e., their Wronskian is nonzero:

W = W (f_{1}, …, f_{i}, …, f_{n}) = | \begin{matrix}f_{1} & … & f_{i} & … & f_{n} \\ f′_{1} & … & f′_{i} & … & f′_{n} \\ … & … & … & … & … \\ f_{1}^{(n−1)} & … & f_{i}^{(n−1)} & … & f_{n}^{(n−1)} \end{matrix} | ≠ 0

Independent solutions f_{i} coincide with basic symmetries in this case: f_{i} = f_{c_{i}}.

Differential constant c_{i} is given by the formula

c_{i}(y, y′, …, y^{(n−1)}) = \frac{W_{i}}{W} ,

where W_{i} is obtained from W by changing the entries of the i-th column of W for y, y′, …, y^{(n−1)} in respective order.

The general form of the symmetry is

φ = ∑_{i=1}^{n} A_{i} (\frac{W_{1}}{W} , …, \frac{W_{i}}{W} , …, \frac{W_{n}}{W} ) f_{i}(x).

Example 4. Linear boundary problem

u_{tt} − u_{xx} = 0, u|_{x=0} = u|_{x=π} = 0

This example is a rather wide generalization of the previous one. Fourier’s general solution on [0, π] for this string is

u = ∑_{n=0}^{∞} sin nx(a_{n} cos nt + b_{n} sin nt),

where a_{n}, b_{n} are constants, but neither differential nor local: the Fourier coefficient formula states that

a_{n} = \frac{2}{π} ∫_{0}^{π} u|_{t=0} sin nx dx, b_{n} = \frac{2}{πn} ∫_{0}^{π} u_{t}|_{t=0} sin nx dx (23)

A general form of the symmetry is given by

φ = ∑_{n=0}^{∞} sin nx [A_{n}(a_{1}, b_{1}, …, a_{i}, b_{i}, …) cos nt+

B_{n}(a_{1}, b_{1}, …, a_{i}, b_{i}, …) sin nt] .

Here A_{n}, B_{n} are arbitrary functions depending on any finite number of a_{i}, b_{j} which are given by (23).

3. Full symmetry algebra for a general control system

3.1. General solution and differential constants. Consider a first order control system

y′ = F(x, y, v(x)), (24)

where y ∈ ℝ^{m} is an m-vector of unknown functions and v(x) ∈ ℝ^{k} in an k-vector of control functions.

With any fixed choice of controls, (24) comes to (6), where n = 1. Thus, the general solution of (24) is of the form

y = f(x, c_{1}, …, c_{m}, v(x)), (25)

where c_{i} are constants. From (25) it follows that there exists (at least an implicit) dependence

c_{i} = c_{i}(x, y(x), y′(x), v(x)), i = 1, …, m (26)

of differential constants c_{i} on x, y(x), y′(x), v(x). Both f and c_{i} are operators on v. Examples show that these operators may be nonlocal.

3.2. Full symmetry algebra. Technically, equation (24) is an equation with two types of dependent variables, that is, with y and v. Put this equation in a form

H(y, v) = y′ − F(x, y, v(x)) = 0.

The symmetry equation in this case is as follows:

(D − F_{y})A − F_{v}B|_{H=0} = 0, (27)

where (A, B) is a symmetry (if it defines a flow, then y_{τ} = A, v_{τ} = B). Besides, F_{y} is an m × m matrix with entries (F_{i})_{y_{j}} and F_{v} is an m × k matrix with entries (F_{i})_{v_{j}}.

It is convenient to put (27) in a vector form,

(D − F_{y}, −F_{v}) · ( \begin{matrix}A \\ B \end{matrix} )|_{H=0} = 0. (28)

The left factor in this formula is the linearization of H, denoted by L_{H} = (D − F_{y}, −F_{v}).

Theorem 2. Partial derivatives vectors

( \begin{matrix}f_{c} \\ 0 \end{matrix} ) , ( \begin{matrix}f_{v} \\ I \end{matrix} ) (29)

form a full functionally independent basis of symmetries for the equation (24).

Proof. In terms of the general solution, the general form of a flow on the set of solutions of equation (24) is given by the formula

y = f(x, c_{1}(τ), …, c_{m}(τ), v(x, τ)), (30)

where τ is the parameter of the flow. Since (30) is a solution for any τ, we have

f′(x, c_{1}(τ), …, c_{m}(τ), v(x, τ)) − F(x, f(x, c_{1}(τ), …, c_{m}(τ), v(x, τ)), v(x, τ)) = 0 Therefore,

\frac{d}{dτ} [f′(x, c_{1}(τ), …, c_{m}(τ), v(x, τ)) −

F(x, f(x, c_{1}(τ), …, c_{m}(τ), v(x, τ)), v(x, τ))] = 0.

It follows that

(31) [(D − F_{y})(f_{c} · c_{τ} + f_{v} · v_{τ}) − F_{v}v_{τ}]_{H=0} =

(D − F_{y}, −F_{v}) · ( \begin{matrix}f_{c} · c_{τ} + f_{v} · v_{τ} \\ v_{τ} \end{matrix} )|_{H=0} =

L_{H} ( \begin{matrix}f_{c} · c_{τ} + f_{v} · v_{τ} \\ v_{τ} \end{matrix} )|_{H=0} = 0.

Thus, the general solution of the symmetry equation is (compare with (17))

( \begin{matrix}f_{c} \\ 0 \end{matrix} ) · c_{τ} + ( \begin{matrix}f_{v} \\ I \end{matrix} ) · v_{τ} (32)

Here f_{c} = (f_{i})_{c_{j}} is an m × m matrix, f_{v} is an m × k matrix and I is the k × k identity matrix.

To obtain the general form of the symmetry for equation (24) it remains to notice that

• v_{τ} is an arbitrary vector-function;

• for any fixed v equation (24) coincides with (6), so c_{i}_{τ} are the components of a vector field on the solution space for the chosen v. Therefore, c_{i}_{τ} = A_{i}(c, v) are arbitrary functions;

• c_{i} are differential constants on solution of (24) given by (26). Finally, we can write down the general form of the symmetry for

(24).

(33) φ = ( \begin{matrix}f_{c} \\ 0 \end{matrix} ) · A(c, v(x)) + ( \begin{matrix}f_{v} \\ I \end{matrix} ) · u(x)

Here A(c, v(x)) and u(x) are arbitrary proper-sized matrices.

Remark 6. Generally, the solution (25) and its derivatives as well as expressions of the type A(c, v(x)) or u(x) are operators on v(x). In the case these are differential operators of order l, we obtain lth order higher symmetries by the formula (33).

3.3. Examples.

Example 5. A linear scalar equation

y′ = xy + v(x). (34)

The general solution in this case is easy to obtain:

y = e^{\frac{x^{2}}{2}} ∫_{x_{0}}^{x} e^{− \frac{t^{2}}{2}} v(t) dt + c · e^{\frac{x^{2}}{2}} .

Thus,

c = y · e^{\frac{−x^{2}}{2}} − I(x), where I(x) = ∫_{x_{0}}^{x} e^{− \frac{t^{2}}{2}} v(t) dt,

is a constant on any solution of (34).

Therefore, from (33) it follows that the general form of the symmetry in this example is

φ = ( \begin{matrix}e^{\frac{x^{2}}{2}} \\ 0 \end{matrix} ) · A(y · e^{\frac{−x^{2}}{2}} − I(x), v(x)) + ( \begin{matrix}e^{\frac{x^{2}}{2}} ∫_{x_{0}}^{x} e^{− \frac{t^{2}}{2}} [ •] dt \\ 1 \end{matrix} ) · u(x) (35)

Here A(c, v(x)) and u(x) are arbitrary operator and function respectively; f_{v} = e^{\frac{x^{2}}{2}} ∫_{x_{0}}^{x} e^{− \frac{t^{2}}{2}} [ • ] dt is an operator acting on u(x) by the formula

(e^{\frac{x^{2}}{2}} ∫_{x_{0}}^{x} e^{− \frac{t^{2}}{2}} [ • ] dt) u(x) = e^{\frac{x^{2}}{2}} ∫_{x_{0}}^{x} e^{− \frac{t^{2}}{2}} u(t) dt

This example shows that, since a general solution f = f(v) of a control system is an operator on controls, f_{v} in the formula (33) is a linearization of this operator.

In Theorem 2 the flow of the control function v is arbitrary, so v is a functional parameter. Suppose it is a subject to some differential constraint v_{τ} = G(x, v, v′, …, v^{(r)}). (This constitutes an alternative approach since v is then considered as an unknown on par with y, cf. [6]).

If r is the maximal order of the derivative of v entering this such a constraint, then y_{τ} can depend on v^{(s)}, s ≤ r − 1 only, cf. [5]. The next example is an illustration of this general statement.

Example 6. v_{τ} = v, c_{τ} = 0.

¿From (28) and (32) we obtain

(D − F_{y}, −F_{v}) · ( \begin{matrix}f_{v}v \\ v \end{matrix} )|_{H=0} = 0. (36)

The highest order derivative of v entering this equation is v′. It enters linearly and its coefficient is f_{v} +vf_{vv} so it must be zero. Solving f_{v} + vf_{vv} = 0 we obtain f_{v} = 1/v and y_{τ} = f_{v}v = 1. In particular, it does not depend on v in a perfect accordance with the result of [5].

References

[1] A. V. Samokhin, Symmetries of Linear Ordinary Differential Equations, Amer. Math. Soc. Transl. (2), 167 (1995).

[2] A. V. Samokhin, Symmetries of Linear and Linearizable Systems of Differential Equations, Acta Appl. Math., 56 (1999), 253–300.

[3] P. W. Doyle, Symmetry and Ordinary Differential Constraints, Int. J. of Nonlinear Mech., 34, (1999), 1089–1102

[4] V. N. Chetverikov, On the structure of integrable C-fields Diff. Geom. Appl. 1 (1991), 309–325

[5] A. V. Samokhin, Symmetries of control systems, Banach Center Publications, 33 (1996), 337–342

[6] P.H.M. Kersten, The general symmetry algebra structure of the underdetermined equation u_{x} = v_{xx}, J. Math. Phys. 32 (1991), 2043–2050

Department of Mathematics, Moscow State Technical University of Civil Aviation, 20 Kronshtadtsky Blvd., Moscow 117331, Russia

  E-mail address: asamohin@online.ru