Abstract
This chapter is devoted to introducing the mathematical basis on which various evolution problems involving impulsive terms rely. Impulsive forces in mechanics are first presented disregarding what they may be produced by. It is shown on simple examples why impulsive mechanics involves only measures (Dirac “functions”), and no distribution of higher degree (derivatives of the Dirac “function”). Various classes of measure differential equations (MDEs), or impulsive systems, are introduced. Then unilaterally constrained dynamical systems are presented, and the differences with the foregoing MDEs are discussed. Variable changes that allow one to transform MDEs into Carathéodory ordinary differential equations (ODEs) or unilaterally constrained mechanical systems into Filippov’s differential inclusions, are described in the last section.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Recall that distributions are indefinitely differentiable, and that the derivatives of the Dirac measure are defined as \(\langle \delta _{t_{k}}^{(m)},\varphi \rangle =(-1)^{m}\varphi ^{(m)}(t_{k})\) for \(m \ge 0\), [1082].
- 2.
- 3.
Note that in the following definition, we do not pretend to define the very basic notion of what a force is. We just set what is meant by a “regular” force, in opposition to an “impulsive” force. For a discussion on the basic definition of what forces are, see for instance [1218] and references therein, who argue that in fact, forces in physics should be defined from basic axioms, just like real numbers are in mathematics.
- 4.
While systems subject to unilateral constraints will be embedded into measure differential inclusions
- 5.
The norm \(||.||^{\star }\) is defined as \(||f||^{\star }=\sum _{i=1}^{n}||f_{i}||\) with \(||f_{i}||=\text{ var }(f_{i},I)+|f_{i}(a^{+})|\) for scalar functions \(f_{i}(\cdot )\) on \(I=[a,b]\).
- 6.
The choice of the interval [0, 1] is arbitrary and could be replaced by any other compact interval without changing significantly the developments.
- 7.
RCLBV in short.
- 8.
See Appendix A.2.
- 9.
A multivalued mapping \(F: \mathbb {R}^{n} \rightrightarrows \mathbb {R}^{n}\) is said outer semicontinuous, if its graph \(\{(x,y) | x \in \mathbb {R}^{n}, y \in F(x)\} \subset \mathbb {R}^{2n}\) is closed. It is locally bounded on C if for each compact set \(S \subset C\) one has F(S) bounded.
- 10.
The codimension of a submanifold (a surface) is the difference between the dimension of the ambient space and the dimension of the submanifold [60]. Recall that there are three ways of defining a surface S of dimension \(n-m\) in an ambient space of dimension n [359]. We make use only of one of them, which consists of defining S through m relationships like \(f_{i}(q_{1},\ldots ,q_{n})=0\). A non-singular point \(q_{0}\) is such that the Jacobian matrix \(\left( \frac{\partial f}{\partial q}(q_{0})\right) \in \mathbb {R}^{m \times n}\) has rank m. Then the three definitions are equivalent in a neighborhood of \(q_{0}\). The codimension of the intersection \(S_{1} \cap S_{2}\) is the sum of the codimensions of \(S_{1}\) and \(S_{2}\), provided the intersection is transversal, (i.e., the tangent hyperplanes to each one of the surfaces at the intersection span the whole ambient space [487, p. 50]. The reader can think of two planes in \(\mathbb {R}^{3}\) (codimension 1 surfaces): either they are parallel, or they intersect transversally and the intersection is a straight line whose codimension is 2).
- 11.
- 12.
From Definition B.7 we may use either the subdifferential of the indicator function, or the normal cone.
- 13.
This may be justified for massless or near-massles cables, where the force exerted on one side of the cable equals the tension in it.
- 14.
Such potentials were introduced by J.J. Moreau who called them superpotential functions [879].
- 15.
In general, the equation in (1.45) possesses several real solutions, and one has to decide which one is the right , e.g., the bouncing ball case in Chap. 7, Eq. (7.7). In the degenerate case, the trajectories in a neighborhood of \(t_{0}\) are on the manifold \(\frac{\partial h}{\partial t}(t_{0}) = 0, h=0, f(q)=0\) and are tangent to the surface \(f(q)=0\) [140]: those orbits are grazing trajectories .
- 16.
Recall that given an ODE: \(\dot{x}(t)=f(x(t))\), its flow is a smooth function of t and \(x_{0}=x(\tau _{0})\), denoted as \(\varphi _{t}(x_{0})\), such that \(\frac{\partial \varphi _{t}(x_{0})}{\partial t}=f(\varphi _{t}(x_{0}))\) and with \(\varphi _{\tau _{0}}(x_{0})=x_{0}\). In other words, a vector field f(x) allows the construction of a flow, and the flow is an integral curve of f(x) (then f(x) is said to generate the flow \(\varphi _{t}(x_{0})\)). A flow may be local or global, and possesses several properties, like invertibilty: \(\varphi _{t}^{-1}(x_{0})=\varphi _{-t}(x_{0})\), and the autonomy (or semi-group) property: \(\varphi _{t+s}(x_{0})=\varphi _{t}(\varphi _{s}(x_{0}))\). There is a bijective relation between the set of flows and that of generating vector fields. This means that given a priori a flow, there is one and only one vector field that generates it.
- 17.
It is justified to speak of the flow between impacts since the dynamics is smooth on those period.
- 18.
Clearly this property is not true in general for non-autonomous systems, since the initial vector field, i.e., the slope of the curve (the orbit) in the two-dimensional case, changes if the initial time changes. Therefore even if the initial state remains unchanged, there is no reason that after a certain amount of time, both solutions coincide.
- 19.
Existence of solutions is a basic property, and we shall come back on existence results in the next chapters (see Theorem 5.3). We take some freedom here with the mathematical logic, since our goal is to highlight the differences between various sorts of measure differential equations.
- 20.
Consequently closedness of graphs with the Hausdorff distance may be the right notion [888].
- 21.
Which could as well be defined with preimpact values.
- 22.
i.e., the space that consists of equivalence classes.
- 23.
To be more specific: the set-valued right-hand side in (1.36) is not a suitable set for contact forces that stem from a complementarity modeling.
- 24.
Positive invariance theory is a field of control theory that deals with the invariance of polyhedral sets under linear- state feedback.
- 25.
- 26.
The Routh’s function is usually introduced for n-degree-of-freedom systems that possess \(n_{c}\) cyclic coordinates [1178] [845, §3.3] (i.e., coordinates \(q_{1},\cdots ,q_{n_{c}}\) that do not appear in the Lagrangian function L or in the Hamiltonian function H). Every cyclic coordinate yields a first integral of the system since the corresponding momenta \(p_{1},\cdots ,p_{n_{c}}\) are invariant. Routh’s method consists of applying a Legendre transformation only in the coordinates \(q_{1},\cdots ,q_{n_{c}}\), i.e., the Routh’s function is equal to \(R=L-\sum _{i=1}^{n_{c}}p_{i}\dot{q}_{i}\). Comparing this formula with (1.76) one sees that the unconstrained coordinates play the role of the cyclic coordinates. The interest of the Routh’s function is that it plays the role of a Hamiltonian function for the cyclic coordinates, i.e., \(\dot{p}_{i}=-\frac{\partial R}{\partial q_{i}}\) and \(\dot{q}_{i}= \frac{\partial R}{\partial p_{i}}\).
- 27.
If the function \(f(\cdot )\) is nonlinear in its third argument, Filippov’s convexification and other frameworks like Utkin’s equivalent control method for discontinuous ODEs may not be equivalent, however.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Brogliato, B. (2016). Impulsive Dynamics and Measure Differential Equations. In: Nonsmooth Mechanics. Communications and Control Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-28664-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-28664-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-28662-4
Online ISBN: 978-3-319-28664-8
eBook Packages: EngineeringEngineering (R0)