Relaxation and Optimization of Impulsive Hybrid Systems

Inspired by Impact Mechanics

Elena Goncharova and Maxim Staritsyn

Matrosov Institute for System Dynamics and Control Theory of the Siberian Branch of the Russian Academy of Sciences,

134, Lermontov St., Irkutsk, Russia

Keywords:

Hybrid Systems, Impulsive Control, Lagrangian Mechanics, Trajectory Relaxation, Approximation.

Abstract:

The paper compresses some results on modeling and optimization in a class of hybrid systems with control

switches of dynamics. The study is motivated by widespread physical phenomena of impulsive nature, faced

in contact dynamics, such as unilateral contacts of rigid bodies and impactively blockable degrees of freedom.

The developed modeling approach is based on a representation of hybrid events as impulsive control actions

produced by distributions or Borel measures under constraints on states before and after the action. Basically,

such systems are described by measure differential equations with states of bounded variation, and the relations

between the trajectory and the control measure are given by a speciﬁc mixed condition of a complementarity

type. The main goal of the study is to describe the closure of the tube of solutions to the addressed system.

For this, we design an approximation of the hybrid property, and develop a speciﬁc singular time-spatial

transformation of the original system. A convexiﬁcation of the transformed system then deﬁnes – after the

inverse transform – the closed set of generalized, limit solutions. The main result concerns the asymptotic

behavior of these generalized solutions, stating that the hybrid property is preserved after the relaxation.

1 INTRODUCTION

Control systems with afﬁne impulses present one of

the most popular objects of research in the mod-

ern control theory (Arutyunov et al., 2011; Bres-

san and Rampazzo, 1993; Bressan and Rampazzo,

1994; Dykhta, 1990; Dykhta, 1997; Dykhta and Sam-

sonyuk, 2000; Dykhta and Samsonyuk, 2001; Gur-

man, 1991; Gurman, 1997a; Gurman, 1997b; Gur-

man, 2004; Krotov, 1960; Krotov, 1961b; Krotov,

1961a; Krotov, 1989; Miller, 2011; Pereira and Vin-

ter, 1986; Vinter and Pereira, 1988; Rishel, 1965;

Warga, 1965; Warga, 1972; Zavalishchin and Sesekin,

1997). From the mathematical viewpoint, such a sys-

tem is the result of a trajectory relaxation (below we

give a formal deﬁnition of this notion) of a control-

afﬁne ordinary system of the form

˙x

dx(t)

= f (t, x) + G(t,x)u (1)

driven by the input signal u = u(·) being a Lebesgue

(or, possibly, Borel) measurable and summable (L

)

function. Here, x(t) ∈ R

and u(t) ∈ R

are state and

control vectors, respectively; f : R

→ R

, G : R

→

n×m

are given vector and matrix functions.

Aimed at studying the system evolution from an

initial state x(0) = x

∈ R

over a certain ﬁnite time

period T

= [0,T ], one can meaningfully suppose an a

priori bound M > 0 on the “total control action” avail-

able during T . In other words, one can impose the

condition

kuk

|u|dt ≤ M. (2)

Under standard regularity assumptions on the func-

tions f and G, this implies a uniform bound on the

total variations of Carath

eodory solutions x = x[u](·)

under all controls satisfying (2). At the same time,

one can design a sequence of Carath

eodory solutions

that tends to a discontinuous function, which is not

admitted by (1). This fact entails that the tube of so-

lutions to the system is not compact in the space C

of continuous functions. As a consequence, one fails

to guarantee the existence of a solution to an optimal

control problem stated for system (1), (2). Still, the

mentioned uniform estimate on the total variation of

trajectories enables us to design a compactiﬁcation

of the trajectory tube (or reachable sets) in a certain

topology of the space BV of functions with bounded

variation. If the function G does not depend on x, the

extension is given by the measure differential equa-

tion

474

Goncharova, E. and Staritsyn, M.

Relaxation and Optimization of Impulsive Hybrid Systems - Inspired by Impact Mechanics.

DOI: 10.5220/0006426004740485

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 474-485

ISBN: 978-989-758-263-9

dx = f (t,x) dt + G(t)µ(dt). (3)

Literally, this is done by multiplying equation (1) by

dt, and further replacing the term udt with a formal

differential dU of a, possibly, discontinuous function

U of bounded variation. The latter deﬁnes the so-

called differential, or a Lebesgue-Stieltjes measure

µ(dt)

= dU(t), which is a ﬁrst order distribution, pos-

sibly, of the Dirac (“δ-”) type.

If G does depend on x (for brevity, in what follows

we get rid of the explicit dependence of functions f

and G on the time variable), the operation G(x)dµ is

formally incorrect, as it is, in general, the product of

a discontinuous function — the composition G ◦ x —

and a Dirac distribution concentrated at discontinuity

points of G ◦ x. In this case, a correct trajectory relax-

ation is provided by a space-time extension (Gurman,

1997a; Gurman, 1997b; Miller, 2011; Warga, 1965),

based on the principle of separation of motions. By

tradition, the relaxed dynamics is also written in the

form of the measure differential equation:

dx = f (x)dt +G(x)µ(dt), (4)

which now is just a formal, conceptual object.

Let us stress that equations (3), (4) describe deter-

ministic processes. The readers familiar with stochas-

tic differential equations can note a certain similarity.

However, stochastic processes are generically driven

not by measures, but by rough paths generated by

functions of unbounded variation (in fact, of bounded

p-variation with a certain p > 1 (Lyons et al., 2007)).

1.1 Mixed Constrained Measure

Differential Equations:

Complementarity Problem

In (Goncharova and Staritsyn, 2012) we ﬁrst ﬁnd the

following complementarity problem for the measure

differential equation (4):

x(t

−

) ∈ Z

−

and x(t) ∈ Z

|µ|-a.e. (5)

Here, x(t

−

) denotes the left one-sided limit of a func-

tion x at a point t (throughout the paper we operate

with right continuous functions; this is just a tech-

nical assumption and does not lead to loss of reason-

able generality); “|µ|-a.e.” means “almost everywhere

with respect to the total variation of the measure µ”;

⊆ R

are given closed (not necessarily bounded)

subsets of the state space. Condition (5) establishes

the hybrid property, i.e., it prescribes possible system

conﬁgurations before and after switches of state. In

convention of hybrid systems theory (Branicky et al.,

1998; Haddad et al., 2006; Teel et al., 2012; van der

Schaft and Schumacher, 2001), the sets Z

are called

the jump permitting and jump destination sets, respec-

tively. Without loss of generality we can suppose

that Z

are deﬁned as Z

= {x ∈ R

| W

(x) = 0},

where W

are certain nonnegative continuous func-

tions R

→ R. Then one can rewrite (5) in the equiv-

alent form:



−

(x) +W

(x)



|µ|

(dt) = 0, and (6)

−



x(τ

−

)





x(τ)



= 0 ∀τ ∈ D

|µ|

. (7)

For absolutely continuous measures µ, the condition

(5) just expresses the orthogonality in L

between the

control u and the composition (W

−

) ◦ x .

Conditions of the sort (5) were originally intro-

duced based on methodological considerations — as

a “natural form” of mixed constraints in problems of

impulsive control — and were promptly recognized

as a useful mathematical framework for a represen-

tation of a class of hybrid systems and for modeling

certain effects in impact mechanics to be discussed in

the following Section. Since system (4), (5) performs

the hybrid feature produced by impulsive control, we

call this class of models impulsive hybrid systems.

1.2 Mechanical Inspiration. Motivating

Examples

Afﬁne impulses naturally appear in the framework of

impact mechanics and contact dynamics as a mathe-

matical formalization of elastic collision with unilat-

eral constraints and dry friction (Acary et al., 2011;

Brogliato, 2016; Glocker, 2001b; Glocker, 2001a;

Kozlov and Treshch

ev, 1991; Moreau, 1966; Moreau,

1979a; Moreau, 1979b; Yunt and Glocker, 2007;

Yunt, 2011).

As is well known, in the smooth case the rigid

body dynamics of generalized coordinates (q, ˙q) can

be described by the classical Lagrange equation



∂L

∂



−

∂L

∂ϕ

= F

with the Lagrangian L = K − P, the total kinetic en-

ergy K = K(q, ˙q) and the total potential energy P =

P(q). Here the term F comprises all external forces

acting on the mechanical system, in particular, control

forces.

If the dynamics is non-smooth (say, admits colli-

sions), the equations of motion are better written in

the form

M(q) ¨q − h(q, ˙q) = f + B(q)u. (8)

Here, the forces are composed of the uncontrolled ex-

ternal force f and the control force u, the (symmetric,

positive deﬁnite) generalized mass matrix M = M(q)

Relaxation and Optimization of Impulsive Hybrid Systems - Inspired by Impact Mechanics

475

is deﬁned as M = ∂

˙q ˙q

K, the function h = h(q, ˙q)

collects ﬁnite smooth forces (spring, damper, cen-

tripetal, gyroscopical, coriolis etc.) and is determined

by h = ∂

˙qq

K − ∂

L, and B = B(q) is a linear mapping

that represents directions of control forces.

J.J. Moreau proposed to generalize the equations

of motion (8) up to the measure-driven system

M(q)d( ˙q) − h(q, ˙q) dt = π(dt) + B(q)µ(dt). (9)

Here, measures π and µ formalize the generalized un-

controlled and control external forces, respectively.

Note that, compared to the previous Section, here

the impulsive behavior is a priori postulated rather

than thought of as an idealized mathematical phe-

nomenon, originated by a sequence of smooth mod-

els.

As an illustration, we consider the following toy

example.

Example 1. “Klapstos” strike in billiard. The model

presents two “balls” (simpliﬁed to material points of

unit mass) moving without friction in the line. The

“klapstos” problem is to strike a ball x by applying a

force ρ ≥ 0 (the billiard cue) such that x collides with

another ball y, and the balls exchange their momenta

at the instant of collision: the incoming ball stops at

the contact position, while the other ball, being in rest

up to this moment, acquires a translational velocity

and starts moving. The simpliﬁed dynamics takes the

form

d( ˙x) = (ρ − µ)(dt), d( ˙y) = k µ(dt),

where k ∈ (0,1) is the coefﬁcient of restitution, and

µ ≥ 0 is the complementary contact force. Assumed

that

(x,y)(0) = (x

), ( ˙x, ˙y)(0) = (0,0),

and y

> x

, the maneuver can be formalized as the

condition

x(t

−

) = y

, ˙x(t) = 0 µ-a.e.

In general, the force µ could also play the part of con-

trol; this case refers us to the framework of impulsive

control in the phase of unilateral contact (Bentsman

et al., 2012).

We can mark out two (to some extent, stan-

dard) approaches to modeling of contact dynamics:

the so-called regularized and non-smooth approaches.

They are well known, we refer to (Brogliato, 2016;

Glocker, 2001b; Moreau, 1966) and the bibliogra-

phy therein. Yet another approach (Bentsman et al.,

2007; Bentsman et al., 2007; Bentsman et al., 2008;

Bentsman et al., 2012; Miller and Bentsman, 2006)

is based on active singularities, and it is mostly close

in the methodological sense to the one we develop in

the present study. Note that these approaches do not

give a solution, if impulse forces of impacts act si-

multaneously with dry friction (Pfeiffer and Glocker,

1996). Such phenomena are known as Painlev

e para-

doxes (Painlev

e, 1895; Stewart, 2000). Successful at-

tempts to overcome this problem are made in (Miller

et al., 2016; Stewart, 1997).

The classical approaches operate with forces or

torques as driving signals. However, in practical

applications these physical quantities are technically

hardly measurable signals. Sometimes, one is com-

pelled to design control signals based on the obser-

vation of their inﬂuence on the system or relying on

some average characteristics of control forces. In

(Bressan and Rampazzo, 1993; Bressan and Wang,

2009; Bressan and Rampazzo, 2010; Bressan et al.,

2013) it is proposed to control Lagrangian systems

directly by a part of state coordinates. This lets one

avoid a straightforward computation of driving forces,

since their effects are already accounted by friction-

less holonomic constraints provided by the control co-

ordinates. Our approach — based on a complementar-

ity problem for the measure differential equation —

enjoys the same advantages: condition (5) contains

the complete information about the system behavior

prior to and after applying control forces. This infor-

mation lets us avoid calculation of effortfully observ-

able signals.

A pretty eloquent example of real-life models,

where control forces produce such a prescribed in-

ﬂuence on the system, is given by Lagrangian systems

with impactively blockable degrees of freedom. The

following example is aimed to illustrate the idea.

Example 2. A double pendulum with a blockable joint.

The model describes planar motion of a double pen-

dulum with links of unit length and mass, actuated

by gravity. Two degrees of freedom of the system are

due to angular positions ϕ = (ϕ

,ϕ

) of the links. The

total kinetic energy is

K =

+ 1/2

+ ϕ

cos∆ϕ,

and the total potential energy is

P = −g(2 cosϕ

+ cos ϕ

where ∆ϕ = ϕ

−ϕ

, and g is the acceleration of grav-

ity.

Denote by ω = (ω

,ω

)

= (

) the vector of

angular velocities; the states are now (ϕ,ω). Suppose

that one can control the pendulum by instantaneously

blocking/releasing the joint between the two links.

Such blocking is provided by an impulsive force, per-

formed by a scalar signed Borel measure µ. In the in-

ertial reference system, the equations of motion then

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

476

should be written as

ϕ = ω,



1 1/2cos ∆ϕ

cos∆ϕ 1



dω





1/2sin ∆ϕ ω

−sin ∆ϕ ω



dt + g



sinϕ



dt =



µ(dt)



while the blocking condition takes the form (5) with

−

= R

, and Z

= {x = (x

,. .. ,x

) ∈ R

: x

= x

i.e., in our terms,

(t) = ω

(t) |µ|-a.e.

The latter condition postulates the conﬁguration

= {(ϕ

,ϕ

,ω

) ∈ R

: ω

= ω

} after any

jump of the trajectory component ω

1.3 Notations and Basic Mathematical

Background

By N we denote the set of positive integers, and R

the n-dimensional arithmetic space with the Manhat-

tan norm | · |

= k · k

, R = R

is the set of real num-

bers, and R

is the cone of vectors with nonnegative

components.

Given a ﬁnite interval T

= [0, T ] ⊂ R

, let C =

C(T ,R

) denote the Banach space of n-dimensional

continuous functions on T with the topology of uni-

form convergence ⇒, and AC = W

1,1

⊂ C denote

the set of absolutely continuous vector functions;

= L

(T , R

) is the Lebesgue quotient space of

summable functions.

The dual C

∗

of C is known to be the space of

vector-valued signed Borel measures, i.e., countably

additive set functions µ : B → R

deﬁned on the

Borel sigma-algebra B = B

of subsets of the inter-

val T . Among all measures we single out the usual

Lebesgue measure on R, denoted by λ. A classical

fact from Analysis claims that any Borel measure ad-

mits a unique Lebesgue-Stieltjes extension, and the

set of extended measures is isomorphic to the space

BV = BV

(T , R

) of right continuous vector func-

tions with bounded variation on [0, T ). The extended

measures are called the Lebesgue-Stieltjes measures.

In what follows, the term “measure” is understood in

the extended sense.

Given a measure µ, we denote by |µ| its total vari-

ation, by supp µ its support, i.e., the minimal closed

subset of T such that µ(supp µ) = µ(T ); by D

the set

{τ ∈ T | |µ|({τ}) > 0} of its atoms, and by F

∈ BV

its distribution function deﬁned as F

(t) = µ([0,t]),

−

) = 0. Any measure µ can be thought of as the

distributional derivative or “differential” of its distri-

bution function, i.e., µ(dt) = dF

(t). Given µ,ν ∈ C

∗

µ  ν indicates that µ(A) ≤ ν(A) for any A ∈ B ;

dµ

dν

stands for the Radon-Nikodym derivative of µ with re-

spect to ν. If

dµ

dν

= 0, the measures µ and ν are said to

be mutually singular. A Lebesgue-Stieltjes measure

admits a unique Lebesgue decomposition into the sum

of the absolutely continuous µ

, singular (with re-

spect to λ) continuous µ

, and singular discrete com-

ponents µ

: µ = µ

+ µ

= µ

+ µ

. If µ is ab-

solutely continuous, then

dµ

dλ

. All the notions,

raised in the context of measures, can be adopted to

functions of bounded variation.

1.4 Weak* Topology of BV . Relaxation

of System’s Trajectory Tube

The weak* topology performs one of the two regu-

lar ways to introduce a “coarse” topological structure

in the dual X

∗

of a linear space X. Sometimes, es-

pecially in Probability Theory, the weak* topology is

called simply the weak topology. In Analysis these

two notions are distinguished: The weak topology

on X

∗

corresponds to the following notion of conver-

gence of a sequence {φ

}

n∈N

⊂ X

∗

to a point φ ∈ X

∗

ψ(φ

) → ψ(φ) for all ψ ∈ X

∗∗

. On the other hand, X

can be embedded into its double dual X

∗∗

by the map-

ping x 7→ T

, where T

(φ)

= φ(x). Thus T : X → X

∗∗

is an injective linear mapping (not necessarily surjec-

tive). The weak* topology on X

∗

is the weak topol-

ogy induced by the image of T , T (X) ⊂ X

∗∗

. A se-

quence {φ

}

n∈N

⊂ X

∗

converges to a point φ ∈ X

∗

the weak* topology iff φ

(x) → φ(x) for all x ∈ X.

Note that the weak* topology of X

∗

is weaker than

the weak topology of this space, and furthermore, it is

the weakest topology such that the maps T

: X

∗

→ R

are continuous.

In this paper we deal with BV -functions. By the

Helley’s selection principle, a sequence of functions

}

n∈N

⊂ BV is pre-compact in BV in the topology

of pointwise convergence iff {x

}

n∈N

is uniformly

bounded and has uniformly bounded variation. Note

that the compactness in the topology of pointwise

convergence implies the compactness in the weak*

topology.

The weak* topology of BV (i.e., of C

∗

) can be

speciﬁed as follows: A sequence {x

}

n∈N

⊂ BV con-

verges to a function x ∈ BV in the weak* topology of

BV (we will write x

+ x) iff x

(t) → x(t) for all con-

tinuity points t of x, and at the boundary points t = 0,

and t = T .

By a trajectory relaxation of a control dynamical

system we mean a closure of the tube of its solutions

(in the accepted sense) in a certain weak topology.

Relaxation and Optimization of Impulsive Hybrid Systems - Inspired by Impact Mechanics

477

Our study will operate with a speciﬁc impulsive solu-

tion concept of a measure differential equation. And

the problem of its trajectory relaxation will be stated

in the weak* topology of BV .

1.5 Notion of Impulsive Control Input,

and Solution Concept for

Measure-Driven Dynamical Systems

We accept the basic assumptions (H): the functions f

and G are uniformly Lipschitz continuous.

Similarly to (Arutyunov et al., 2011; Karamzin

et al., 2015), by an impulsive control we mean a col-

lection

= (µ,ν, {u

}

τ∈D

Here,

• µ ∈ C

∗

(T , R

) and ν ∈ C

∗

(T , R) are measures

with

|µ|  ν, |µ|

= ν

, and ν(T ) ≤ M. (10)

• {u

}

τ∈D

is a family of Borel measurable func-

tions

: T

= [0,T

] → R

= ν({τ}), parameterized by atoms of the mea-

sure ν and meeting the constraints

(θ)| = 1 λ-a.e. on T

, (11)

(θ)dθ = µ({τ}) (12)

for all τ ∈ D

The “attached” controls u

are jump-driving pa-

rameters (see the comment below). Since control

measures µ are signed, the limit of a sequence |µ

of total variations of weakly* converging measures

+ µ may not coincide with |µ|. The nonnegative

measure ν appears here to overcome this problem. We

set |ϑ|

= ν and formally regard |ϑ| as the total varia-

tion of impulsive control.

Let Θ denote the set of admissible controls, i.e.,

all collections ϑ satisfying (10)–(12).

Given ϑ = (µ, ν,{u

}) ∈ Θ, and x

∈ R

, by a so-

lution to (4) under the control input ϑ with the ini-

tial condition x(0

−

) = x

we mean a function x ∈

(T , R

) meeting the following integral relation

for each t ∈ T :

x(t) = x

f (x) dθ +

G(x)µ

(dθ)

∑

τ∈D



x(τ) − x(τ

−

)



. (13)

The integration with respect to the measure µ

is un-

derstood in the Lebesgue-Stieltjes sense; jump exit

points x(τ) of a function x at the instants τ ∈ D

impulses are deﬁned as x(τ) = κ

), where κ

is a

Carath

eodory solution of the limit control system

dς

κ(ς) = G(κ(ς)) u

(ς), κ(0) = x(τ

−

) (14)

on the time interval T

= [0,T

Assumptions (H) guarantee the existence and

uniqueness of a solution x[ϑ] to (13), (14) for any in-

put ϑ ∈ Θ (see, e.g., (Miller, 2011)).

The employed concept of impulsive solution is

dictated by the fact (generic, if we deal with vector-

valued measures) that the result of impulsive action –

i.e., the value of the jump of a state – depends on the

way of approximation of Dirac distributions by ordi-

nary controls. In other words, the mapping “control

measure 7→ trajectory” is, typically, multivalued, and

the selection of this map is implemented by the choice

of an attached control u

. The input-output map is

single-valued when the matrix function G enjoys the

so-called Frobenius correctness property (Rampazzo,

1999), which corresponds to the commutativity of the

vector ﬁelds deﬁned by the columns G

of the matrix

G. In this case, the terms ν and {u

} in the deﬁnition

of impulsive control can be omitted, and the solution

concept can be simpliﬁed (Miller, 2011).

1.6 Final Model Statement

In view of the given notion of impulsive control, we

have to rewrite conditions (5) in the correct form:

x(t

−

) ∈ Z

−

and x(t) ∈ Z

|ϑ|-a.e. (15)

A pair σ

= (x,ϑ) with ϑ ∈ Θ and x = x[ϑ] is called

an impulsive control process. Note that ordinary con-

trol processes (x,u) of system (1), (2) are embedded

into the measure-driven system (4) (or (13), (14)) by

setting µ = u dλ.

By Σ we denote the set of all impulsive control

processes satisfying the complementarity conditions

(5) and the following endpoint constraints:

x(0

−

) = x

, x(T ) ∈ Ω, (16)

where x

∈ R

is a ﬁxed initial position, and Ω ⊆ R

is a given closed set representing a desired terminal

conﬁguration of the modeled object. We are to as-

sume that Σ 6=

Our ﬁnal goal is to design a proper relaxation of

the set Σ in connection with related optimization prob-

lems.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

478

2 TRAJECTORY RELAXATION

OF IMPULSIVE HYBRID

SYSTEMS

The study is arranged as follows:

• First, we give a correct deﬁnition of ε-

approximation (perturbation) of solutions to sys-

tem (13)–(16).

• As a basic tool for further analysis, we develop a

speciﬁc singular transformation of the impulsive

hybrid system, and establish the equivalence re-

sult between the original and transformed models.

• By passing to the weak* limit in BV , the set of

perturbed solutions will be extended, and this ex-

tension will be found to be greater than Σ. Tech-

nically, this will be done by a convexiﬁcation of

the transformed system with a consequent inverse

(discontinuous) reparameterization of time.

• Finally, we investigate the asymptotic behavior of

the hybrid property (15), and discover its limit

form.

2.1 Approximation of Impulsive Hybrid

Systems

We start with the following important

Deﬁnition 1. Given ε > 0, an impulsive control pro-

cess σ = (x = x[ϑ], ϑ = (µ,ν,{u

})), ϑ ∈ Θ, is said

to be an ε-approximate solution of the complemen-

tary system (13)–(15), if there exists another process

= (

= x[

ϑ],

= (

µ,

ν,{

})) such that the follow-

ing relations hold:

1. (

x,F

) belongs to an ε-neighborhood of (x, F

) in

the weak* topology of BV , i.e.,



(x,F

)(t) − (

x,F

)(t)



≤ ε for

t ∈



(0,T ) \ D



∪ {0} ∪ {T }.

(17)

2. Processes meet the following “ordering” condi-

tion:

Q(F

)dν

Q(F

∑

τ∈D

Q(F

(τ),F

(τ))ν({τ})

∑

τ∈D

Q(F

(τ),F

(τ))

ν({τ}) ≤ ε. (18)

Here, Q = Q(η

,η

−

) is an arbitrary ﬁxed con-

tinuous nonnegative function R

→ R vanishing

only on the set {(η

,η

−

) ∈ R

: η

−

≤ η

3. The perturbed version of the complementarity

condition (15) is satisﬁed:

−

(

x)dν +

(x)d

ν ≤ ε. (19)

We remark on conditions (18), (19): inequality (18) is

a relaxed version of the pointwise state constraint

≤ F

. (20)

In the asymptotic respect, (20) serves to distinguish

jumps Z

−

→ Z

compared to Z

→ Z

. If Z

−

= Z

this demand is ambiguous, and (18) can be dropped.

Relation (19) establishes ε-complementarity of a

process with respect to its small weak* perturbation

(rather than with respect to itself). As we will see

below, this is an important invention.

Let us stress that the part of ε-solutions can be

played by ordinary control processes deﬁned by (1),

(2) (see the forthcoming Example 3). In this respect,

Deﬁnition 1 solves the problem of a correct approxi-

mation of the impulsive hybrid behavior by a regular,

ordinary system.

Example 3. On the time interval [0, 1], consider the

following model:

dx = dµ, x(0

−

) = 0,

µ  0, µ([0,1]) ≤ 1,

x(t) = 1 µ-a.e.

(any control action should steer the state to the set

= {1}).

Note that, for any ordinary control dµ = u dt with

kuk

= 1, the complementarity condition does not

hold, even approximately:

[0,1]

(1 − x)u dt = x(1) −

(1)

= 1/2.

On the other hand, consider the following two fami-

lies of ordinary control processes:

(t)

= (x,u)



(

), t ∈ [0,ε),

(1,0), t ∈ [ε,1],

(t)

= (







(0,0), t ∈ [0,ε),

(

t − 1,

), t ∈ [ε,2ε),

(1,0), t ∈ [2ε,1].

The pair (σ

) satisﬁes the assumptions of Deﬁni-

tion 1, in particular, (19) collapses into

[0,1]

(1 − x

) ˜u

dt = 0.

Let

X denote the set of functions x ∈ BV

(T , R

)

such that there exists a sequence {σ

}

ε>0

of impul-

sive control processes σ

= (x, ϑ)

with the following

properties:

Relaxation and Optimization of Impulsive Hybrid Systems - Inspired by Impact Mechanics

479

• for any ε > 0, σ

satisﬁes constraints (16) with

accuracy to within ε, i.e., |x

(0) − x

| ≤ ε and

(T ) ∈ U

(Ω), where U

stands for the ε-

neighborhood of a set;

• σ

is an approximate ε-solution of (4), (15) in the

sense of Deﬁnition 1, and

• x

+ x as ε → 0.

The next step is to design a constructive representa-

tion of X.

3 TRAJECTORY EXTENSION:

SEPARATION OF MOTIONS

AND DISCONTINUOUS

SPACE-TIME TRANSFORM

In this Section, we develop an equivalent transforma-

tion of the impulsive hybrid system to an ordinary

control system with bounded inputs and absolutely

continuous states. The main technical trick is a spe-

ciﬁc Lipschitzian parameterization of the time vari-

able t named the discontinuous time change (Bres-

san and Rampazzo, 1994; Miller, 2011; Warga, 1965;

Warga, 1972).

The idea originates in the general approach of

“separation of motions”. When operating with sys-

tem (13)–(15) we implicitly separate the following

essences:

• Separation in time – “slow” motions versus “fast”

dynamics. Informally speaking, the continuous

component of a solution to the measure differen-

tial equation (4) (performing slow motions) and

solutions to the limit system (14) “live” in differ-

ent time scales.

• Separation in space – the left and right one-sided

limits of an impulsive solution play the parts of

different (independent) states. Here one can re-

turn to Deﬁnition 1 and observe that it operates

with a couple of control processes. It is not hard

to see that – asymptotically – ˜x corresponds to

the left one-sided limit of a generalized solution,

while x describes the right one.

The following time-space transformation puts all

the separated objects in a common scale: the instants

of impulses will be extended into intervals and the di-

mension of the state space will increase.

On a time interval S

= [0, S], S > T , we consider

the following auxiliary ordinary control system:

ξ = α,

= α f (y

) + G(y

)β

, (21)

= |β

|, (22)

ζ = α



∆

η + |∆





|β

| + |β

−



Q(η

,η

−

)

+|β

−

) + |β

−

), (23)

(0) = x

, (ξ,η

,η

−

,ζ)(0) = 0 ∈ R

, (24)

ξ(S) = T, ∆

(y,η)(S) = 0 ∈ R

n+1

, ζ(S) = 0, (25)

(S) ∈ Ω, η

(S) ≤ M, (26)

(α,β

,β

−

) ∈ U. (27)

Here, s is a new “extended” time; the control set U

U(S) is formed by Borel measurable control functions

= (α,β

,β

−

), where α,β

: S → R are such that

α(s) ≥ 0 and α(s)+ |β

(s)| + |β

−

(s)| = 1 λ-a.e. over

the interval S .

States of the reduced system are presented by

= (ξ,y

= (y

−

),η

= (η

,η

−

),ζ) ∈ R

2n+4

, where

(s) ∈ R

, and ξ(s),η

(s),ζ(s) ∈ R

The operation ∆

applied to a vector c of the struc-

ture (c

−

), c

∈ R

, returns the vector c

−c

−

, and

∆

∓

= −∆

By x[u] we denote a Carath

eodory solution of sys-

tem (21)–(24) on the interval S , under control u ∈ U.

Given an impulsive control ϑ ∈ Θ, we denote ˆµ

λ + 2|ϑ| and introduce a strictly increasing function

ϒ : T → [0, ˆµ(T )]

as follows:

ϒ(t) = F

ˆµ

(t), t ∈ T .

Since ϒ is strictly monotone, there exists its inverse

[0, ˆµ(T )] → T , which we denote by υ.

The following theorem, proved in (Goncharova

and Staritsyn, 2015), claims that systems (13)–(16)

and (21)–(27) are equivalent to each other.

Theorem 1 1) Let ϑ ∈ Θ be such that the solution

x = x[ϑ] of (13), (14) meets conditions (15) and (16).

Then, there exist a real S ≥ T and a control u ∈ U(S)

such that the solution x = x[u] of control system (21)–

(24) satisﬁes the terminal constraints (25), (26), and

−

◦ ϒ = y

◦ ϒ = x, υ = ξ. (28)

2) Assume that S ≥ T and u ∈ U(S) are chosen

such that the related solution x

= (ξ, y,η, ζ)[u] of sys-

tem (21)–(24) satisﬁes constraints (25), (26). Deﬁne

the function x ∈ BV

(T , R

) by the composition

x = y

◦ Ξ on T , (29)

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

480

where Ξ : T → S is given by

Ξ(t) = inf{s ∈ S : ξ(s) > t}, t ∈ [0,T ),

Ξ(T ) = S.

(30)

Then, x satisﬁes (13), (14) together with constraints

(15), (16).

Note that the function Ξ deﬁned in (30) is strictly

monotone increasing, and right continuous. Further-

more, it is a pseudo-inverse of ξ, that is, the compo-

sition ξ ◦ Ξ coincides with the identity mapping Id on

T , while Ξ◦ ξ = Id for continuity points t = ξ(s) of Ξ

(Miller, 2011).

The proof of Theorem 1 is based on the following

formulas for the direct and inverse transforms.

Direct transformation. Given ϑ = (µ,ν,{u

}) ∈

Θ, put S

= ϒ(T ), and introduce the control functions

α(s)

(

◦ υ)(s), υ(s) ∈ supp ν

0, otherwise;

(31)

(s)











◦ θ

τ±

)(s), ∃τ ∈ D

s.t. s ∈ S

τ±

◦ υ)(s)α(s), υ(s) ∈ supp ν

◦ υ)(s), υ(s) ∈ supp ν

0, otherwise.

(32)

Here, m

dλ

d ˆµ

, m

dµ

dλ

, m

dµ

d ˆµ

(the deriva-

tives are regarded in the Radon-Nikodym sense);

τ+

= ϒ(τ

−

) + ν({τ}), S

τ−

= [ϒ(τ

−

),ϒ(τ)] \ S

τ+

;

τ+

(s)

= s − ϒ(τ

−

), s ∈ ϒ(τ

−

) + [0,ν({τ})], and

τ−

(s)

= s − θ

τ+

(s), s ∈ (ϒ(τ

−

) + ν({τ}), ϒ(τ)].

Inverse transformation. Given u = (α,β

−

) ∈

U(S), deﬁne a desired impulsive control ϑ =

(µ,ν, {u

}) ∈ Θ as follows:

(µ,ν)

= d(F

), (33)

−

) − 0, F

= η

◦ Ξ, (34)

−

) = 0, F

(t) =

Ξ(t)

(s)ds, t ∈ T , (35)

= β

◦ s

, (36)

where s

(θ)

= inf{S ∈ [Ξ(τ

−

),Ξ(τ)] : θ

(s) > θ}, and

(s) = η

(s) − η

(Ξ(τ

−

)), s ∈ [Ξ(τ

−

),Ξ(τ)].

Now we proceed to the convexiﬁcation of the re-

duced system. Let F denote the right-hand side of

system (21)–(24). Consider the relaxed dynamics

dx(s)

∈ co





x,u



| u ∈ U



(37)

with the initial state x(0) deﬁned by (24). Here, co A

denotes the closed convex hull of a set A.

Theorem 2. The set X coincides with the trajectory

tube of (37), (24)–(26), up to a discontinuous time

change. Namely:

1) For any x ∈ X, there exist S ≥ T and a solu-

tion x = (ξ,y

−

,η

−

,ζ) of the terminally con-

strained differential inclusion (37), (24)–(26), such

that relations (28) hold.

2) Let x = (ξ,y

−

,η

−

,ζ) be a solution to

the Cauchy problem for differential inclusion (37) on

a time interval S = [0,S], S ≥ T , such that conditions

(25), (26) hold. Deﬁne x by (29). Then, x ∈ X.

This theorem is a straightforward generalization

of a similar result proved in (Goncharova and Starit-

syn, 2017) for absolutely continuous approximations

)

ε>0

. The generalization consists in admitting

all approximations, not only absolutely continuous.

As the proof is not essentially different, but quite

lengthy, it is omitted here because of the lack of space.

Note that Theorem 2 does not claim that a limit so-

lution satisﬁes the measure-driven system (13), (14).

On the other hand, we can prove that limit solu-

tions somehow preserve the complementarity rela-

tions (15).

4 LIMIT BEHAVIOR OF

IMPULSIVE HYBRID

PROCESSES

Our goal now is to show that any limit solution x ∈

X, even not satisfying the impulsive hybrid system,

enjoys the hybrid property (15).

We will need the following simple assertion.

Lemma 1. Let {φ

}

ε>0

⊂ C and {ψ

}

ε>0

⊂ AC be two

sequences of functions S → R such that {|

ε>0

uniformly essentially bounded by a certain constant

independent of ε, (φ

,ψ

) ⇒ (φ, ψ) as ε → 0, and ψ ∈

AC. Then,

dt →

ψφ dt as ε → 0.

Proof: Consider the difference



ψφ −



dt =



ψ −



φdt +



φ − φ



dt = I

+ I

Thanks to the uniform boundedness of derivatives

{

}, the strong convergence ψ

⇒ ψ implies the

weak convergence of the sequence

ψ. As ε → 0,

this entails the convergence I

→ 0.

In its turn, the convergence I

→ 0 follows from

the uniform convergence φ

⇒ φ, and, again, the uni-

form estimation on

Theorem 3. Let x ∈

X. Consider an approximat-

ing sequence of impulsive processes σ

= (x

,ϑ

Relaxation and Optimization of Impulsive Hybrid Systems - Inspired by Impact Mechanics

481

= (µ, ν,{u

})

, x

= x[ϑ

], proposed by the deﬁ-

nition of the set X. Then, the sequence ν

converges

in the weak* topology of C

∗

to a nonnegative mea-

sure ν such that ν(T ) ≤ M, and x satisﬁes (15) with

the measure ν.

Proof: Consider the monotone increasing map ϒ

(t)

= t + F

(t) + F

(t), t ∈ T ,

and let υ

denote its inverse. Set S

= ϒ

(T ). On the

time intervals S

= [0,S

], we can deﬁne a control u

by formulas similar to (31), (32):

(

◦ υ

, υ

∈ supp(ν

)

0, otherwise;











|+|

◦ θ

, ∃τ ∈ D

s.t. s ∈ S

◦ υ

, υ

∈ supp ν

◦ υ

, υ

∈ supp ν

0, otherwise,

−











|+|

◦ θ

, ∃τ ∈ D

s.t. s ∈ S

2−

◦ υ

, υ

∈ supp

3−

◦ υ

, υ

(s) ∈ supp

0, otherwise,

where m

dλ

d(λ + ν

)

, m

dµ

dλ

, m

dµ

d(λ + ν

)

, m

2−

dλ

, m

3−

d(λ + ν

)

(s)

= s + ϒ

(τ

−

), s ∈ S

; S

= [ϒ

(τ

−

),ϒ

(τ)].

As is easily checked, the deﬁned control sat-

isﬁes α

(s) ≥ 0 and α

+ |β

(s)| + |β

−

(s)| = 1,

and the respective solution x

= (ξ,y

−

,η

−

,ζ)

to the transformed system (21)–(24) meets termi-

nal conditions (25), (26) with accuracy to within

ε. Furthermore, by a change of variable under

the sign of the Lebesgue-Stieltjes integral it holds

−

,η

−

)

◦ ϒ

= (x

Consider the extension of solutions x

to the com-

mon interval S

= [0,S

= sup{S

: ε > 0}] by constant

values: x

(s) = x

) on (S

,S]. For the extended

functions, we keep the same notation x

By assumptions (H) and usual arguments based

on the Gronwall’s inequality, the sequence {x

}

ε>0

is equicontinuous and uniformly bounded. Then,

by the Arzel

a-Ascoli selection principle, there is

a uniformly converging subsequence. Let x

(ξ,y

−

,η

−

,ζ) be the limit point.

Introduce the inverse time changes Ξ

: Ξ

(t)

inf{s ∈ S : ξ

(s) > t}, Ξ

(T ) = S, and Ξ : Ξ(t)

inf{s ∈ S : ξ(s) > t}, Ξ(T) = S (note that Ξ

= ϒ

for

all ε > 0). Thanks to the equalities Ξ

= ϒ

, we get

◦Ξ

= x

. At the same time, by arguments similar

to (Miller, 2011, Theorem 2.13), one easily derives

that y

◦ Ξ

+ y

◦ Ξ. On the other hand, x

+ x by

the assumption of the theorem. Thus,

◦ Ξ = x. (38)

Analogously, setting η

◦ Ξ = F

, we obtain ν

+ ν.

Recall that x

meets all the terminal conditions with

accuracy to within ε. The estimate η

(S) ≤ M + ε

implies that η

(S) ≤ M. Thus, ν(T ) ≤ M, as desired.

Consider the estimation ζ

(S) ≤ ε. Thanks to the

non-negativity of the velocities

and the initial

condition ζ

(0) = 0, the considered estimation can

be decomposed into the following system of integral

equalities:



∆

+ |∆



ds ≤ ε, (39)



−



Q(η

)ds ≤ ε, (40)



+ε

−

) +

−

+ε

)



ds ≤ ε.(41)

Note that x is a solution to differential inclusion (37),

(24) on S (Aubin and Cellina, 1984), and, therefore, it

is absolutely continuous. The compositions W

◦ y

and Q ◦ (η

,η

−

) are continuous. Then, by Lemma 1

1ε

→ J



∆

η + |∆



ds,

2ε

→ J



−



Q(η)ds,

3ε

→ J



−

) +

−

)



as ε → 0, which, together with (39)–(41), establishes

= J

= 0.

Let us analyze these relations:

(i) The equality J

= 0 implies ∆

(y,η) = 0 λ-a.e.

over supp

ξ. Thanks to the continuity of (y,η), this

equality, in fact, holds for all s ∈ S \

[Ξ(τ

−

),Ξ(τ)].

Furthermore,

0 = J

Ξ(T )



∆

η + |∆



dξ(s) =



∆

η + |∆



◦ Ξ dt,

which yields ∆

(y,η)◦Ξ = 0 λ-a.e. over T . By conti-

nuity of the latter composition over T \D

dΞ

and (25),

we then obtain: (y,η)

◦Ξ = (y,η)

−

◦Ξ for all t ∈ T .

(ii) Note that, thanks to the deﬁnition of the func-

tion Q, the relation J

= 0 together with dynamics (22)

and the endpoint conditions η

(0) = η

−

(0) = 0 and

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

482

(S) = η

−

(S) imply the pointwise relation η

−

≤ η

on S. This is clear, since the functions η

have no

growth points beyond the domain, where η

−

(s) ≤

(s).

(iii) Now we can show that y

(Ξ(τ

−

)) ∈ Z

−

, and

(Ξ(τ)) ∈ Z

, which completes the proof of the

statement. To do this, let us focus on the remaining

equality: J

= 0. The integral here is separated into

the series of integrals with nonnegative integrands.

The ﬁrst term gives:

0 =

S \

[Ξ(τ

−

),Ξ(τ)]

)dη

∓

(s) =

◦Ξ)dν

(t),

which yields the continuous part of the desired com-

plementarity (5). Consider the remaining conditions

Ξ(τ)

Ξ(τ

−

)

|β

−

)ds =

Ξ(τ)

Ξ(τ

−

)

|β

−

)ds, τ ∈ D

dΞ

We now claim that y

−

(Ξ(τ)) ∈ Z

−

, and y

(Ξ(τ)) ∈

for all discontinuity points of Ξ, and, due to obser-

vation (i), this would ﬁnally imply the hybrid property

(15). Assume, ad absurdum, that, say, y

−

(Ξ(τ)) /∈ Z

−

for some τ ∈ D

dΞ

. Then, by continuity, W

−

(s)) >

0 on an interval

S = [Ξ(τ

−

), ˆs). This immediately im-

plies that β

= 0 λ-a.e. on

S , and therefore

= 0,

−

= (1 −|β

|) = 1 λ-a.e. on

S , i.e., η

stays in rest,

while η

−

increases. Since η

(Ξ(τ

−

)) = η

−

(Ξ(τ

−

))

by (i), this fact contradicts estimation (ii). Inclusions

(Ξ(τ)) ∈ Z

are validated by similar arguments.

Due to (38), observing that discontinuity points of

the function x

= y

◦ Ξ are concentrated within the

set of jump points of the function Ξ, we immediately

obtain the discrete part of inclusions (5), as desired.

In other words, the hybrid property is preserved

under the designed relaxation.

5 OPTIMAL CONTROL

Consider the following optimal impulsive control

problems:

(P) Minimize I(σ) = ϕ(x(T )) over σ ∈ Σ;

(P) Minimize ϕ(x(T )) over x ∈ X.

Here, ϕ is a given terminal cost being a lower semi-

continuous function R

→ R.

The ordinary counterparts or these problems are

(RP) Minimize ϕ(y

(S)) subject to (21)–(27);

(RP) Minimize ϕ(y

(S)) subject to (37), (24)–(26).

As a theoretical application of the developed model

transformation, we establish the equivalence between

problems (P) and (RP) (respectively, between (P)

and (RP)), which would open the possibility to treat

the impulsive, non-regular models by ordinary analyt-

ical methods or existing software.

Theorem 4. 1) For problems (P) and (RP), the exis-

tence of a solution to one of them implies the exis-

tence of a solution to the other, and inf(P) = inf(RP).

2) Solutions to problems (P) and (RP) do exist,

furthermore, min(P) = min(RP).

Proof: We do the ﬁrst assertion, as the second

one is its corollary. Consider a minimizing sequence

for {σ

= (x

,ϑ

)} ⊂ Σ of problem (P). Assume, ad

absurdum, that inf(P)

= lim

k→∞

ϕ(x

(T )) > inf(RP).

Then, there should exist

S ≥ T and

u ∈ U(

S) such that

the respective solution

x = (

ξ, ˆy

, ˆy

−

ζ,

ι) of

system (21)–(24) satisﬁes rightpoint conditions (25),

(26), and inf(P) > ϕ( ˆy

(

S)). Thanks to Theorem

1, we can deﬁne a control

ϑ ∈ Θ such that ˆx =

ϑ] solves (13), (14) and is related with x

the inverse transform (29). Then, the inequality

lim

k→∞

ϕ(x

(T )) > ϕ( ˆx(T )) contradicts the deﬁnition

of the sequence x

. Thus, inf(P) ≤ inf(RP). By dis-

carding the dual hypothesis, we obtain the inverse in-

equality inf(RP) ≤ inf(P). Next, by contradiction, we

prove that a minimizer for one problem is a minimizer

for the other one, up to direct or inverse transform.

6 CONCLUSIONS

The addressed model statement comprises a large va-

riety of reasonable hybrid systems arising in mechan-

ics. In our opinion, the developed approach can be

efﬁcient in modeling and analysis of biomechanical

systems, where impulsive effects such as “almost in-

stantaneous” blocking/releasing degrees of freedom,

associated with joints of limbs of biological beings,

are, sometimes, not really meaningfully described in

terms of friction or unilateral contact.

A challenging option for further study is the exten-

sion of the obtained results to systems with quadratic

impulses, representing fast vibration of invisibly

small amplitude (Bressan and Rampazzo, 1994).

ACKNOWLEDGEMENTS

The work is partially supported by the Russian Foun-

dation for Basic Research, grants nos 16-31-60030,

16-08-00272, and 17-08-00742, and Council for

grants of the President of Russia, support of leading

scientiﬁc schools, project No NSh-8081.2016.9.

Relaxation and Optimization of Impulsive Hybrid Systems - Inspired by Impact Mechanics

483

REFERENCES

Acary, V., Bonnefon, O., and Brogliato, B. (2011). Nons-

mooth modeling and simulation for switched circuits,

volume 69 of Lecture Notes in Electrical Engineering.

Springer, Dordrecht.

Arutyunov, A., Karamzin, D., and Pereira, F. L. (2011). On

a generalization of the impulsive control concept: con-

trolling system jumps. Discrete Contin. Dyn. Syst.,

29(2):403–415.

Aubin, J.-P. and Cellina, A. (1984). Differential inclusions,

volume 264 of Grundlehren der Mathematischen Wis-

senschaften [Fundamental Principles of Mathemati-

cal Sciences]. Springer-Verlag, Berlin. Set-valued

maps and viability theory.

Bentsman, J., Miller, B. M., and Rubinovich, E. Y. (2008).

Dynamical systems with active singularities: in-

put/state/output modeling and control. Automatica J.

IFAC, 44(7):1741–1752.

Bentsman, J., Miller, B. M., Rubinovich, E. Y., and

Mazumder, S. K. (2012). Modeling and control of

systems with active singularities under energy con-

straints: single- and multi-impact sequences. IEEE

Trans. Automat. Control, 57(7):1854–1859.

Bentsman, J., Miller, B. M., Rubinovich, E. Y., and Zheng,

K. (2007). Hybrid dynamical systems with controlled

discrete transitions. Nonlinear Anal. Hybrid Syst.,

1(4):466–481.

Branicky, M. S., Borkar, V. S., and Mitter, S. K. (1998).

A uniﬁed framework for hybrid control: model and

optimal control theory. IEEE Trans. Automat. Control,

43(1):31–45.

Bressan, A., Han, K., and Rampazzo, F. (2013). On the con-

trol of non holonomic systems by active constraints.

Discrete Contin. Dyn. Syst., 33(8):3329–3353.

Bressan, A. and Rampazzo, F. (1993). On differential sys-

tems with quadratic impulses and their applications

to Lagrangian mechanics. SIAM J. Control Optim.,

31(5):1205–1220.

Bressan, A. and Rampazzo, F. (2010). Moving constraints

as stabilizing controls in classical mechanics. Arch.

Ration. Mech. Anal., 196(1):97–141.

Bressan, A. and Wang, Z. (2009). On the controllability of

Lagrangian systems by active constraints. J. Differen-

tial Equations, 247(2):543–563.

Bressan, Jr., A. and Rampazzo, F. (1994). Impulsive con-

trol systems without commutativity assumptions. J.

Optim. Theory Appl., 81(3):435–457.

Brogliato, B. (2016). Nonsmooth mechanics. Commu-

nications and Control Engineering Series. Springer,

[Cham], third edition. Models, dynamics and control.

Dykhta, V. (1997). Second order necessary optimality

conditions for impulse control problem and multi-

processes. In Singular solutions and perturbations

in control systems (Pereslavl-Zalessky, 1997), IFAC

Proc. Ser., pages 97–101. IFAC, Laxenburg.

Dykhta, V. A. (1990). Impulse-trajectory extension of de-

generated optimal control problems. In The Lya-

punov functions method and applications, volume 8

of IMACS Ann. Comput. Appl. Math., pages 103–109.

Baltzer, Basel.

Dykhta, V. A. and Samsonyuk, O. N. (2000). Optimal-

noe impulsnoe upravlenie s prilozheniyami. Fizmatlit

“Nauka”, Moscow.

Dykhta, V. A. and Samsonyuk, O. N. (2001). The max-

imum principle in nonsmooth optimal impulse con-

trol problems with multipoint phase constraints. Izv.

Vyssh. Uchebn. Zaved. Mat., (2):19–32.

Glocker, C. (2001a). On frictionless impact models in rigid-

body systems. R. Soc. Lond. Philos. Trans. Ser. A

Math. Phys. Eng. Sci., 359(1789):2385–2404. Non-

smooth mechanics.

Glocker, C. (2001b). Set-valued force laws, volume 1 of

Lecture Notes in Applied Mechanics. Springer-Verlag,

Berlin. Dynamics of non-smooth systems.

Goncharova, E. and Staritsyn, M. (2012). Optimization

of measure-driven hybrid systems. J. Optim. Theory

Appl., 153(1):139–156.

Goncharova, E. and Staritsyn, M. (2017). On bv-extension

of asymptotically constrained control-afﬁne systems

and complementarity problem for measure differential

equations. submitted to Discrete Contin. Dyn. Syst.

Goncharova, E. V. and Staritsyn, M. V. (2015). Optimal

impulsive control problem with state and mixed con-

straints: the case of vector-valued measure. Autom.

Remote Control, 76(3):377–384. Translation of Av-

tomat. i Telemekh. 2015, no. 3, 13–21.

Gurman, V. (1991). Extensions and global estimates for

evolutionary discrete control systems. In Modelling

and inverse problems of control for distributed param-

eter systems (Laxenburg, 1989), volume 154 of Lec-

ture Notes in Control and Inform. Sci., pages 16–21.

Springer, Berlin.

Gurman, V. I. (1997a). Printsip rasshireniya v zadachakh

upravleniya. Fizmatlit “Nauka”, Moscow, second edi-

tion.

Gurman, V. I. (1997b). Singularization of control sys-

tems. In Singular solutions and perturbations in con-

trol systems (Pereslavl-Zalessky, 1997), IFAC Proc.

Ser., pages 5–12. IFAC, Laxenburg.

Gurman, V. I. (2004). The method of multiple maxima and

optimality conditions for singular extremals. Differ.

Uravn., 40(11):1486–1493, 1581–1582.

Haddad, W. M., Chellaboina, V., and Nersesov, S. G.

(2006). Impulsive and hybrid dynamical systems.

Princeton Series in Applied Mathematics. Princeton

University Press, Princeton, NJ. Stability, dissipativ-

ity, and control.

Karamzin, D. Y., de Oliveira, V. A., Pereira, F. L., and

Silva, G. N. (2015). On the properness of an impulsive

control extension of dynamic optimization problems.

ESAIM Control Optim. Calc. Var., 21(3):857–875.

Kozlov, V. V. and Treshch

ev, D. V. (1991). Billiards, vol-

ume 89 of Translations of Mathematical Monographs.

American Mathematical Society, Providence, RI. A

genetic introduction to the dynamics of systems with

impacts, Translated from the Russian by J. R. Schu-

lenberger.

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

484

Krotov, V. F. (1960). Discontinuous solutions of varia-

tional problems. Izv. Vys

s. U

cebn. Zaved. Matematika,

1960(5 (18)):86–98.

Krotov, V. F. (1961a). On discontinuous solutions in varia-

tional problems. Izv. Vys

s. U

cebn. Zaved. Matematika,

1961(2 (21)):75–89.

Krotov, V. F. (1961b). The principle problem of the calcu-

lus of variations for the simplest functional on a set

of discontinuous functions. Dokl. Akad. Nauk SSSR,

137:31–34.

Krotov, V. F. (1989). Global methods to improve control

and optimal control of resonance interaction of light

and matter. In Modeling and control of systems in

engineering, quantum mechanics, economics and bio-

sciences (Sophia-Antipolis, 1988), volume 121 of Lec-

ture Notes in Control and Inform. Sci., pages 267–298.

Springer, Berlin.

Lyons, T. J., Caruana, M., and L

evy, T. (2007). Differential

equations driven by rough paths, volume 1908 of Lec-

ture Notes in Mathematics. Springer, Berlin. Lectures

from the 34th Summer School on Probability Theory

held in Saint-Flour, July 6–24, 2004, With an intro-

duction concerning the Summer School by Jean Pi-

card.

Miller, B. M. (2011). Controlled systems with impact in-

teractions. Sovrem. Mat. Fundam. Napravl., 42:166–

178.

Miller, B. M. and Bentsman, J. (2006). Optimal control

problems in hybrid systems with active singularities.

Nonlinear Anal., 65(5):999–1017.

Miller, B. M., Rubinovich, E. Y., and Bentsman, J. (2016).

Singular space-time transformations. Towards one

method for solving the Painlev

e problem. J. Math.

Sci. (N.Y.), 219(2, Problems in mathematical analysis.

No. 86 (Russian)):208–219.

Moreau, J. J. (1966). Quadratic programming in mechanics:

Dynamics of one-sided constraints. SIAM J. Control,

4:153–158.

Moreau, J.-J. (1979a). Application of convex analysis to

some problems of dry friction. In Trends in applica-

tions of pure mathematics to mechanics, Vol. II (Sec-

ond Sympos., Kozubnik, 1977), volume 5 of Mono-

graphs Stud. Math., pages 263–280. Pitman, Boston,

Mass.-London.

Moreau, J.-J. (1979b). Application of convex analysis to

some problems of dry friction. In Trends in applica-

tions of pure mathematics to mechanics, Vol. II (Sec-

ond Sympos., Kozubnik, 1977), volume 5 of Mono-

graphs Stud. Math., pages 263–280. Pitman, Boston,

Mass.-London.

Painlev

e, P. (1895). Sur le lois frottement de glissemment.

C. R. Acad. Sci. Paris. Hermann.

Pereira, F. and Vinter, R. B. (1986). Necessary conditions

for optimal control problems with discontinuous tra-

jectories. J. Econom. Dynam. Control, 10(1-2):115–

118.

Pfeiffer, F. and Glocker, C. (1996). Multibody dynamics

with unilateral contacts. Wiley Series in Nonlinear

Science. John Wiley & Sons, Inc., New York. A

Wiley-Interscience Publication.

Rampazzo, F. (1999). Lie brackets and impulsive controls:

an unavoidable connection. In Differential geometry

and control (Boulder, CO, 1997), volume 64 of Proc.

Sympos. Pure Math., pages 279–296. Amer. Math.

Soc., Providence, RI.

Rishel, R. W. (1965). An extended Pontryagin principle for

control systems whose control laws contain measures.

J. Soc. Indust. Appl. Math. Ser. A Control, 3:191–205.

Stewart, D. E. (1997). Existence of solutions to rigid body

dynamics and the Painlev

e paradoxes. C. R. Acad. Sci.

Paris S

er. I Math., 325(6):689–693.

Stewart, D. E. (2000). Rigid-body dynamics with friction

and impact. SIAM Rev., 42(1):3–39.

Teel, A. R., Sanfelice, R. G., and Goebel, R. (2012). Hybrid

control systems. In Mathematics of complexity and dy-

namical systems. Vols. 1–3, pages 704–728. Springer,

New York.

van der Schaft, A. J. and Schumacher, J. M. (2001). Mod-

elling and analysis of hybrid dynamical systems. In

Advances in the control of nonlinear systems (Murcia,

2000), volume 264 of Lecture Notes in Control and

Inform. Sci., pages 195–224. Springer, London.

Vinter, R. B. and Pereira, F. L. (1988). A maximum princi-

ple for optimal processes with discontinuous trajecto-

ries. SIAM J. Control Optim., 26(1):205–229.

Warga, J. (1965). Variational problems with unbounded

controls. J. Soc. Indust. Appl. Math. Ser. A Control,

3:424–438.

Warga, J. (1972). Optimal control of differential and func-

tional equations. Academic Press, New York-London.

Yunt, K. (2011). An augmented Lagrangian based shoot-

ing method for the optimal trajectory generation of

switching Lagrangian systems. Dyn. Contin. Discrete

Impuls. Syst. Ser. B Appl. Algorithms, 18(5):615–645.

Yunt, K. and Glocker, C. (2007). Modeling and optimal

control of hybrid rigidbody mechanical systems. In

Hybrid systems: computation and control, volume

4416 of Lecture Notes in Comput. Sci., pages 614–

627. Springer, Berlin.

Zavalishchin, S. T. and Sesekin, A. N. (1997). Dynamic im-

pulse systems, volume 394 of Mathematics and its Ap-

plications. Kluwer Academic Publishers Group, Dor-

drecht. Theory and applications.

Relaxation and Optimization of Impulsive Hybrid Systems - Inspired by Impact Mechanics

485