Table Of ContentLecture Notes in Control and Information Sciences 477
Aram Arutyunov
Dmitry Karamzin
Fernando Lobo Pereira
Optimal
Impulsive
Control
The Extension Approach
Lecture Notes in Control and Information
Sciences
Volume 477
Series editors
Frank Allgöwer, Stuttgart, Germany
Manfred Morari, Zürich, Switzerland
Series Advisory Board
P. Fleming, University of Sheffield, UK
P. Kokotovic, University of California, Santa Barbara, CA, USA
A.B. Kurzhanski, Moscow State University, Russia
H. Kwakernaak, University of Twente, Enschede, The Netherlands
A. Rantzer, Lund Institute of Technology, Sweden
J.N. Tsitsiklis, MIT, Cambridge, MA, USA
Thisseriesaimstoreportnewdevelopmentsinthefieldsofcontrolandinformation
sciences—quickly, informally and at a high level. The type of material considered
for publication includes:
1. Preliminary drafts of monographs and advanced textbooks
2. Lectures on a new field, or presenting a new angle on a classical field
3. Research reports
4. Reports of meetings, provided they are
(a) of exceptional interest and
(b) devoted to a specific topic. The timeliness of subject material is very
important.
More information about this series at http://www.springer.com/series/642
Aram Arutyunov Dmitry Karamzin
(cid:129)
Fernando Lobo Pereira
Optimal Impulsive Control
The Extension Approach
123
AramArutyunov Dmitry Karamzin
Moscow State University Federal ResearchCenter “Computer
Moscow,Russia ScienceandControl”of the Russian
Academyof Sciences
and Moscow,Russia
Institute of Control Sciences of the Fernando LoboPereira
Russian Academy ofSciences FEUP/DEEC
Moscow,Russia Porto University
Porto, Portugal
and
RUDN University
Moscow,Russia
ISSN 0170-8643 ISSN 1610-7411 (electronic)
Lecture Notesin Control andInformation Sciences
ISBN978-3-030-02259-4 ISBN978-3-030-02260-0 (eBook)
https://doi.org/10.1007/978-3-030-02260-0
LibraryofCongressControlNumber:2018957638
©SpringerNatureSwitzerlandAG2019
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission
orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors, and the editorsare safeto assume that the adviceand informationin this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor
for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
The idea for writing this book emerged from the need to put together a number of
researchresultsonoptimalimpulsivecontrolproblemsobtainedbytheauthorsover
the past years. The class of impulsive dynamic optimization problems stems from
thefactthatmanyconventionaloptimalcontrolproblemsdonothaveasolutionin
the classical setting. The absence of a classical solution naturally invokes the
so-called extension, or relaxation, of a problem and leads to the concept of gen-
eralized solution including the notion of generalized control and trajectory.
Herein, we consider several extensions of optimal control problems within the
framework of optimal impulsive control theory. In such a framework, the feasible
arcs are permitted to have jumps, and therefore, conventional continuous solutions
may fail to exist. Various types of results derived by the authors, essentially cen-
tered on the necessary conditions of optimality in the form of Pontryagin’s maxi-
mumprinciple,andexistence theorems,whichshape asubstantial bodyofoptimal
impulsivecontroltheory,arebroughttogether.Atthesametime,optimalimpulsive
control theory is presented in a certain unified framework, while the paradigms
of the different problems are introduced in increasing order of complexity. More
precisely, the rationale underlying the book consists in addressing extensions of
increasing complexity, starting from the simplest case provided by linear control
systems and ending with the most general case of totally nonlinear differential
control system with state constraints.
This book reflects rather long and laborious cooperative research work under-
taken by the authors over the past years, most of which was developed through
strong networking between the authors’ institutions, the emphasis being on the
Research Unit SYSTEC of Engineering Faculty of the University of Porto. Such
work certainly would not have been possible, if it were not for the multilateral
support of a whole team of researchers and staff. In this regard, we would like to
express our gratitude to a number of people who, through their constant research,
administrativeorlogisticsupport,helpedusinourefforttowritethisbook.Firstof
v
vi Preface
all, we are sincerely grateful to our teachers, co-authors, and colleagues. We
especiallythank ProfessorRichard Vinterfrom ImperialCollegeLondon.Weowe
muchtotheconstantassistanceofthestaffofDEEC/FEUP,notably,PauloManuel
Lopes,IsidroRibeiroPereira,PedroLopesRibeiro,andJoséAntónioNogueira.We
gratefully acknowledge the proofreading of Alison Goldstraw Fernandes which
contributed to the valuable readability of the book. We also thank Oliver Jackson,
Manjula Saravanan and Komala Jaishankar from Springer for their excellent sup-
port throughout the publishing process.
Finally,weacknowledgethevaluablesupportfromvarioussciencefoundations,
both in Russia and Portugal. This work was supported by the Russian Foundation
for Basic Research during the projects 16-31-60005 and 18-29-03061, by the
RussianScienceFoundationduringtheproject17-11-01168,andbytheFoundation
for Science and Technology (Portugal) during the projects FCT R&D Unit
SYSTEC—POCI-01-0145-FEDER-006933 funded by ERDF j COMPETE2020 j
FCT/MEC j PT2020 extension to 2018, NORTE-01-0145-FEDER-000033 funded
by ERDF j NORTE 2020, and POCI-01-0145-FEDER-032485 funded by ERDF j
COMPETE2020.
Porto, Portugal Aram Arutyunov
July 15, 2018 Dmitry Karamzin
Fernando Lobo Pereira
Contents
1 Linear Impulsive Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Existence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2 Impulsive Control Problems Under Borel Measurability . . . . . . . . . 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Impulsive Control Problems Under the Frobenius Condition . . . . . . 39
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Second-Order Optimality Conditions for a Simple Problem. . . . . . 56
3.6 Second-Order Necessary Conditions Under the Frobenius
Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4 Impulsive Control Problems Without the Frobenius Condition . . . . 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Problem Statement and Solution Concept. . . . . . . . . . . . . . . . . . . 77
4.3 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
vii
viii Contents
4.4 Existence of Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.5 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5 Impulsive Control Problems with State Constraints . . . . . . . . . . . . . 99
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3 Maximum Principle in Gamkrelidze’s Form. . . . . . . . . . . . . . . . . 104
5.4 Nondegeneracy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6 Impulsive Control Problems with Mixed Constraints . . . . . . . . . . . . 121
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.2 Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3 Problem Formulation and Basic Definitions . . . . . . . . . . . . . . . . . 126
6.4 Basic Constructions and Lemmas . . . . . . . . . . . . . . . . . . . . . . . . 130
6.5 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
7 General Nonlinear Impulsive Control Problems . . . . . . . . . . . . . . . . 153
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
7.3 Extension Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.4 Generalized Existence Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . 161
7.5 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
7.6 Examples of Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 173
Notation
Rk Euclidean space of dimension k
R Real line: R¼R1
CðT;RkÞ Space of continuous vector-valued functions
xðtÞ¼ðx1ðtÞ;x2ðtÞ;...xkðtÞÞ defined on T with values in Rk, w.r.t.
the norm kxk ¼ max jxðtÞj
C t2T
C(cid:2)ðT;RkÞ Dual space to CðT;RkÞ, or the space of Borel vector-valued
measures l¼ðl1;l2;...;lkÞ wRith values in Rk, w.r.t. theRnorm of
total variation klk¼sup hf;dli, where notation hf;dli
kfkC¼1PTR T
standsforthesumofintegrals fidli,beingfi thecomponents
i T
of f, i¼1;2;...;k
BVðT;RkÞ Space of vector-valued functions xðtÞ of bounded variation defined
on T with values in Rk, w.r.t. the norm kxk ¼Varxj
BV T
L ðT;RkÞ Spaceofmeasurablevector-valuedfunctionsxðtÞdefinedonT with
p
values in Rk s.t., jxðtÞjp is integrable, 1(cid:3)p\1, w.r.t. the norm
R
kxk ¼ð jxðtÞjpdtÞ1=p, while for p¼1, xðtÞ is essentially
Lp T
bounded, w.r.t. the norm kxk ¼esssup jxðtÞj
L1 t2T
W1;pðT;RkÞ Spaceofabsolutelycontinuousvector-valuedfunctionsxðtÞdefined
on T with values in Rk, and such that dx2L ðT;RkÞ
dt p
l Vector-valued Borel measure
l Continuous component of measure l
c
l Discrete, or atomic, component of measure l
d
l Absolute continuous component of measure l
ac
l Singular continuous component of measure l
sc
rangel Set of all possible values of measure l
suppl Support of measure l
jlj Total variation measure of measure l
klk Norm given by total variation of measure l on T, that is
klk¼jljðTÞ
lðsÞ Value of measure l on the set fsg, that is lðfsgÞ
ix
