Algebraic formalism of differential p-forms and vector fields for nonlinear control systems on homogeneous time scales

The paper develops further the algebraic formalism for nonlinear control systems defined on homogeneous time scales. The delta derivative operator is extended to differential p-forms and vector fields.


INTRODUCTION
Algebraic formalism for nonlinear control systems based on differential one-forms has been developed separately for continuous-time systems [1] and discrete-time systems [2][3][4].In [5] a single common formalism based on time scale calculus has been introduced.It covers both the continuous-and discretetime cases in such a manner that those are the special cases of the formalism.However, it has to be stressed that in [5] the discrete-time system is described in terms of the difference operator unlike in the majority of papers where the system is described via the shift operator (see for example [2][3][4]6,7]).
This paper may be understood as the continuation of paper [5], which developed the algebraic formalism of differential one-forms associated with the nonlinear control system defined on a homogeneous time scale.An inversive σ f -differential field K of meromorphic functions in system variables equipped with two operators, delta derivative ∆ f , and forward jump σ f , was constructed under the nonrestrictive assumption which guarantees the submersivity of the system.In the continuous-time case the delta derivative is just an ordinary time derivative and the forward jump is an identity operator.In the discrete-time case the delta derivative is the forward difference and the forward jump is the forward shift operator.Moreover, a vector space E (over K ) of differential one-forms was introduced, the operators ∆ f and σ f were extended to E and some of their properties were studied.The developed formalism has later been used in [8][9][10][11] to study different modelling and analysis problems.In [12] the results of [5] were partly extended for the case of regular but nonhomogeneous time scales.In the case of nonhomogeneous time scales the delta derivative and shift operator do not commute.However, the main difficulty is related to the fact that the additional time variable t appears in functions from the differential rings associated with the control system.Therefore, in the construction of the inversive closure of the differential ring the new variables depend on t and have to be chosen to be smooth at each dense point of the time scale (see [12]).Moreover, the differential ring associated with the considered system can have zero devisors, so it is impossible to construct its quotient field (see [12]).One can easily observe that taking the nonhomogeneous time scale the graininess function which depends on point t from time scale T may not be continuous and consequently not delta differentiable, so delta differentiability of the graininess function is the problem that one can encounter for nonhomogeneous time scales.Then the computation of the higher-order delta derivatives of functions is not always possible.
The goal of this paper is to unify the calculus of p-forms, extend the operators of the delta derivative and forward jump to p-forms and prove some of their properties.Moreover, we also introduce the dual space of vector fields and extend the operators of the delta derivative and forward jump to vector fields.

TIME SCALE CALCULUS AND DIFFERENTIAL FIELD
For a general introduction to the calculus on time scales, see [13].Here we recall only those notions and facts that will be used later.
A time scale T is a nonempty closed subset of R. We assume that the topology of T is induced by R. The forward jump operator σ : T → T is defined as σ (t) := inf {s ∈ T | s > t}, σ (max T) = max T if there exists a finite max T, the backward jump operator ρ(t) : T → T is defined as ρ(t) := sup {s ∈ T | s < t}, ρ(min T) = min T if there exists a finite min T. The graininess functions µ : T → [0, ∞) and ν : T → [0, ∞) are defined by µ(t) = σ (t) − t and ν(t) = t − ρ(t), respectively.A time scale is called homogeneous if µ and ν are constant functions.
Let T κ denote a truncated set consisting of T except for a possible left-scattered maximal point.The reason for omitting a maximal left-scattered point is to guarantee uniqueness of f ∆ , defined below.
Definition 2.1.The delta derivative of a function f : T→R at t ∈ T κ is the real number f ∆ (t) (provided it exists) such that for each ε > 0 there exists a neighbourhood U(ε) of t, U(ε) ⊂ T such that for all τ ∈ U(ε), For a function f : T → R we can define the second delta derivative f [2] := f ∆ ∆ provided that f ∆ is delta differentiable on T κ 2 := (T κ ) κ with the derivative f [2] : T κ 2 → R. Similarly we define higher-order delta derivatives f [n] : , n 1.Note that for homogeneous time scale T κ = T, i.e. there is no left-scattered maximal point in T, so f [n] , n 1 are uniquely defined for all t ∈ T. From now on we assume that T is homogeneous.
If f and f ∆ are delta differentiable functions, then for homogeneous time scale one has f σ ∆ = f ∆σ .Consider now the control system, defined on a homogeneous time scale T, where ) and assume1 that there exists a map ϕ : U → R m such that Φ = ( f , ϕ) T is an analytic diffeomorphism from the set U onto U.This means that from ( x, z) = f (x, u), ϕ(x, u) = Φ(x, u) we can uniquely compute (x, u) as an analytic function of ( x, z).For µ = 0 this condition is always satisfied with ϕ(x, u) = u.
The basic types of operators covered by the delta derivative are the time derivative, and the difference operator [ f (t + 1) − f (t)]/µ.So, in the special case T = R, equation (1) becomes an ordinary differential equation and when T = hZ, for h > 0, it becomes the difference equation.
For notational convenience, (x 1 , . . ., x n ) will simply be written as x, and u as u [k] , for k 0.
For i k, let u [i...k] := u [i] , . . ., u [k] .We assume that the input applied to system (1) is infinitely many times delta differentiable, i.e. u [0...k] exists for all k 0. Consider the infinite set of real (independent) indeterminates C = x i , i = 1, . . ., n, u [k] j , j = 1, . . ., m, k 0 and let K be the (commutative) field of meromorphic functions in a finite number of the variables from the set C .Let σ f : K → K be an operator defined by where F ∈ K depends on x and u [0...k] , u [0...k] σ = u [0...k] + µu [1...k+1] , for k 0 and by (1) We assume that (x, u) ∈ U and the other variables are restricted in such a way that σ f is well defined.Under the assumption about the existence of ϕ such that Φ = f , ϕ is an analytic diffeomorphism, σ f is an injective endomorphism.
The field K can be equipped with a delta derivative operator where F ∈ K depends on x and u [0...k] .The more compact notations F σ f and F ∆ f will be sometimes used instead of σ f (F) and ∆ f (F).
The delta derivative ∆ f satisfies, for all F, G ∈ K , the conditions An operator satisfying the generalized Leibniz rule is called a "σ f -derivation" and a commutative field endowed with a σ f -derivation is called a σ f -differential field [14].Therefore, under the assumption about the existence of ϕ such that Φ = f , ϕ is an analytic diffeomorphism, K endowed with the delta [14].Since σ f is an injective endomorphism, it can be extended to K * so that σ f : K * → K * is an automorphism.It was shown in [5] that for µ = 0 the inversive closure of K may be constructed as the field of meromorphic functions in a finite number of the independent variables 1}, where the new variables are related by σ f as follows: , where ψ is a certain vector-valued function, determined by f in (1) and the extension z = ϕ(x, u).Although the choice of variables z is not unique, all possible choices yield isomorphic field extensions.We extend the operator ∆ f to new variables by The extension of operator ∆ f to K * can be made in analogy to (2).Such operator ∆ f is now a σ f -derivation of K * .A practical procedure for the construction of K * (for µ = 0) is given in [5].
From now on Consider the infinite set of symbols dC * = {dζ i , ζ i ∈ C * } and define E := span K * dC * .Any element of E is a vector of the form where only a finite number of coefficients B jk and C s are nonzero elements of K * .The elements of E are called differential one-forms.Let d : K * → E be defined in the standard manner: One says that ω ∈ E is an exact one-form if ω = dF for some F ∈ K * .We will refer to dF as to the total differential (or simply the differential) of F. and The operator σ f : E → E is invertible and the inverse operator ρ It was shown in [5] that for the homogeneous time scale T we have where F ∈ K * .For one-forms similarly as for functions the more compact notations ω ∆ f and ω σ f will be used instead of ∆ f (ω) and σ f (ω).

THE DUAL SPACE OF VECTOR FIELDS
Let E be the dual vector space of E , i.e. the space of linear mappings from E to K * .The elements of E are of the form where a i , b jk , c s ∈ K * and are called the vector fields.Taking ∈ E and the vector field X ∈ E of the form (5), we get Note that even if the linear combination ( 5) is infinite, it nevertheless defines an element of E because, for all ω ∈ E , X, ω may be written as a sum with only finitely many nonzero terms; see (6).
The delta-derivative X ∆ f and forward-shift X σ f of X ∈ E may be defined uniquely by the equations and respectively, where ω is an arbitrary one-form.Note that X, σ −1 f (ω) ∈ K * , so X, σ −1 f (ω) σ f and X, σ −1 f (ω) ∆ f are well defined.Evaluating (7) and (8) with the elements of canonical basis (i.e. with the elements from the set dC * ), we obtain two systems of equations that define X ∆ f and X σ f , respectively.
Then by ( 7) and ( 8) we get From the invertibility of operator σ f : E → E we get σ f (E ) = E .Therefore by (10) the relation (9) holds.Moreover, from ( 7) and ( 9) we get We show below on a simple example how to compute X ∆ f and X σ f .Example 3.2.Consider the system described by For µ > 0 the system can be rewritten as Then the inversive closure of K can be chosen as the field of meromorphic functions in a finite number of variables ).We construct below the field extension in three different ways (choosing z as u, x 2 or x 1 , respectively) and compute X ∆ f and X σ f on different canonical bases of E , corresponding to three choices of the variable z.
Let X = ∂ ∂ x 2 be an element of E .Note that X ∆ f and X σ f have the following forms: and Case 1 (z = u).Since X ∆ f , X σ f have the form (11), (12), respectively, σ f and Then b k = c = 0, for k 0, 1, a 1 = − x 1 1+µx 2 , a 2 = 0 and bk = c = 0, for k 0, 1, ã1 = −µx 1 1+µx 2 , ã2 = 1.Therefore in particular, for µ = 0 we have Case 2 (z = x 2 ).Alternatively, the inversive closure can be chosen as a field of meromorphic functions in a finite number of variables ).Since X ∆ f has the form (11) k 0, we get equations ( 13), ( 14), (15), respectively.For the considered vector field X and the differential one-form corresponding to new variable x −1 2 we have which is different from (16) given in Case 1 for the vector field X and the differential of new variable u −1 , but taking X, dx ∆ f , 1, we get equations (17), i.e. c = 0, 1.Similarly, since X σ f have the form (12) , (20), respectively.For X, dx which is different from (21) in Case 1, but taking X, dx σ f , 1, we get equations (22), i.e. c = 0, 1.Then, as in Case 1, we get c = 0 and c = 0, for 1, but are different from coefficients given in Case 1. Therefore and Note that for µ = 0 the vector fields X ∆ f and X σ f coincide with (23).
Case 3 (z = x 1 ).A third possibility is to choose z = x 1 and define inversive closure K * as a field of meromorphic functions in a finite number of variables ).Since X ∆ f has the form (11) ∆ f , k 0, we get equations ( 13), ( 14), (15), respectively.For X, x which is different from ( 16) and ( 21) given for the vector field X and the differential of new variable u −1 in Case 1 and , we get equations (17), i.e. c = 0, 1.Similarly, since X σ f have the form (12) k 0, we have equations ( 18), ( 19), (20), respectively.For X, dx which is again different from ( 21) and ( 24) in Cases 1 and 2, but taking X, dx and c , 1, bk , k 0, ã1 , ã2 are the same as in Case 2, so X ∆ f and X σ f have the form ( 25) and ( 26), respectively.Moreover, for µ = 0, X ∆ f and X σ f coincide with (23).The fact that for µ = 0 the vector fields X ∆ f and X σ f are the same in all considered cases is related to the fact that σ f = id for µ = 0 and consequently, K = K * .Note that even if the vector field X is given by the finite linear combination (5), as in Example 3.2 where we have X = ∂ ∂ x 2 , it may happen that X σ f and X ∆ f are the infinite vector fields as in Cases 2 and 3 of Example 3.2.

p-FORMS
In this section we unify the calculus of p-forms, extend the operators ∆ f and σ f to p-forms, and prove some of their properties.
For any integer p, p 1, consider the infinite set of symbols and denote by ∧ p E the vector space spanned over K * by the elements of ∧ p dC * : In ∧ p E , p 2, we consider the equivalence relation R defined by the equalities where {i 1 , i 2 , . . ., i p } = { j 1 , j 2 , . . ., j p } and k is the signature of the permutation i 1 i 2 . . .i p j 1 j 2 . . .j p .The vector space ∧ p E mod R will be denoted 2 by E p .Its elements are called forms of degree p or simply p-forms.Every p-form α ∈ E p has a unique representative of the form 2 Wedge of p one-forms.
where A i 1 ...i p ∈ K * .Usually such a representative will be used.For instance, if chosen as a representative of the considered 2-form.By the constructions described above we obtain a sequence of vector spaces E 0 := K * , E 1 := E , E 2 , E 3 , . .., E p , . . .The exterior product (alternatively called the wedge product) of a p-form representative ω 1 = ∑ k i=1 F i dζ i 1 ∧ . . .∧ dζ i p and a q-form representative ω 2 = ∑ j=1 G j dζ j 1 ∧ . . .∧ dζ j q , denoted as ω 1 ∧ ω 2 , is defined by a (p + q)-form representative in the following way: where F i , G j ∈ K * and ζ i l , ζ j s ∈ C * , l = 1, . . ., p, s = 1, . . ., q.This definition does not depend on the choice of the representative in the equivalence class.It can be easily verified that the exterior product is bilinear and associative, moreover, it induces a map ∧ : E p × E q → E p+q , p, q 0, given by ∧(ω 1 , ω 2 ) = ω 1 ∧ ω 2 for some representatives ω 1 and ω 2 in equivalence classes.In general, the exterior product for representatives ω 1 and ω 2 is not commutative, since (27) implies A differential p-form α ∈ E p is said to be closed if dα = 0 and exact if there exists a differential (p − 1)-form β ∈ E p−1 such that α = dβ .An exact differential form is closed.
A subspace V ⊂ E 1 is said to be closed (or completely integrable) if V admits (locally) a basis composed of closed forms.To check whether the subspace V is integrable, one may use the Frobenius Theorem (see for instance [15]).Theorem 4.1 (Frobenius).Let V be the subspace of E 1 generated by the one-forms {ω 1 , . . ., ω r }.V is closed if and only if for any i = 1, . . ., r.
Example 4.2.Let x = 0.The one-form ω = xdu − udx is not closed (therefore neither exact) since dω = 2dx ∧ du.However, the vector space span K * {ω} is integrable since dω ∧ ω = 0 and one may choose the integrating factor An element ω ∈ E can be written in a unique way as for some N 0, where ω p ∈ E p is called the pth component of ω.E is called the exterior algebra over E .It has a structure of a graded algebra with multiplication given by the exterior product ∧.
Note that the space of forms with the exterior product is a ring.A subring Observe that α ∈ I implies β ∧ α ∈ I for all β ∈ E. Thus algebraic ideals are two-sided ideals.An exterior differential system is an algebraic ideal I that is stable with respect to exterior differentiation.
Note that i X (dζ i ) = X, dζ i so that for a differential 2-form ϑ = ∑ i, j a i j dζ i ∧ dζ j we have i X ϑ = ∑ i, j a i j ( X, dζ i dζ j − X, dζ j dζ i ) .
Taking the delta derivative of both sides in this identity yields, by (7), The characteristic vector fields associated with an exterior differential system I are the elements of the set A(I ) = X ∈ E | i X (I ) ⊂ I .
The annihilator C(I ) of A(I ) is the characteristic system of I .The characteristic system is completely integrable, see [15].

CONCLUSIONS
The paper extends further the algebraic formalism of differential one-forms, described in [5] for nonlinear control systems defined on homogeneous time scale.First, we unify the calculus of p-forms by extending the main concept of time scale calculus -delta-derivative -to p-forms and prove a number of its properties.In particular, we prove that the operators of the exterior derivative and the delta derivative commute when applied to p-forms.Second, we introduce the dual space of the vector fields over the field of meromorphic functions.The vector fields may be interpreted as the linear mappings from the space of one-forms into the field of meromorphic functions.A collection of Mathematica functions has been developed in order to simplify the computations with vector fields.These functions are part of the larger package NLControl, addressing various nonlinear control problems.Moreover, the functions are made available on the NLControl website [16], so that everyone can use them via the internet browser.