Appendix A Distributions, Measures, Functions of Bounded Variations Sections A.1 and A.2 are given for the sake of completeness because some notions are used in Chaps. 1 and 2, but may be safely skipped since their implication on understanding Nonsmooth Mechanics is weak. By contrast, Sects. A.3 and B survey useful mathematical tools that cannot be ignored. A.1 Schwartz’ Distributions A.1.1 The Functional Approach In this section we first briefly introduce the functional notion of a distribution as defined in [1082]. Definition A.1 D is the subspace of smooth1 functions ϕ : Rn → C, with bounded support. Thus a function ϕ(·) on Rn belongs to D, if and only if ϕ(·) is smooth, and there n exists a bounded set Kϕ of R outside of which ϕ ≡ 0. As an example, L. Schwartz gives the following function [1082, Chap. 1,§2], with n = 1, Kϕ =[−1, 1]: 0if|t|≥1 ϕ(t) = −1 (A.1) e 1−t2 if |t| < 1 Definition A.2 A distribution D is a continuous linear form defined on the vector space D. This means that to any ϕ ∈ D, D associates a complex number D(ϕ), noted D, ϕ. The space of distributions on D is the dual space of D and is noted D . The functions in D are sometimes called test-functions. 1i.e., indefinitely differentiable. © Springer International Publishing Switzerland 2016 535 B. Brogliato, Nonsmooth Mechanics, Communications and Control Engineering, DOI 10.1007/978-3-319-28664-8 536 Appendix A: Distributions, Measures, Functions of Bounded Variations Two distributions D1, D2 are equal on an open interval Δ if D1 − D2 = 0 on Δ, i.e., if for any ϕ ∈ D whose support Kϕ is contained in Δ, then D1 − D2, ϕ=0. In fact, one can generate a distribution from any locally integrable f f (x)ϕ(x)dx function , via the integral Kϕ . However, some distributions cannot be generated by locally integrable functions, like for instance, the Dirac distribution and its derivatives. They are called singular distributions (or sometimes generalized functions). Contrarily to functions, all distributions (i.e., elements of D ) are infinitely dif- ferentiable. The mth derivative of T ∈ D is given by T (m), ϕ=(−1)m T, ϕ(m), for all m ∈ N. An important feature of distributions is that, in general, the product of two distributions does not define a distribution. A.1.2 The Sequential Approach Schwartz’ distributions can be defined via the sequential approach [51, §4.3]. Roughly, one starts by defining fundamental sequences of continuous functions on a fixed interval (a, b), and then a relation of equivalence between fundamental se- quences. Distributions can thus be defined as limits of sequences of continuous functions, but there are sequences of continuous functions that do not converge to- ward a function, as it is well known for the Dirac distribution. This shows that the space of functions has to be completed by other mathematical objects, which one calls distributions. Definition A.3 A sequence { fn(·)} of continuous functions defined on (a, b) is fun- damental if there exist a sequence of functions {Fn(·)} and an integer k ∈ N such that • (k)( ) = ( ) ∈ ( , ) Fn x fn x for all x a b . •{Fn(·)} converges almost uniformly A sequence of functions converges almost uniformly (a.u.) on (a, b) if it converges [ , ]⊂( , ) { x } uniformly on any interval c d a b (for instance, n converges a.u. toward 0 on (−∞, +∞). Before defining distributions, one needs to define equivalent funda- mental sequences: Definition A.4 Two fundamental sequences { fn(·)}, {gn(·)} are equivalent if there exist {Fn(·)}, {Gn(·)} and k ∈ N such that • (k)( ) = ( ) (k)( ) = ( ) Fn x fn x and Gn x gn x . •{Fn(·)} and {Gn(·)} converge a.u. toward the same limit. One denotes { fn(·)}∼{gn(·)}. A distribution on (a, b) is an equivalent class in the set of all fundamental sequences on (a, b). For instance, the Dirac distribution has to be seen as the limit of a sequence of continuous functions δn(·) with support Kn =[tc, tc + Δtn], Δtn → 0asn →+∞, and: Appendix A: Distributions, Measures, Functions of Bounded Variations 537 • (i) δn(·) ≥ 0 • ∈ N +∞ δ (τ) τ = (ii) for all n , −∞ n d 1 • (iii) for all a > 0, limn→+∞ δn(τ)dτ = 0 |τ|>a+tc δ (·) → δ →+∞ Then n tc as n , and we know that there exists a sequence of contin- ( ) (k) = δ { } uous functions Pn t and k such that Pn n, and Pn converges uniformly. Note that such a fundamental sequence (called a delta-sequence) that determines the Dirac δ measure is by far not unique. Examples of functions n that belong to this equiva- δ = n − x2 = sin(nx) = lent class are given in [51, Chaps. 1,2]: t=0 π exp n π 2 2 x n (− | |) = 1 n 2 exp n x π exp(nx)+exp(−nx) . Distributions have derivatives of any order: Definition A.5 If a fundamental sequence { fn(·)} consists of functions with contin- ∈ m ( , ) [ (m)(·)] uous mth derivatives (i.e., fn C a b ) then the distribution fn is the mth derivative of the distribution [ fn(·)]. Each distribution has derivatives of all orders as one can always choose a fundamental sequence of functions which are differentiable up to an arbitrary order. For instance, ( ) = n − x2 [ (m)(·)]=δ(m) if fn x 2π exp n 2 , fn t=0.From[51, Theorem 2.2.5], the mth δ [δ(m)] {δ } derivative of the Dirac distribution 0 is given by n where n is a delta sequence with continuous first m derivatives. A.1.3 Notions of Convergence Some existence of solution results presented in Chap.2 use the notion of strong and weak convergence. Let us explain here what this means. For simplicity, we restrict ourselves to the spaces D and D . The definitions also exist for other spaces of functions and their dual spaces like Sobolev spaces, see [191]. Definition A.6 A sequence of functions ϕn ∈ D is weakly convergent to ϕ ∈ D if for each T ∈ D one has lim T, ϕn=T, ϕ (A.2) n→+∞ The sequence ϕn ∈ D is convergent to ϕ(·) in the topology of D if their supports are ϕ → ϕ ϕ(k) → ϕ(k) contained in a fixed compact set, n uniformly and all derivatives n uniformly, for all k ≥ 1. Definition A.7 A sequence of functionals Tn ∈ D is weakly convergent to T ∈ D if for each ϕ ∈ D one has lim Tn, ϕ=T, ϕ (A.3) n→+∞ It is also possible to define a strong convergence in D . However, in D and D , strong and weak convergences coincide [1082][488, §6.3,Theorem 2]. 538 Appendix A: Distributions, Measures, Functions of Bounded Variations The notation weak is to recall that this applies to elements of D . As an example, ( ) = ( ) | ( )|= consider the sequence of functions fn x n cos nx . Clearly, supx∈R fn x n − 1 ( ), ϕ¨= so that this sequence does not converge uniformly. However, n cos nx sin(nx), ϕ˙=n cos(nx), ϕ=fn, ϕ→0asn →+∞. Hence { fn}→0ina p weak sense. One says that fn converges strongly to f in L if || fn − f ||L p → 0 p as n →+∞. One says that fn converges weakly in L to f if fnϕ → f ϕ as →+∞ ϕ ∈ q 1 + 1 = < +∞ < < +∞ n ,for L , p q 1, p .For1 p , weak and strong convergences are the same. Consider the proofs of existence of solutions based on discretization of the mea- sure differential inclusions, like those in [867]or[1142]. The constructed discrete- time solutions are such that the acceleration is a function. Now, a sequence of func- tions converges strongly in the sense of measures toward a limit that is also a function. Hence, the only way to get a limit that is a singular measure (thus not identifiable with a function) is to consider its weak convergence, because weak convergence permits functions (considered as measures) to tend to singular measures. Without this notion of convergence it would be hopeless to get a limit with discontinuous velocity. A.2 Measures and Integrals We have seen that a proper statement of nonsmooth shock dynamics involves to consider bounded forces as density with respect to the Lebesgue measure dx of the contact impulse measure, whereas contact percussions are atoms of the contact impulse measure, and the impulse magnitude is the density of these atoms with δ respect to the Dirac measure at the impact time tk . The aim of this appendix is to introduce all these notions. Let us start by defining abruptly what is meant by a measure [477]: Definition A.8 Let (X, R) be a measurable space. A positive measure (or simply measure) on (X, R) is a mapping μ : R →[0, +∞] with the following properties: • μ( ∅) = 0. • μ ∪ = μ( ) { } R ∩ n≥1 An n≥1 An for any sequence An of subsets of , with An Am =∅for n = m. Such a mapping that satisfies the second property is called countably additive. Engineers should recall that a measure is defined as a function of sets of X that belong to a family of sets R, i.e., it assigns to a set a positive real number. Remark A.1 In fact it would be preferable to denote (X, R, μ) a measurable space, to emphasize that it is attached to a measure μ. An example of measurable space is (N, R) where R is a σ-ring of subsets of N, and the measure is defined as μ(A) =(the number of elements of A), with A ⊂ R.One has for any A and B ∈ R [477, p.78]: Appendix A: Distributions, Measures, Functions of Bounded Variations 539 μ(A) + μ(B) = μ(A ∪ B) + μ(A ∩ B) (A.4) μ(A − B) = μ(A) − μ(A ∩ B) Now we are ready to introduce what is called the Lebesgue measure: Theorem A.1 [477] There exists a unique measure λ on the measurable space (R, B) such that λ [[a, b)] = b − a for all couples (a, b) of real numbers, with a ≤ b.