Hybrid Dynamics Comprising Modes Governed by Partial Differential Equations:
Modeling, Analysis and Control for Semilinear Hyperbolic Systems in One Space Dimension
Der Naturwissenschaftlichen Fakult¨at
der Friedrich-Alexander-Universit¨at Erlangen-N ¨urnberg zur
Erlangung des Doktorgrades Dr. rer. nat.
vorgelegt von Falk Michael Hante
Erlangen-N ¨urnberg
Tag der m ¨undlichen Pr ¨ufung: 14. Juli 2010 Vorsitzender der
Promotionskommission: Prof. Dr. Eberhard B¨ansch Erstberichterstatter/in: Prof. Dr. G ¨unter Leugering Zweitberichterstatter/in: Prof. Dr. Hans Josef Pesch
Abstract
Hybrid dynamical systems are considered as a design paradigm or multiscale model for networked transport problems. Evolution in time is governed by switching among a family of solutions to vector-valued, semi-linear hyperbolic partial differential equa-tions on a bounded interval in space with reflecting boundary condiequa-tions. Existence of generalized solutions and appropriate continuous dependency properties are studied on the discrete-continuous level. In particular, it is shown how the Zeno phenomenon can be avoided, despite the propagation of discontinuities along characteristics re-sulting from instantanously changing boundary conditions and their interaction with switching rules evaluating pointwise boundary obervation. Moreover, for a given cost function including switching costs, the problem of optimal switching is studied. For scalar equations with boundary switching control, optimality is characterized in terms of a first order necessary condition based on a formula for switching time sensitivity. The results are validated numerically on examples.
Kurzzusammenfassung in deutscher Sprache
Es werden diskret-kontinuierliche dynamische Systeme als Konstruktions-Paradigma oder Multiskalen-Modell f ¨ur vernetzte Transportprozesse betrachtet. Die zeitliche Ent-wicklung wird bestimmt durch Schalten zwischen L ¨osungen von vektorwertigen, se-milinearen hyperbolischen partiellen Differentialgleichungen auf einem beschr¨ankten Ortsintervall mit reflektierenden Randbedingungen. Existenz von verallgemeinerten L ¨osungen und geeignete stetige Abh¨angigkeitseigenschaften werden auf dem diskret-kontinuierlichen Niveau untersucht. Insbesondere wird gezeigt, wie das Zeno’sche Ph¨anomen vermieden werden kann, trotz der Fortpflanzung von Unstetigkeiten ent-lang Charakteristiken, die auf abrupten Wechsel der vorgeschriebenen Randbedin-gungen zur ¨uckgehen, sowie deren Interaktion mit Schaltregeln wenn sie punktweise Randbeobachtung auswerten. Dar ¨uberhinaus werden f ¨ur gewisse Kostenfunktionen, die Schaltkosten beinhalten, das Problem des optimalen Schaltens untersucht. F ¨ur ska-lare Gleichungen mit Rand-Schalt-Steuerung wird Optimalit¨at im Sinne notwendiger Bedingungen charakterisiert, die auf einer Formel f ¨ur Schaltzeit-Sensitivit¨aten basiert. Die Ergebnisse werden numerisch an Beispielen validiert.
Preface
Today’s technological problems more than ever demand the study of dynamical systems beyond a completely continuous or completely dis-crete description. As a well known example, a model of a temperature control system consisting of a heater and a thermostat would certainly include as a variable the room temperature and the operating mode of the heater (ON or OFF) where it is common practice to take the former as real-valued and the latter as discrete. For the control system to be effective there needs to be a coupling between these variables, e. g., the operating mode to be switched to ON when the room temperature falls below a certain threshold. A fully discretized model may serve for a simulation, but without a mathematical description of the system on the hybrid level, for instance, the convergence properties of any algo-rithm based only on the discrete model is often meaningless. On the other hand, relaxation of the discrete variables easily lead to physically infeasible solutions if mathematics would then, for instance, suggest to operate the heater half ON (or half OFF) for maximal performance at minimal cost.
The activities following a systematic way of dealing with dynami-cal systems whose evolution depends on a coupling between variables taking values in a continuum and variables that take values in a finite or countable set let to the field of hybrid dynamical systems, though a general theory is not yet identifiable. One reason is that various scien-tific communities mainly in the field of computer science and control engineering contribute each with different goals and with there own approaches to this area. Hybrid dynamical systems therefore became a very interdisciplinary field.
As a matter of fact, distributed parameter systems as governed by partial differential equations are seldom considered in this context, al-though many examples easily come into mind. Observe that a reason-able model even for the temperature control system would involve the heat equation. More demanding and technologically challenging
ap-Acknowledgments are due to Prof. Dr. G ¨unter Leugering who en-visions and suggests the treatment of such problems by the concept of hybrid dynamical systems. Since this opens up a new branch with a strong emphasize on mathematics in this very interdisciplinary field, it is clear that we can at this stage only scratch at the surface of this big subject, restricting our analysis at first for instance to systems of semilinear hyperbolic equations. But even the primary theory devel-oped here brings in new aspects for both, hybrid dynamical systems and partial differential equations.
I am indebted to many people who influenced my thinking and of-fered valuable suggestions and advice for this work. Especially I like to thank my supervisor Prof. Dr. G ¨unter Leugering for his encour-agement and support. He also suggested to me the candidacy for the International Doctorate Program Identification, Optimization and Control with Applications in Modern Technologieswhere this work elaborated. The research position and financial support granted by the Elite-Network of Bavaria and the mentoring from my advisors Prof. Dr. Hans Josef Pesch and Prof. Dr. Lars Gr ¨une in the program is gratefully acknowl-edged. Many results were obtained in fruitful discussion and coorpora-tion with Prof. Dr. Thomas Seidman, my sincere thanks to him for this exceptional collaboration.
I like to thank my colleagues at the University of Erlangen-N ¨urn-berg and the members of the doctorate program for creating a very stimulating environment, in particular to PD Dr. Martin Gugat, Prof. Dr. Michael Stingl and Christoph Schumacher for many helpful discus-sions. Also I like to thank Prof. Dr. Alexandre Bayen and his research group members, first and foremost Saurabh Amin, for openly deliv-ering insight into their research and the valuable experiences I could make during my stay at the Department of Civil and Environmental Engineering at the University of California at Berkeley.
I am grateful to my parents who incessantly support me wherever possible and to Brigitte Guldan for her love and the understanding she is showing me. She also took the burden of proof-reading the manuscript.
Contents
1 Introduction and preliminaries 1
1.1 Hybrid systems governed by semilinear hyperbolic PDEs 1 1.2 Motivation: Multiscale modeling for networked
dynami-cal systems . . . 7 1.3 Solution concepts: Broad solutions and hybrid time
evo-lution . . . 14 1.4 The scalar equation and piecewise continuous broad
so-lutions . . . 21 1.5 Systems of equations and a BV setting . . . 25 1.6 Outline . . . 32 2 The direct problem: BV broad solutions for switching as data 35 3 Open loop optimal switching control 65 3.1 Existence of global minimizers . . . 65 3.2 A characterization of optimal switching boundary control 69 4 Closed loop switching control by feedback 79 4.1 Global existence of solutions . . . 83 4.2 Continuous dependency on data . . . 88 5 Refined analysis in a modified BV setting 91 6 Computation of optimal switching boundary controls 99 6.1 An indirect approach using gradient information . . . 100 6.2 Direct Approach: A mixed integer formulation . . . 106 6.3 Numerical results for two model problems . . . 107
A Appendix 117 A.1 Properties of the variation and BV-functions . . . 117
Summary 127
Zusammenfassung in deutscher Sprache 131
Basic Terminology and Notation 135
List of Figures 137
1 Introduction and preliminaries
1.1 Hybrid systems governed by semilinear
hyperbolic PDEs
Despite the aphorism “natura non facit saltus”, it is frequently use-ful to work, either prescriptively or descriptively, with simplified mod-els which involve instantaneous switching between different modes of evolution. The constitutional idea behind this concept is illustrated in Figure 1.1 on the following page. Roughly speaking, asolution of such a system is anticipated with a sequence
(q0,δ0,x0)→(q1,δ1,x1)→(q2,δ2,x2)→ · · · (1.1.1) where each (qk,δk,xk) consists of a mode qk in a finite set of all
pos-sible modesQ, a nonnegative time δk representing the duration of the
system in that modeqk and a solutionxk defined on a time interval of
lengthδkwhile the modeqkis fixed. So we necessarily have a sequence (q0,τ0)→(q1,τ1)→(q2,τ2)→ · · · (1.1.2)
with switching times τk = Pkκ=1δκ where the mode discontinuously
changes from qk to qk+1 defining a switching signal. This switching
signal may be given, subject to optimization (minimizing some perfor-mance cost depending on the solution) or may even be constructed dur-ing the system’s evolution by given rules, typically in form of a suitable partition of the state space/output space, each corresponding to a mode q∈Qand triggering a mode switch when the state/output enters such a partition.
We wish to understand this hybrid dynamical systems simultane-ously as a design paradigm and as approximating multiscale problems in which the implementation of switching is merely on a faster scale
Modeq: Fq(x,xt, . . .) =0 y(t) =Gq(x, . . .) Modeq’: Fq′(x,xt, . . .) =0 y(t) =Gq′(x, . . .) qyq′ q′ yq ·yq qy· q′y· ·yq′
Figure 1.1: The concept of a hybrid dynamical system. The sys-tem evolves in a mode q according to the differential equation Fq(x,xt, . . .) = 0 with an output y(t) = Gq(x, . . .) and the
expec-tation of modal transitions from a modeq to q′ at certain time in-stances, where the system then continuous evolving according to Fq′(x,xt, . . .) = 0 with an outputy(t) = Gq
′
(x, . . .) and, proceeding
this way, similar transitions to and from other modes, not shown in the figure, are possible. Such mode transitionsq y q′ may for in-stance occur at given switching times as part of the system’s data, may be controlled and subject to optimization in order to minimize a given performance cost or may be triggered from switching rules depending ony(t)and being associated with each mode.
than what is being considered otherwise. Such interactions among com-ponents possibly operating at distinct time scales is a challenging and important area of research and — though having great practical conse-quences — is not yet understood in its full complexity. One scenario in this context is a continuous time dynamical process on a slow scale coupled with a (possibly observation based) controller acting on a much faster time-scale which we will then be approximating as instantaneous. The effect of control decisions on the fast scale then largely shows up as switching. On alternative modeling grounds, either total discretization or relaxation is often used, where one then ignores one of the difficulties
1.1 Hybrid systems governed by semilinear hyperbolic PDEs
and solutions easily become meaningless.
On the hybrid level, this kind of systems are already an intensively studied area in case the continuous dynamics in each mode are gov-erned by ordinary differential equations, although much of the under-lying theory still remains open. For an introduction to the topic of hybrid dynamical systems in the context of ODEs we refer to [60] or the more recent tutorial [23]. As an example for the large quantity of pub-lications in this multifaceted field we mention the proceedings of the workshop seriesInternational Conference on Hybrid Systems: Computation and Control[16, 9, 30, 45, 2, 41, 58, 14, 39, 59, 29].
With very few exceptions, noting [49], [4] and [26], similar systems involving partial differential equations have seldom been considered in this context.
This thesis is primarily concerned with hybrid dynamical systems when each mode is governed by a n-dimensional system of semilin-ear hyperbolic partial differential equations of one space variable s, in matrix form given by
∂
∂tx(t,s) +A
q(t,s) ∂
∂sx(t,s) =f
q(t,s,x(t,s)), (1.1.3)
for a matrixAqand a (possibly non-linear) functionfq, with initial data
prescribed along an interval of the forma6s6band linear reflecting boundary conditions
CqL(t)x|s=a=uqL(t), CqR(t)x|s=b =uqR(t) (1.1.4)
at the left and right boundary of the strip [a,b]×{t> 0} with bound-ary data CqL, CqR, uqL and uqR, respectively, and with the expectation of switching the modeqas timetevolves. As an outputy(·)of the system, we consider
y(t) =CSx|s=b (1.1.5)
at the output ends=b.
This setting is largely motivated by the urge for mathematical mod-els of networked dynamical systems involving the interaction of com-ponents that operate at distinct time scales. In particular we aim at applications from civil engineering, where pumps, valves and other actuators in networked dynamical systems interact. Such a modeling approach will be discussed in more detail in Section 1.2.
In the following, we will study three specific problems for a hybrid dynamical system with modes governed by (1.1.3)–(1.1.5):
1. The direct problem, i. e., well-posedness of the system on an arbi-trary finite time horizon[0,T]when a switching signal is given by specifying a finite sequence of switching times in
{(τ0, . . . ,τK)∈RK+1:0=τ06τ1 6· · ·6τK=T} (1.1.6)
with an assignment of a modeqk ∈Qon each interval[τk−1,τk].
2. Open loop optimal switching, i. e., existence and characterization of switching signals as above when these are subject to optimiza-tion in order to minimize a given cost of the form
J=J(x,(τ1,q1yq2),(τ2,q2y q3), . . .), (1.1.7) involving the solutionx(·,·)of the system.
3. Switching by feedback, i. e., well-posedness of the dynamical sys-tem with modes governed by (1.1.3)–(1.1.5) and switching is ac-complished by rules of the form
If one is in the modeqat timet, then:
switchingqyq′is permitted (only) ify(t)∈C(qyq′), staying in mode qis permitted (only) ify(t)∈A(q).
(1.1.8) where, for each q∈ Q, the setsA(q), {C(q y q′) :q′ , q} cover the space in which the outputy(·)takes its values.
These specific problems are motivated by the concept of supervisory control. Switching among candidate controllers orchestrated by a high-level decision maker is called asupervisor. When the controller selection is performed in a discrete fashion, one is no longer forced to construct continuously parameterized families of controllers (which may be a dif-ficult task, especially when using advanced controllers) and performing optimal parameter estimation. The switching approach rather allows for handling process models that are nonlinearly parameterized over nonconvex sets and avoids potential loss of stabilizability of the esti-mated model. These are well-known difficulties in continuous adaptive
1.1 Hybrid systems governed by semilinear hyperbolic PDEs
control. Moreover, supervisory control by switching provides greater flexibility in applications, where one often wishes to utilize existing control structures suitably orchestrated. For example, the operation of a compressor in a gas network is carried out by a continuous con-troller specifically designed to run the compressor with a compromise of maximum performance and longevity whenever it is switched ON. The supervisory control task consists of switching the compressor ON and OFF. Several other examples illustrating the advantages of switch-ing supervisory control in the context of ODEs can be found in [36] and the references therein. Examples for distributed parameter systems eas-ily come into mind. For instance, open loop optimal control as well as feedback control in terms of switching rules play an important role in so called supervisory control and data acquisition systems(short: SCADA systems) with large modeling uncertainties. Typically such SCADA sys-tems involve centralized devices which monitor and control large plants spread out over areas that range up to continents, for example in the case of pipelines, and most control actions are performed automatically based on sensor measurements, see for instance [11].
Clearly, the above problems that we wish to study are non-standard in view of classical mathematical system and control theory, even in an ODE setting. Nevertheless we anticipate that they present interesting theoretical challenges as well as being important for many real world problems. Because of their standard nature, they also bring in non-standard difficulties not appearing — for instance — in classical control theory. First and foremost, for control in a mixed discrete and continu-ous fashion, we will see that it is important to consider the possibility of this leading to the Zeno phenomenon1 which is associated with the accumulation points of discrete events or switching times. This phe-nomenon is best illustrated by an example.
Example 1.1.1: (Thomson’s Lamp [57])
“There are certain reading lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off. So if the lamp was originally off, and you
1The name refers to the paradox in “Achilles and the Tortoise” of the Greek philosopher
pressed the button an odd number of times, the lamp is on, and if you pressed the button an even number of times the lamp is off. Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half minute, and so on, according to Russell’s recipe. After I have completed the whole infinite sequence of jabs, i.e. at the end of the two minutes, is the lamp on or off?”
One can now argue that it is impossible to answer this question: It can-not be on, because one did can-not ever turn it on without at once turning it off. On the other hand, it cannot be off, because if one did in the first place turn it on, and thereafter one never turned it off without at once turning it on. But the lamp must be either on or off. This is a
contradiction.
The Zeno phenomenon is one of the main technical difficulties in obtaining global existence results, even for hybrid dynamical systems governed by ODEs. In the context of switching control, a practical so-lution to avoid the Zeno phenomenon is to implement adwell-time, i. e., a mechanism that forbids any further switchings for a fixed amount of time after a switch occurred. The drawback of a dwell-time is that, in case of open loop optimal switching, there are no guarantees that the performance of the system in the chosen mode will not deteriorate to an unacceptable level before the next switch is permitted. Similarly, for the case of switching feedback control, it may become impossible for the controller to react to system failures during that dwell time inter-val. Therefore, we will not consider dwell-time and must, for a sound theory, pay attention to the avoidance of the Zeno phenomenon.
A significant difference to the theory of hybrid dynamical systems governed by ODEs resides in the fact that we must, in addition to the evolution in time, be strongly concerned with the regularity of solutions in the coupled space variable. Moreover, due to the distributed nature of the dynamics, we also see the necessity of including an output map in the analysis which has partial state information only. In an ODE set-ting, typically, full state feedback is assumed. Therefore, an appropriate theory beyond the ODE perspective paying particular attention to these aforementioned aspects is needed.
1.2 Motivation: Multiscale modeling for networked dynamical systems L L R L L R L R R L L R R
Figure 1.2: Example of a graph modeling a networked dynamical sys-tem. The direction of the arcs coincide with the parameterization along the edge, but not necessarily with the direction of flow along the edge. The grey vertices are sensor nodes, the diamond shaped vertices represent nodes with actuators.
1.2 Motivation: Multiscale modeling for networked
dynamical systems
Consider a network of pipes (or wires, roads, strings, beams, etc.) rep-resented by a graph G = (E,V) with edgesE = {ej}j=1,...,m, each edge
ej corresponding to a pipe parameterized by an interval[a(j),b(j)], and
with vertices V ={vi}, i=1, . . . ,nwhere pipes may be interconnected.
Assume that the network is consistent in the sense that the graph is bipartite, i. e., 2-colorable for instance with labelsleftandright, directed (coinciding with the direction from leftto right and the positive direc-tion of the parameterizadirec-tion of each edge), and weakly connected (i. e., replacing all of its directed edges with undirected ones, then for each vertex there exist a path to each other vertex). Note that these assump-tion are not restrictive for applicaassump-tions such as gas, sewer or traffic net-works, because one may, for instance, introduce auxiliary vertices to make a graph bipartite while modeling the same physical system.
Now suppose that the networked system under consideration in-volves actuators and sensors placed at selected vertices. This is ubiqui-tous for example in SCADA systems mentioned above.
Assuming that the sensors and actuators are non-collocated in such a network, this implies that the set of vertices consists of three disjoint subsetsV=VS∪VA∪VN, where
• VS contains all vertices with sensors,
• VAcontains all vertices with actuators,
• VNcontains all vertices with neither sensors nor actuators.
To each edgeej,j=1, . . . ,mwe associate a dynamical system
∂tx(j)+A˜(j)∂sx(j)=f˜(j)(x) (1.2.1)
forx(t,s) = (x1(t,s),x2(t,s), . . . ,xN(t,s))∈RN in the regiona(j) < s <
b(j), t >0. The choice of this particular dynamic is strongly motivated by applications, noting that such systems model a variety of complex physical systems in networks ranging from the vibrating string or vi-brating beam [50] over conterflow processes in chemical engineering [51] up to dynamical flows in communication or logistic areas [33]. Fur-ther, (1.2.1) can be obtained by appropriate linearization of non-linear balance equations around (steady-state) solutions modeling gas flow in pipelines by means of the isothermal Euler gas equations [8], water flow in open channels by means of the St. Venant equations [35] or traffic flow on highways by means of the Lighthill-Whitham-Richards model [21].
Common scaling
It is useful to parameterize all edges ej, j = 1, . . . ,Nwith a single
pa-rameters over a common interval [a,b]. For the j-th edge, this entails a parameter change s 7→ b(j)−a(j)
b−a s and the dynamical system (1.2.2)
corresponding to that edge becomes
∂tx(j)+A(j)∂sx(j)=f(j)(x), s∈(a,b) (1.2.2) with A(j)= b (j)−a(j) b−a A˜ (j) , f(j)(t,s,x(j)) =f˜(t,b (j)−a(j) b−a s,x (j)) (1.2.3)
1.2 Motivation: Multiscale modeling for networked dynamical systems
Nodal conditions
To each vertex vi, i = 1, . . . ,n with connected incoming pipes δ−vi ⊂ E
and outgoing pipes δ+vi ⊂ E, we associate nodal conditions. Typically, these consist of conditions for
• continuity of a quantityxκ at a node, given as
x(κk)(t,b(k)) =x(k ′) κ (t,a(k ′) )for allk∈δ−vi, k′∈δ+vi (1.2.4) for someκ∈{1, . . . ,N}.
• a balance equation for a quantityxκ′ at a node, given as
X k∈δ−vi ωikx(κk′)(t,b(k)) − X k′∈δ+ vi ωik′x(k ′) κ′ (t,a(k ′) ) =ϕ(κi′)(t) (1.2.5)
for someκ′ ∈{1, . . . ,N}with appropriate weightsωik∈[0, 1]such
that Pk∈δ−
viωik =1 and
P
k′∈δ+
viωik′ =1 and prescribed values
ϕ(i)= (ϕ(i)
κ1 , . . . ,ϕ
(i)
κ
n(i))modeling sources/demands.
Using that the equations (1.2.4) and (1.2.5) are linear in xκ and xκ′,
respectively, any combination of these with κ , κ′ imply two linear equation systems for each edgeej
C(Lj)x(k)(t,a(k)) =u(Lj), C(Rj)x(k)(t,b(k)) =u(Rj) (1.2.6) with coefficient matrices C(L/Rj) involving the respectiveωik and
right-hand-sideu(L/Rj) involving the respectiveϕ(i).
Sensors
Physics commonly suggests that the solution of distributed parameter systems can only be observed partially, e. g., temperature can often only be measured at (parts of) the boundary of a physical system. Here, we take account of that by assuming that some components of the solution can be observed at certain nodes in the graph, i. e., at sensor nodes. It is not restrictive to assume that all sensor nodes are labeled with ’right’ by
introducing auxiliary nodes if needed. So for each edgee(k), we adjoin
an output equation of the form y(k)(t) =X
κ
σ(κk)x
(k)
κ (t,b) (1.2.7)
with the convention that non-zero outputs are only obtained from those edges connected to a sensor nodevi∈VS, i. e.,
σ(κk)∈R ifk∈δ−vi, vi∈ W S, σ(κk)=0 else. (1.2.8) A typical example would be the projection of the solution at the end-point of a pipe onto its component associated with pressure in corre-spondence with a physical measurement.
With coefficient matricesC(Sk) = diag(σ(κk)), we may write (1.2.7) as
the linear equation system
y(k)(t) =CS(k)x(k)(t,b). (1.2.9) Actuators
For eachviin the subset of vertices with actuatorsVA, we assume that
ωik=ωqik, andϕ
(i) =ϕ(i),q (1.2.10)
with q∈ Q(i) for some finite setQ(i). We will takeq as an additional ‘state’ of the networked dynamical system, with the physical interpre-tation that q indicates the mode of an inlet, outlet, valve, pump, etc., e. g.,Q(i)≃{ON, OFF}.
Example 1.2.1: Suppose that at some vertexvi,i∈1, . . . ,n, a
compres-sor dynamically increases the pressurepi(corresponding to some
quan-tityxκ′forκ′∈{1, . . . ,N}) by a gain ofβi, where the gainβiis governed
by an ODE
d
dtβi(t) =πi(βi(t),pi(t)), (1.2.11) coupled with the dynamics on the network by setting
ϕ(κi′)(t) =
βi(t) if ON
1.2 Motivation: Multiscale modeling for networked dynamical systems
in (1.2.5) and which may undergo a gain reset with every modal switch ON yOFF or OFFyON at switching timeτk, such as
β(τk) =0. (1.2.13)
Note that additional nodal dynamics as in Example 1.2.1 requires a careful analysis of each respective compressor model (1.2.11), (1.2.13). But at this point, we do not want to focus on developing a realistic (descriptive) model of a compressor rather than investigating the theo-retical implications of a modeling alike involving logical decisions such as ON and OFF at the nodes. In particular we wish to study funda-mental wellposedness properties of aforementioned abstract systems in (optimal) open loop or closed loop operation. So we continue writing (1.2.12) asϕ(κi′)=ϕ
(i),q κ′ with
ϕ(κi′),ON(t) =βi(t), ϕ(κi′),OFF(t) =0, (1.2.14)
effectively assuming ϕ(κi′),ON/OFF(t) as given. As with the sensors, it
is not restrictive to assume that the boundary actuators are located at nodes labeled withleftby introducing auxiliary nodes if needed, so the left boundary conditions (1.2.6) for each edgeejbecomes
C(Lj),qx(k)(t,a(k)) =u(Lj),q. (1.2.15) We will be allowing that, along with the nodal conditions, the con-vection and reaction term may depend on a discrete parameterq, i. e.,
A(t,s) =Aq(t,s)andf(t,s,x) =fq(t,s,x) (1.2.16)
in (1.2.2). This permits the modeling of numerous complex physical systems rarely considered in the literature so far.
Examples 1.2.2:
• Consider a family of traffic flow models on a highway associated with a mode q ∈ Q = {1, 2, 3, 4} each corresponding to a pre-scribed recommended speed of 60, 80, 100 or 120 km/h on dy-namical speed limit signs flected by a different advection function Aq. The control task consists of prescribing the optimal mode in order to avoid congestion. For an application in the context of civil engineering, see [49].
• In chemical engineering, reactive transport may involve materi-als that instantaneously switch between dormant and active, e. g., depending on the availability of another material concentration. Obviously, this can be modeled by different reaction terms fq.
For an application in the context of bio-remediation, see [54].
If the sign of the eigenvalues in Aq depend onq, it might also be
nec-essary for the well-posedness of the boundary conditions to include a dependency onC(Rj)andu(Rj) onq, so the right boundary conditions in general become
C(Rj),qx(k)(t,b(k)) =u(Rj),q. (1.2.17) It is clear that the implementation of switching the state of any such actuators, e. g., opening a valve or inlet, turning on a pump, compressor, etc., will actually consume some time. However, in many applications, the time scale in which these control elements operate is significantly finer than the time scale used for modeling the transportation process within the network. Therefore we consider these timescales distinct and
approximatethe dynamics on the fine scale as effectively instantanous. It is now precisely the interaction of the discrete and continuous dynamics resulting from this multi-scale model that we wish to study.
Reformulation to a single system in matrix form
The systems (1.2.2) can be arranged to a single system in matrix form ∂tx+Aq∂sx=fq(x) (1.2.18)
inx=x(1) x(2) · · · x(n)⊤with the following block structures
Aq= ˜ A(1),q 0 ˜ A(2),q . .. 0 A˜(n),q , fq(t,s,x) = f(1)(t,s,x(1),q) f(2)(t,s,x(2),q) .. . f(n)(t,s,x(n),q) .
The boundary conditions in the variablexbecome CqLx=uqL, CqRx=uqR
1.2 Motivation: Multiscale modeling for networked dynamical systems with CqL = C(L1),q 0 C(L2),q . .. 0 C(Ln),q , CqR = C(R1),q 0 C(R2),q . .. 0 C(Rn),q , and dqL = d(L1),q d(L2),q .. . d(Ln),q , dqR= d(R1),q d(R2),q .. . d(Rn),q .
The sensor output in the variable xbecomes
y(t) =CSx(t,b) with CS = C(S1) 0 C(S2) . .. 0 C(Sn) .
Without loss of generality we may take the discrete parameterq as global for the graph (rather than individualq(i)for each node and edge) by introducing sufficiently many auxiliary modes inQ.
Thus we have derived a quite general hybrid model in the form of (1.1.3)–(1.1.5) for a broad class of networked dynamical system with multi scale character. Moreover, in addition to the applications already mentioned, the above model provides a systematic theoretical founda-tion for studying the coupling of different models in a single network, for example in order to increase the computational efficency for the simulation of large gas networks as proposed in [7].
1.3 Solution concepts: Broad solutions and hybrid
time evolution
Both the partial differential equation (1.1.3) and the hybrid nature of the model require a clarification of what is meant by asolutionof the system. For the PDE, we will consider generalized solutions defined in a broad sense, following the pattern in [13] to be restricted by the requirement that they and their relevant derivatives represent idealizations by limit-ing processes. The hybrid time evolution introduced below is based on similar principles.
Broad solutions for systems of hyperbolic partial differential equations
Consider a semilinear hyperbolic system of n equations in two inde-pendent variables
∂x
∂t +A(t,s) ∂x
∂s =f(t,s,x) (1.3.1) wherextakes values inRn,Ais ann×nmatrix for allt,s andftakes values in Rn for all t,s,x. Assuming that the system is strictly hyper-bolic, i. e., eachA(t,s)hasnreal distinct eigenvaluesλ1, . . . ,λn, one can
select a bases of left and right eigenvectors {l1, . . . ,ln} and {r1, . . . ,rn}
such that
liA=λili, Ari=λiri (1.3.2)
at every point(t,s). These eigenvectors can be normalized according to
|ri|≡1, lj·ri=
1 ifi=j
0 ifi,j. (1.3.3)
Observe that (1.3.2) and (1.3.3) imply ˜ x= n X i=1 (li·x˜)ri, Ax˜ = n X i=1 λi(li·x˜)ri (1.3.4) for every ˜x∈Rn.
By aclassical solutionof the hyperbolic system (1.3.1) we mean a con-tinuously differentiable functionx(·,·)which satisfies (1.3.1) for all(t,s)
1.3 Solution concepts: Broad solutions and hybrid time evolution
in a broader sense and consider weaker solutions that do not need to be continuously differentiable everywhere as commonly accepted for the theory of conservation or balance laws. We will follow classical methods of partial differential equations [13] and will interprete the semilinear hyperbolic system in the sense of broad solutions, meaning that the solution components
xi(t,s) =li·x(t,s) (1.3.5)
with a componentwise right-hand-side fi=li·f+ [ ∂ ∂tli+λi ∂ ∂sli]·x, x= n X i=1 xiri, (1.3.6)
satisfy the (non-linear) ordinary differential equation d
dtxi(t,si(t;τ,σ)) =fi(t,si(t;τ,σ),x(t,si(t;τ,σ))) (1.3.7) along the integral curves si(·;τ,σ)in the vector fieldvi = (1,λi)going
through (τ,σ) in the domain of consideration. This notion of solution is motivated by the observation that, for sufficient regular data and by using the chain rule, (1.3.7) and the definition of si(·;τ,σ) together
imply d dtxi(t,si(t;τ,σ)) = ∂ ∂txi(t,si(t;τ,σ)) + d dtsi(t;τ,σ)xi(t,si(t;τ,σ)) (1.3.8) = ∂ ∂txi(t,si(t;τ,σ)) +λi(t,s)xi(t,si(t;τ,σ)) (1.3.9) =fi(t,si(t;τ,σ),x(t,si(t;τ,σ))). (1.3.10)
Moreover, this notion of solution is known as physically meaningful and it is easy to see that any classical solution is also a solution in this broad sense. From (1.3.7), we then see for instance that discontinuities, if involved in the solution, can only propagate along the characteristic curvessi(·;τ,σ).
Though the theory developed in this thesis will strongly build on the notion of broad solutions, at this point, we note that there are various other ways to define generalized solutions for the system (1.3.1) and we list some of them without claiming completeness:
• Limit solutions, meaning that there exists a sequence of classical solutionsxνto (1.3.1) with boundary datauν satisfying
uν→uinL1loc asν→∞ (1.3.11) and the sequence of solutions satisfying
xν→xinL1loc asν→∞, (1.3.12) see, e. g., [13].
• Distributional solutions, meaning that for every test function ϕ ∈ C1c one has " x∂ϕ ∂t +A(t,s)x ∂ϕ ∂s −f(x,t,s)ϕ dx dt=0, (1.3.13) see, e. g., [37].
• Mild solutions, in the sense thatxsatisfies the integral equation x(t) =U(t,τ)x+
Zt τ
U(t,ξ)f(ξ,x(ξ))dξ, t>τ (1.3.14) where U(t,τ) is a continuous linear evolution process {U(t,τ) :
t > τ} such that for every τ ∈ R and x in the domain of the (unbounded) linear operator
A(t)x(s) := −A(t,s) ∂
∂sx(s) (1.3.15) the functionx(t) =U(t,τ)xis the determined solution of the op-erator equation
dx
dt =A(t)x, (1.3.16)
satisfyingx(τ) =x, see, e. g., [47].
• Weak solutions in the sense that the solution is piecewise smooth and satisfies I ∂Ω −ξds+Λξdt= " Ω g(t,s,ξ) + ∂Λ ∂s(t,s)ξ ds dt (1.3.17) for any domain Ω ⊂ [a,b]×{t > 0} with piecewise Lipschitz boundary ∂Ω, where Λ = diag(λ1, . . . ,λn), g = (g1, . . . ,gn) and
1.3 Solution concepts: Broad solutions and hybrid time evolution
Some of these generalized solutions — though obtained by different concepts — may coincide and it may be of interest to investigate this since some concepts have advantages over others.
Time evolution for hybrid dynamical systems
A conceptual difficulty of hybrid dynamical systems is the formal treat-ment of time evolution. As already indicated in the sketch of the so-lution concept (1.1.1), time evoso-lution is broken up into a sequence of closed intervals which may even be reduced to single points.
Therefore an approach widely considered in the literature of hybrid dynamical systems is the formal treatment called hybrid time evolution
using a set oftime events, see [60] or [53, 38, 24, 25] with similar concepts. A time event consists on anevent timet∈Rtogether with a multiplicity m(t) and can be denoted by a sequence
(t0,t1,t2, . . . ,tm(t)) (1.3.18) specifying the sequentially ordered discrete transition times at the same continuous time instantt∈R(the event time).
A time event with multiplicity equal to 1 is just given by a pair(t0,t1)
with the interpretation of denoting the time instants “just before” and “just after” the event has taken place. If the multiplicity of the time event is larger than one (a multiple time event) then there are some in-termediate time instants (all at the same event time t) ordering the se-quence of discrete transitions taking place att.
So a sequence of time events with event times (t1 < t2 < t3 < · · ·)
implies a sequence of discrete transition times orswitching times (τk)k∈N= (t11, . . . ,t m(t1) 1 ,t12, . . . , tm(t2) 2 ,t13, . . . ,t m(t3) 3 , . . .) (1.3.19) with a mode q(t) = qk (and a continuous state x(t), output y(t), etc.)
defined fortin theinterswitching intervals[τk,τk+1]for two consecutive
switching timesτk,τk+1in (1.3.19).
The embedding of this hybrid time evolution in anormal timeinterval T ⊂Rcan then realized by an augmentation
so we haveT∗ ⊂R×Z. Solutions of hybrid dynamical systems are then functions defined on this partially ordered structure T∗. A difficulty with this hybrid time evolution and augmentation for a set of solutions is that the relevantT∗will vary with the solution.
The time evolution is closely connected with the appropriate notion of a switching signal. Therefore, we include at this point the following basic definitions we will work with subsequently.
Definition 1.3.1: On a finite time horizon [0,T] for some T > 0 and a finite set of modes Q a switching signal is determined by a finite se-quence of pairs (qk,tk)Kk=1 ∈ Q×R+, K ∈ N such that
PK
k=1tk = T.
Here,qk denotes the activemode andtk the length of theinterswitching interval in time where this mode is active. This defines switching times
τk=Pkκ=1tκwhere the switching signal discontinuously changes from
qk toqk+1, in the following denoted byqkyqk+1. We denote the
col-lection of all such switching signals byS([0,T];Q).
We note the following remarks concerning the above definition.
Remarks 1.3.1:
1. Finiteness of the sequences(qk,tk)Kk=1has close ties with the Zeno
phenomenon. Similarly as with Zeno’s paradoxes, we have seen in Example 1.1.1 that an infinite series of switches in a finite pe-riod of time may result in contradiction. Therefore we exclude that possibility in the definition of a switching signal, but point out that this only avoids the Zeno phenomenon when the switch-ing signal is a-priori known. For our analysis of the problem when the switching signal as a control is given implicitly by optimality or feedback rules, we must make sure that the a-priori unknown switching signal is then in accordance with this definition. We note that there are physical systems whose modeling idealization involves switching with infinitely many switches on finite time horizons which are not covered by the Definition 1.3.1, for in-stance in modeling a buzzer. Solutions of such chattering systems can be defined in the sense of Filippov [19], replacing a differen-tial equation dtdx(t) = f(x) with a discontinuous righthand side
1.3 Solution concepts: Broad solutions and hybrid time evolution
f(·)by a differential inclusion d
dtx(t)∈F(x), (1.3.21) where F(x) is defined to be the smallest convex closed set con-taining all limit values of the function f(·) at x. The Zeno phe-nomenon then shows up as the possibility of non-unique solutions of (1.3.21), cf. paragraph 10.4 in [19]. Indeed, Filippov hypothe-sizes in Theorem 2.10.4 in [19] the non-Zenoness of the system as a sufficient condition for unique solutions, but without providing conditions how to verify this a-priori. The theory developed here provides such conditions for the system (1.1.3)–(1.1.5) in case of optimal switching and switching by feedback, but does not cover such chattering systems unless one uses some averaging (homog-enization) to redefine this behavior as a single mode.
As a possible extension of the theory developed here, we note that the notion of hybrid time evolution as well as the notion of a switching signal are able to cover some situations in which switching times accumulate by allowing m(t),K ∈ N∪{∞}. It is then natural to require that for K = m(t) = ∞, the correspond-ing sequence of modes q1,q2,q3, . . . becomes stationary and that limk→∞x(t
k) exists in an appropriate sense.
2. The definition of switching signals allows tk = 0 and,
conse-quently, a switching signal may have cascaded switches
qkyqk+1y· · ·yqk+L (short:qk yyqk+L) (1.3.22)
at the same switching time τk = τk+1 = · · · = τk+L for some
L ∈ N. The allowance of such cascaded switches coincides with a multiple time event in hybrid time evolution and has close ties with desired limit properties for sequences of switching signals as we will see in Definition 1.3.2 below. As with the hybrid time evolution, the difficulty with this retention of these 0-length in-terswitching intervals is that a switching signal as such is not a function of time with embeddings of the switching times τk in [0,T]and the modesQin [1,|Q|]as a subsets of Rrather than
Somewhat abusing notation, we will nevertheless write q(·) for switching signals (qk,tk)Kk=1 ∈ S([0,T];Q) when the
interpreta-tion ofq(·) as a sequence is clear from the context.
For the considerations of well-posedness and optimal switching in an appropriate sense for hybrid dynamical system, we also need a topol-ogy for S([0,T];Q). The multi-scale perspective of hybrid dynamical
systems suggests the following notion for the convergence of switching signals.
Definition 1.3.2: Letqν(·) =
(qk,tk)Kk=1
ν
be a sequence of switching signals in S([0,T];Q). We say that
(qk,tk)Kk=1 ν converges to a limit switching signalq∞(·) =(q k,tk)Kk=1 ∞ inS([0,T];Q), if for largeν K∞=Kν tνk →t∞ k, k=1, . . . ,K∞ qνk =q∞ k, k=1, . . . ,K∞. (1.3.23)
We do denote this convergence byqν(·)→q∞(·)inS([0,T];Q).
For a switching signalq(·) = (qk,tk)kK=1∈S([0,T];Q)one can define
projectionsπ0tol∞(R)andπ ∗tol∞(R)by π0(q(·)) = (τ1, . . . ,τK, 0, 0, 0, . . .), τk= k X κ=1 tκ (1.3.24) and π∗(q(·)) = (q1, . . . ,qK, 0, 0, 0, . . .). (1.3.25)
It is then easy to see that the topology defined implicity in Defini-tion 1.3.2 is induced by the distance defined, for two switching signals q1(·), q2(·)∈S([0,T];Q), by
dS(q1(·),q2(·)) =kπ0(q1(·)) −π0(q2(·))k∞ +
kπ∗(q1(·)) −π∗(q2(·))k∞.
(1.3.26) Moreover, we see that any sufficiently small neighborhood of anyq(·)∈ S([0,T];Q) in this topology contains only switching signals with the
same number of switching times.
For an illustration of this sense of convergence see Figure 1.3 on page 31.
1.4 The scalar equation and piecewise continuous broad solutions
1.4 The scalar equation and piecewise continuous
broad solutions
Consider the constitutional scalar equation ∂
∂tx(t,s) +λ
q(t,s) ∂
∂sx(t,s) =f
q(t,s,x(t,s)) (1.4.1)
fors∈(a,b),t >0, and the initial-boundary condition
x(0,s) =x¯(s), x(t,a) =uq(t) (1.4.2) for s ∈ (a,b) and t > 0, respectively, with given data ¯x(s) and uq(t)
prescribed on thegnomon2
G:={(t,s) :t>0, s=0}∪{(t,s) :t=0, a6s6b}. (1.4.3) As output of the scalar system above, we considery(t) =x(t,b).
For well-posedness of (1.4.1)–(1.4.2), suppose that for all q ∈ Q the given functions λq(·,·) are continuous in t and continuously differen-tiable ins and that there are bounds 0<
¯
λ <λ¯ such that ¯
λ6λq(t,s)6 ¯
λ. (Thus, by compactness, there exists a Lipschitz-constantLλs such that
|λq(·,s) −λq(·,s′)| 6Lλs|s−s′| for alls,s′ ∈ [a,b].) Moreover, assume
that for allq∈Qthe given functionsfq(·,·,·)are continuous, that there exists a constantCfsuch that|fq(·,·,x)| 6Cf(1+x)and that there exists
a Lipschitz-constantLfx such that|fq(·,·,x) −fq(·,·,x′)|6Lfx|x−x′|for
allx,x′∈R.
As motivated above, we will consider solutions of the hyperbolic equations in a broad sense. The interpretation of (1.4.1)–(1.4.2) using themethod of characteristics simplifies in the scalar case to characteristic curves being obtained as solutions of the ordinary differential equations
d
dts(t) =λ
q(t
,s(t)). (1.4.4)
2Geometrically, a gnomonis the L-shaped piece of a parallelogram remaining when a
similar parallelogram is excised from its corner. We are modifying this usage to con-sider together the bottom and left side of the infinite rectangle[0,∞)×[a,b], visualizing
flow aslefttorightalong an edge associated with the interval[a,b], so this gnomon pre-cisely contains the input data for transport along such an edge modeled by (1.4.1) when
λq(·,·)>
Setting ˆx=x(t,s(t)), (1.4.1) becomes the ODE d
dtxˆ(t) =f
q(t,s(t), ˆx(t)). (1.4.5)
The broad solutionx(·,·) of (1.4.1)–(1.4.2) is thus given by the family of ODEs (1.4.5) with initial data ¯x(s),uq(t)on the gnomonG.
The following example shows that we must accept the introduction and propagation of discontinuities in the solution of the system (1.4.1)– (1.4.2) when the modeqdiscontinuously changes.
Example 1.4.1: Consider the transport dynamics (1.4.1) taken simply as ∂
∂tx(t,s) + ∂
∂sx(t,s) =0 (1.4.6) and with constant initial data ¯x≡1. In a first modeq =1 suppose we
usex(t, 0) =u1(t)≡1, so the solution remainsx≡1. But suppose we would discontinuously switch from modeq=1 to a new modeq=2 at
a switching timet=τwith keeping the dynamics fixed but now using input datax(t, 0) =u2(t) ≡2. Then a jump discontinuity is introduced at the input boundary and propagates (along the characteristics=t−τ)
into the edge.
Moreover, both for the boundary conditions (1.1.4) arising from nodal coupling if we consider transport on a network and for our results on feedback switching control based on sensor observations at nodes, we must be strongly concerned with the regularity of boundary traces for the broad solution:
Example 1.4.2: Continuing Example 1.4.1, suppose we would observe the solution at the output node corresponding to s = 1. We would see a constant value y(t) = x(t, 1) ≡ 1 fort < τ+1, a constant value y(t) =x(t, 1)≡2 fort > τ+1 and a jump discontinuity att=τ+1.
Now consider the following non-standard space of piecewise contin-uous functions.
Definition 1.4.1: Apiecewise continuous functiong(·)on an interval[a,b]
taking values in Ris constructed by specifying finitely many partition
points
1.4 The scalar equation and piecewise continuous broad solutions
k ∈ N, and on each partition subinterval [pk−1,pk], k ∈ {1, . . . ,K}
as-signing a continuous function gk∈C([0, 1];R). For degenerate 0-length
subintervals we impose the restriction that the assigned gk(·) must be
constant. As a function on [a,b], we have g(s) =gk s−pk−1 pk−pk−1 , fors∈(pk−1,pk) (1.4.8)
andg(s) = [gk−1(1),gk(0)]ifsis one of the partition pointspk. We may
even have
g(s) = [gk−1(1),gk, . . . ,gk+l−1,gk+l(0)] (1.4.9)
at coalesced partition points pk = . . . = pk+l, noting that such a
mul-tiple assignment is not a set but preserves the sequence order. The set of all such piecewise continuous functions on[a,b]will be denoted by
Cpw([a,b]).
Furthermore, suppose we topologize the setCpw([a,b])of piecewise continuous functions by the convergence of sequences similar to those of switching signalsS([0,T];Q) in Definition 1.3.2.
Definition 1.4.2: Let {gν(·)} be a sequence in C
pw([a,b]). We say that {gν(·)}converges to some{g∞(·)}∈C pw([a,b]), if for largeν K∞=Kν pνk →p∞ k, k=1, . . . ,K∞ gνk →g∞ k uniformly inC([0, 1]), k=1, . . . ,K∞. (1.4.10)
We do denote this convergence bygν(·)→g∞(·)inC
pw([a,b]).
Now observe that for a given q(·) ∈ S([0,T];Q), the assumption on λq(·,·)andfq(·,·,·)ensure by the Picard-Lindel ¨of theorem the existence and uniqueness of solutions to the equations (1.4.4) and (1.4.5) when q is replaced by q(t). Moreover, using that these solutions depend continuously on the initial data, the resulting coordinate transformation
(t,s)↔(t,s(t))is a homeomorphism between the domain[0,T]×[a,b]
and the corresponding portion of the gnomonG. Hence, if the boundary datauq(·) is continuous for allq ∈Q, we haveuq(·) ∈Cpw([0,T])and
we obtain a unique broad solution on[0,T]withx(t,·)∈Cpw([a,b])for
that the output y(·) = x(·, 1) is in Cpw([0,T]). Finally, standard well-posedness theory for ODEs shows that the solutionx(t,·)∈Cpw([a,b])
(and the outputy(·)∈Cpw([0,T])) depend continuously (as topologized in Definition 1.3.2 and 1.4.2) on ¯x(·),uq(·)andq(·).
In addition to that we have the following compactness result.
Lemma 1.4.1. The subspace SK([0,T];Q) ⊂ S([0,T];Q) corresponding to a bound onKin Definition 1.3.1 is sequentially compact as topologized in Defi-nition 1.3.2.
Proof. Let qν(·) be a sequence in SK([0,T];Q). Then, with a bound on
the number of switching points τk, we may extract a subsequence
(re-indexed byν), such that
Kν=Kν′ =K˜ 6K (1.4.11)
for allν,ν′. With ˜Kfixed, a switching sequence is equivalent to a point in the compact set
{(τ0, . . . ,τK˜)∈R ˜ K+1 :0=τ 06τ16· · ·6τK˜ =T}×Q ˜ K. (1.4.12)
Moreover, we remark that the definition of piecewise continuous func-tions in Definition 1.4.1 and the topology in Definition 1.4.2 is here pri-marily intended for the description of the spatial regularity of broad solutions for the scalar transport equation (1.4.1) and the temporal reg-ularity of the system’s output y(·) in time. But it bears analogy to attempts in developing topological structures for hybrid ODE trajec-tories, mostly used in the context of stability analysis: Similarly as withS([0,T];Q), one can also metrizeCpw([a,b])by defining, forg(·)∈
Cpw([a,b]), the projectionπ†to l∞(R)by
π†(g(·)) = (p0, . . . ,pK, 0, 0, 0, . . .) (1.4.13)
and, forg1(·),g2(·)∈Cpw([a,b]), the distance
dCpw(g 1(·),g2(·)) =kπ †(g1(·)) −π†(g2(·))k∞ + max k>1 kg 1 k(·) −g2k(·)k∞. (1.4.14)
For pairs [q(·),g(·)] ∈ S([0,T];Q)×Cpw([0,T]) as solutions of hybrid
1.5 Systems of equations and a BV setting
by the sum of (1.3.26) and (1.4.14) is known as the Tavernini metric [56]. Morover, this metric induces the Skorohod J1 topology, defined,
for c`adl`ag functionsf1,f2∈D([0,T],R), by dS-J1(f1,f2) = inf λ∈Λmax{kf 1◦λ−f2k ∞,kf 1−ek ∞}, (1.4.15)
where Λis the group of all strictly order preserving homeomorphisms of[0,T]andeis the identity, see [25].
TheJ1-topology is coarser than the compact open topology and finer
than the Lebesque topologies. For example among real functions on the unit intervals, the characteristic function of [0,12) is δ-close to the indicator function of [0,12 +δ) in the Skorohod J1 distance, but in the
uniform metric the distance between these functions is one. On the other hand, the characteristic function of [0,δ)isδ-close to zero in any of the Lebesgue metrics, but the Skorohod J1 distance between them is
one.
We also remark that the Tavernini metric (and the Skorohod J1metric)
are known to be incomplete, see [25] (and [10]).
Nevertheless, we have developed, with these tools, a well-posedness theory for the scalar equation (1.4.1)–(1.4.2), can consider open loop optimal switching for quite general cost functions as well as closed loop switching by feedback with mild assumption on the partitionsAq, C(q y q′) of the observation space — indeed also for a scalar
net-worked system. For details, see [27]. But a generalization of the results in [27] for the constitutional scalar equation to (fully coupled) systems of equations cannot be obtained by a direct use of Banach’s fixed point theorem in an incomplete vector valued space Cpw([a,b];Rn). This is
one reason why we investigate a different approach for the matrix case as discussed in the next Section.
1.5 Systems of equations and a BV setting
Motivated by applications, we wish a theory for switching (fully cou-pled) systems of hyperbolic partial differential equations, i. e., the ma-trix case (1.1.3)–(1.1.5).
For that case, continuous dependency on input data becomes delicate for systems withn>2, even when the equations are just coupled at the
boundary as we may see from the following example.
Example 1.5.1: Consider a Y-graph with edges e1,e2 directed toward
the central node and another edgee3directed out; we take
∂ ∂tx+
∂
∂sx=0 (1.5.1)
on each edge so, using Kirchhoff’s Law for the nodal coupling in (1.2.5), we have
x3(t,s) =x1(0,s−t+1) +x2(0,s−t+1) for 0< s <1, 0< s−t+1<1.
As initial data we take ¯x1 =Hǫ, ¯x2=1−H0with
Hǫ(s) ={0 fors <1/2+ǫ; 1 fors >1/2+ǫ}
so att=1 we havex3(1,·) =Hǫ−H0+1. For smallǫ >0 we then have
x3(1,·) ≡ 1 except on the infinitesimally small interval (1/2, 1/2+ǫ)
where x3(1,·) ≡ 2. On the other hand, forǫ < 0 we have x3(1,·) ≡ 1 except on (1/2+ǫ, 1/2) where, now, x3(1,·) = 0. Taking the limit as ǫ → 0 for the initial data, without regard for sign, we obtain two limit solutions distinguished by the retention of either 2 or 0 as assigned to the degenerate interval corresponding to treating ǫ alternatively as a
positive or negative signed infinitesimal.
Thus we cannot expect unique broad solutions defined everywhere on
[a,b]in the sense of piecewise continuous functions while maintaining a classical continuous dependency property. Nevertheless, we will see that this non-uniqueness is limited to a set ofL1-measure zero on[a,b]
which suggests to consider broad solutions defined almost everywhere. While it seems possible to develop, despite all the difficulties al-ready mentioned, a complete theory based on piecewise continuous functions as in Definition 1.4.1 also for the matrix case with an appro-priate interpretation of well-posedness in the sense that the solution set of piecewise continuous broad solutions is non-empty and upper-semicontinuous in its dependence on the data (noting the results in [27] and [46]), this seems impractical in view of an eventual treatment of fully non-linear hyperbolic partial differential equations replacing (1.1.3), where then in addition to the discontinuities introduced at the
1.5 Systems of equations and a BV setting
boundary, shocks and shock-interactions generate discontinuities also in the interior of the domain.
Therefore, we will in the following investigate for a similar theory in the matrix case the usage of functions of bounded variation (short: BV functions), i. e., functions in L1((a,b);Rn) whose distributional deriva-tive is a measure. This concept was introduced by De Georgi in the 1950s, see [3] for a comprehensive survey and historical notes. In 1964, Miranda characterized BV functions as those f ∈ L1((a,b);Rn) whose
variation b _ a f(s)δs:= sup ϕ∈C1 c((a,b),Rn) kϕk∞61 Zb a f(s)· ∇ϕ(s)ds (1.5.2)
is finite. Note that, for smooth functions f∈C∞((a,b);Rn), integration
by parts yields b _ a f(s)δs= sup ϕ∈C1 c((a,b),Rn) kϕk∞61 − Zb a ∇f(s)·ϕ(s)ds = Zb a |∇f(s)|ds. (1.5.3)
We will denote the set of all functions inL1((a,b);Rn)having bounded variation (1.5.2) with BV((a,b);Rn). It is well-known, see, e. g., [3], that BV((a,b);Rn)endowed with the norm
kfkBV((a,b);Rn):= Zb a |f(s)|ds+ b _ a f(s)δs (1.5.4) is a Banach space. Since we are concerned with BV functions on inter-vals only, observe that there always exists a representative ¯fin theL1 -equivalence class off∈BV((a,b);Rn)such that the variationWb
af(s)δs
as defined above coincides with the pointwise variation in the classical Jordan-sense ˙ _b af¯(s)δs=a<s sup 1<···<sN<b N>2 N−1 X i=1 |f¯(si+1) −f¯(si)| . (1.5.5)
As usual, we say that a sequence of functions fν in BV((a,b);Rn) converges stronglyto somef∞ in BV((a,b);Rn) if
kfν−f∞k
and will denote this norm convergence byfν →f∞ in BV. We will also
use the following two weak convergence processes in BV. We say that the sequencefνconverges strictlytof∞ in BV((a,b);Rn)if
fν→f∞in L1((a,b);Rn) b _ a fν(s)δs→ b _ a f∞(s)δsinR (1.5.7)
and will denote this strict convergence by fν { f∞.3 This sense of
convergence is weaker than norm convergence (as an example consider fν(s) = arctan(νs)), but the topology induced by strict convergence is useful to prove properties in BV by smoothing arguments, cf. Theo-rem A.1.1 and TheoTheo-rem A.1.2.
We say that the sequencefνconverges weaklytof∞in BV((a,b);Rn)if
fν→f∞inL1((a,b);Rn)
Dfν⇀∗ Df∞ (1.5.8)
where Dfν ⇀∗ Df∞ denotes weak∗ convergence for the sequence of
distributional derivativesDfν associated with eachfν, i. e.,
lim ν→∞ Zb a φd Dfν= Zb a φd Df for allφ∈C0((a,b)). (1.5.9) We denote weak convergence in BV by fν ⇀f∞, noting that this sense
of convergence is even weaker than strict convergence (as an example consider fν(s) = sin(ννs)), but the topology induced by weak conver-gence is useful for its compactness properties.4
Clearly, any scalar function in Cpw([a,b]) is also a function in the
space BV((a,b);R) and we have collected in Appendix A.1 some
prop-erties of the variation and BV-functions that we make use of subse-quently.
3Strict convergence is sometimes also calledintermediate convergence, cf., e. g., [6]. 4The notion of theweak topologyandweak convergenceas used here is not to be confused
with the initial topology of BV with respect to its topological dual which is also often called “weak topology”.
1.5 Systems of equations and a BV setting
Switching Signals in theL1-equivalence class
For a switching signal in S([0,T];Q), recall that this may involve a cas-cade qk yyqk+L, for instance occurring for sometνk (or possible
sev-eral of these) converging to t∞
k = 0 as topologized in Definition 1.3.2.
Due to the assumed non-Zenoness, cascaded switches can only occur on a set ofL1-measure zero in the embedding in[0,T]and we will there-fore consider at first the simplification of working withL1-equivalence classes of switching signals then becoming functions on [0,T] taking values in Q. So in view of the graph representation illustrated in Fig-ure 1.1, we will identify with that treatment all paths [q → q′ → q′′]
and [q → qˆ′ → q′′] with q,q′, ˆq′,q′′ ∈ Q when the time the system evolves in mode q′ or ˆq′ is zero. This is of course only reasonable when going though mode q′ or ˆq′ in zero time effectively leads to the same solution. Later we will see examples where this is not the case and we will refine our analysis distinguishing such paths in Chapter 5.
The identification just discussed suggests the following Definition. Definition 1.5.1: For T > 0, the set of essential switching signals q(·) is defined as
Sess([0,T];Q) =BV([0,T];Q) ={q(·) : [0,T]→Q}∩BV([0,T];R). (1.5.10)
The next Lemma motivates the terminology ‘essential switching sig-nal’ and substantiates the relation ofq(·)∈Sess([0,T];Q)with switching signalsq(·)∈S([0,T];Q)as of Definition 1.3.1.
Lemma 1.5.1. Assume thatQis finite andT >0. Then,q(·)∈Sess([0,T],Q) if and only if there existN∈N,τi∈R,i=1, . . . ,Nwith
0=τ0< τ1<· · ·< τN< τN+1=T (1.5.11) andqk,k=0, . . . ,N, such that
q|(τk,τk+1) =qk L1-a. e. (1.5.12) fork=0, . . . ,N.
Proof. If there exist N ∈ N, τi ∈ R, i = 1, . . . ,N with 0 = τ0 < τ1 <
· · · < τN < τN+1 = T andqk, k= 0, . . . ,N, such that (1.5.12) holds for
k=0, . . . ,N, then, clearly,q: [0,T]→Qand by Lemma A.1.5,
kq(·)kBV((0,T);R)62(N+1) N X k=0 kq(·)kBV((τk,τk+1);R) =2(N+1) N X k=0 kqkkBV((τk,τk+1);R)<∞, (1.5.13)
thusq(·)∈Sess([0,T],Q)by Definition 1.5.1.
Now letq(·) ∈ Sess([0,T];Q). As noted in Lemma A.1.6, there exists
a representative ¯q(·) in theL1-equivalence class such thatWb
aq(t)δt =
˙
Wb
aq¯(t)δt with an at most countable set of discontinuities and which
we may assume to be left-continuous. Thus we have a representation ¯ q(·) = ∞ X k=0 qkχ[τk,τk+1)(t) (1.5.14)
where τ0 < τ1 < τ2. . . are the points of discontinuities and qk ∈ Q,
k=1, 2, . . .. So we have b _ a q(t)δt=_˙ b a ∞ X k=0 qkχ[τk,τk+1)(t)δt=2 ∞ X k=0 qk (1.5.15) A bound on Wb
aq(t)δt now implies that the sum in (1.5.14) is finite
which completes the proof.
Topologies for Essential Switching Signals
We will work with the following two notions of convergence for essen-tial switching signals rooted in the concepts of strict convergence and weak convergence in BV.
Definition 1.5.2: Let (qν(·))ν∈N be a sequence inSess([0,T];Q). We say
that (qν(·))ν∈N converges strictly to some limit switching signal q∞ ∈
Sess([0,T],Q) if
qν(·){q∞(·)in BV((0,T);R) (1.5.16)
and shall also writeqν(·){q∞ inS
1.5 Systems of equations and a BV setting b b b b b b b b b b q=0 q=1 0 τν 1 τν2 τν3 τν4 T b b b b b b b b b b q=0 q=1 0 τ1 τ2 τ3 τ4 T (a) b b b b b b q=0 q=1 0 τν 1 τν2 T b b b b q=0 q=1 0 τ1=τ2 T (b)
Figure 1.3: Illustration of the topologies for switching signals in the caseQ= {0, 1}. In subfigure (a), the sequenceqν (left) converges to q(right) in S([0,T];Q) as τνk → τk, k = 1, . . . , 4. The sequence also
converges inSess([0,T];Q), both strictly and weakly. In subfigure (b), qν (left) converges to q (right) in S([0,T];Q) as τνk → τk, k = 1, 2.
It also converges in Sess([0,T];Q) weakly but not strictly (because
WT
0 qν(t)δt=2 for allν, but
WT
0 q(t)δt=0).
Definition 1.5.3: Let (qν(·))
ν∈N be a sequence inSess([0,T];Q). We say
that (qν(·))
ν∈N converges weakly to some limit switching signal q∞ ∈
Sess([0,T],Q) if
qν(·)⇀q∞(·) in BV((0,T);R) (1.5.17)
and shall also writeqν(·)
⇀q∞ inSess([0,T],Q).
Note that Lemma 1.5.1 suggests the following relation:qν(·)⇀q∞in
Sess([0,T];Q)if and only if ¯qν(·) →q¯∞(·) inS([0,T];Q)for appropriate
‘representatives’ ¯qν and ¯q∞ associated with qν(·) and q∞(·),
respec-tively.
For an illustration of the topologies for switching signals in the case Q={0, 1}see Figure 1.3.
Remark 1.5.1: We have defined the set of switching signals and conver-gence of sequences for finite time intervals [0,T]only. When we con-sider infinite horizon problems, we set
