Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat.

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1 Hybrid Dynmics Comprising Modes Governed by Prtil Differentil Equtions: Modeling, Anlysis nd Control for Semiliner Hyperbolic Systems in One Spce Dimension Der Nturwissenschftlichen Fkultät der Friedrich-Alexnder-Universität Erlngen-Nürnberg zur Erlngung des Doktorgrdes Dr. rer. nt. vorgelegt von Flk Michel Hnte us Dtteln

2 Als Disserttion genehmigt von der Nturwissenschftlichen Fkultät der Friedrich-Alexnder-Universität Erlngen-Nürnberg Tg der mündlichen Prüfung: 14. Juli 2010 Vorsitzender der Promotionskommission: Erstberichtersttter/in: Zweitberichtersttter/in: Prof. Dr. Eberhrd Bänsch Prof. Dr. Günter Leugering Prof. Dr. Hns Josef Pesch

3 Abstrct Hybrid dynmicl systems re considered s design prdigm or multiscle model for networked trnsport problems. Evolution in time is governed by switching mong fmily of solutions to vector-vlued, semi-liner hyperbolic prtil differentil equtions on bounded intervl in spce with reflecting boundry conditions. Existence of generlized solutions nd pproprite continuous dependency properties re studied on the discrete-continuous level. In prticulr, it is shown how the Zeno phenomenon cn be voided, despite the propgtion of discontinuities long chrcteristics resulting from instntnously chnging boundry conditions nd their interction with switching rules evluting pointwise boundry obervtion. Moreover, for given cost function including switching costs, the problem of optiml switching is studied. For sclr equtions with boundry switching control, optimlity is chrcterized in terms of first order necessry condition bsed on formul for switching time sensitivity. The results re vlidted numericlly on exmples. Kurzzusmmenfssung in deutscher Sprche Es werden diskret-kontinuierliche dynmische Systeme ls Konstruktions-Prdigm oder Multisklen-Modell für vernetzte Trnsportprozesse betrchtet. Die zeitliche Entwicklung wird bestimmt durch Schlten zwischen Lösungen von vektorwertigen, semilineren hyperbolischen prtiellen Differentilgleichungen uf einem beschränkten Ortsintervll mit reflektierenden Rndbedingungen. Existenz von verllgemeinerten Lösungen und geeignete stetige Abhängigkeitseigenschften werden uf dem diskretkontinuierlichen Niveu untersucht. Insbesondere wird gezeigt, wie ds Zeno sche Phänomen vermieden werden knn, trotz der Fortpflnzung von Unstetigkeiten entlng Chrkteristiken, die uf brupten Wechsel der vorgeschriebenen Rndbedingungen zurückgehen, sowie deren Interktion mit Schltregeln wenn sie punktweise Rndbeobchtung uswerten. Drüberhinus werden für gewisse Kostenfunktionen, die Schltkosten beinhlten, ds Problem des optimlen Schltens untersucht. Für sklre Gleichungen mit Rnd-Schlt-Steuerung wird Optimlität im Sinne notwendiger Bedingungen chrkterisiert, die uf einer Formel für Schltzeit-Sensitivitäten bsiert. Die Ergebnisse werden numerisch n Beispielen vlidiert. i

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5 Prefce Tody s technologicl problems more thn ever demnd the study of dynmicl systems beyond completely continuous or completely discrete description. As well known exmple, model of temperture control system consisting of heter nd thermostt would certinly include s vrible the room temperture nd the operting mode of the heter (ON or OFF) where it is common prctice to tke the former s rel-vlued nd the ltter s discrete. For the control system to be effective there needs to be coupling between these vribles, e. g., the operting mode to be switched to ON when the room temperture flls below certin threshold. A fully discretized model my serve for simultion, but without mthemticl description of the system on the hybrid level, for instnce, the convergence properties of ny lgorithm bsed only on the discrete model is often meningless. On the other hnd, relxtion of the discrete vribles esily led to physiclly infesible solutions if mthemtics would then, for instnce, suggest to operte the heter hlf ON (or hlf OFF) for mximl performnce t miniml cost. The ctivities following systemtic wy of deling with dynmicl systems whose evolution depends on coupling between vribles tking vlues in continuum nd vribles tht tke vlues in finite or countble set let to the field of hybrid dynmicl systems, though generl theory is not yet identifible. One reson is tht vrious scientific communities minly in the field of computer science nd control engineering contribute ech with different gols nd with there own pproches to this re. Hybrid dynmicl systems therefore becme very interdisciplinry field. As mtter of fct, distributed prmeter systems s governed by prtil differentil equtions re seldom considered in this context, lthough mny exmples esily come into mind. Observe tht resonble model even for the temperture control system would involve the het eqution. More demnding nd technologiclly chllenging pplictions re for instnce the cost-efficient opertion of pumps in gs iii

6 pipelines or dynmiclly trvel time optimizing trffic lights in rod networks. Acknowledgments re due to Prof. Dr. Günter Leugering who envisions nd suggests the tretment of such problems by the concept of hybrid dynmicl systems. Since this opens up new brnch with strong emphsize on mthemtics in this very interdisciplinry field, it is cler tht we cn t this stge only scrtch t the surfce of this big subject, restricting our nlysis t first for instnce to systems of semiliner hyperbolic equtions. But even the primry theory developed here brings in new spects for both, hybrid dynmicl systems nd prtil differentil equtions. I m indebted to mny people who influenced my thinking nd offered vluble suggestions nd dvice for this work. Especilly I like to thnk my supervisor Prof. Dr. Günter Leugering for his encourgement nd support. He lso suggested to me the cndidcy for the Interntionl Doctorte Progrm Identifiction, Optimiztion nd Control with Applictions in Modern Technologies where this work elborted. The reserch position nd finncil support grnted by the Elite-Network of Bvri nd the mentoring from my dvisors Prof. Dr. Hns Josef Pesch nd Prof. Dr. Lrs Grüne in the progrm is grtefully cknowledged. Mny results were obtined in fruitful discussion nd coorportion with Prof. Dr. Thoms Seidmn, my sincere thnks to him for this exceptionl collbortion. I like to thnk my collegues t the University of Erlngen-Nürnberg nd the members of the doctorte progrm for creting very stimulting environment, in prticulr to PD Dr. Mrtin Gugt, Prof. Dr. Michel Stingl nd Christoph Schumcher for mny helpful discussions. Also I like to thnk Prof. Dr. Alexndre Byen nd his reserch group members, first nd foremost Surbh Amin, for openly delivering insight into their reserch nd the vluble experiences I could mke during my sty t the Deprtment of Civil nd Environmentl Engineering t the University of Cliforni t Berkeley. I m grteful to my prents who incessntly support me wherever possible nd to Brigitte Guldn for her love nd the understnding she is showing me. She lso took the burden of proof-reding the mnuscript. Erlngen, December 1, 2009 Flk Hnte iv

7 Contents 1 Introduction nd preliminries Hybrid systems governed by semiliner hyperbolic PDEs Motivtion: Multiscle modeling for networked dynmicl systems Solution concepts: Brod solutions nd hybrid time evolution The sclr eqution nd piecewise continuous brod solutions Systems of equtions nd BV setting Outline The direct problem: BV brod solutions for switching s dt 35 3 Open loop optiml switching control Existence of globl minimizers A chrcteriztion of optiml switching boundry control 69 4 Closed loop switching control by feedbck Globl existence of solutions Continuous dependency on dt Refined nlysis in modified BV setting 91 6 Computtion of optiml switching boundry controls An indirect pproch using grdient informtion Direct Approch: A mixed integer formultion Numericl results for two model problems Conclusions 113 v

8 Contents A Appendix 117 A.1 Properties of the vrition nd BV-functions Summry 127 Zusmmenfssung in deutscher Sprche 131 Bsic Terminology nd Nottion 135 List of Figures 137 Bibliogrphy 139 vi

9 1 Introduction nd preliminries 1.1 Hybrid systems governed by semiliner hyperbolic PDEs Despite the phorism ntur non fcit sltus, it is frequently useful to work, either prescriptively or descriptively, with simplified models which involve instntneous switching between different modes of evolution. The constitutionl ide behind this concept is illustrted in Figure 1.1 on the following pge. Roughly speking, solution of such system is nticipted with sequence (q 0, δ 0, x 0 ) (q 1, δ 1, x 1 ) (q 2, δ 2, x 2 ) (1.1.1) where ech (q k, δ k, x k ) consists of mode q k in finite set of ll possible modes Q, nonnegtive time δ k representing the durtion of the system in tht mode q k nd solution x k defined on time intervl of length δ k whilethemode q k is fixed. Sowenecessrilyhvesequence (q 0, τ 0 ) (q 1, τ 1 ) (q 2, τ 2 ) (1.1.2) with switching times τ k = k κ=1 δ κ where the mode discontinuously chnges from q k to q k+1 defining switching signl. This switching signl my be given, subject to optimiztion (minimizing some performnce cost depending on the solution) or my even be constructed during the system s evolution by given rules, typiclly in form of suitble prtition of the stte spce/output spce, ech corresponding to mode q Q nd triggering mode switch when the stte/outputenterssuch prtition. We wish to understnd this hybrid dynmicl systems simultneously s design prdigm nd s pproximting multiscle problems in which the implementtion of switching is merely on fster scle 1

10 1 Introduction nd preliminries q q q q Mode q: F q (x, x t,...) = 0 y(t) = G q (x,...) Modeq : F q (x, x t,...) = 0 y(t) = G q (x,...) q q q q Figure 1.1: The concept of hybrid dynmicl system. The system evolves in mode q ccording to the differentil eqution F q (x, x t,...) = 0 with n output y(t) = G q (x,...) nd the expecttion of modl trnsitions from mode q to q t certin time instnces, where the system then continuous evolving ccording to F q (x, x t,...) = 0 with n output y(t) = G q (x,...) nd, proceeding this wy, similr trnsitions to nd from other modes, not shown in the figure, re possible. Such mode trnsitions q q my for instnce occur t given switching times s prt of the system s dt, my be controlled nd subject to optimiztion in order to minimize given performnce cost or my be triggered from switching rules depending on y(t) nd being ssocited with ech mode. thn wht is being considered otherwise. Such interctions mong components possibly operting t distinct time scles is chllenging nd importnt re of reserch nd though hving gret prcticl consequences is not yet understood in its full complexity. One scenrio in this context is continuous time dynmicl process on slow scle coupled with (possibly observtion bsed) controller cting on much fster time-scle which we will then be pproximting s instntneous. Theeffect of controldecisions on thefst scle then lrgely showsup s switching. On lterntive modeling grounds, either totl discretiztion or relxtion is often used, where one then ignores one of the difficulties 2

11 1.1 Hybrid systems governed by semiliner hyperbolic PDEs nd solutions esily become meningless. On the hybrid level, this kind of systems re lredy n intensively studied re in cse the continuous dynmics in ech mode re governed by ordinry differentil equtions, lthough much of the underlying theory still remins open. For n introduction to the topic of hybrid dynmicl systemsin thecontextof ODEs we referto[60] or the more recent tutoril [23]. As n exmple for the lrge quntity of publictions in this multifceted field we mention the proceedings of the workshop series Interntionl Conference on Hybrid Systems: Computtion nd Control [16, 9, 30, 45, 2, 41, 58, 14, 39, 59, 29]. With very few exceptions, noting [49], [4] nd [26], similr systems involving prtil differentil equtions hve seldom been considered in this context. This thesis is primrily concerned with hybrid dynmicl systems when ech mode is governed by n-dimensionl system of semiliner hyperbolic prtil differentil equtions of one spce vrible s, in mtrix form given by t x(t, s) + Aq (t, s) s x(t, s) = fq (t, s, x(t, s)), (1.1.3) for mtrix A q nd(possiblynon-liner) function f q, with initil dt prescribed long n intervl of the form s b nd liner reflecting boundry conditions C q L (t)x s= = u q L (t), Cq R (t)x s=b = u q R (t) (1.1.4) t the left nd right boundry of the strip [, b] {t 0} with boundry dt C q L, Cq R, uq L nd uq R, respectively, nd with the expecttion of switchingthemode qstime tevolves. Asnoutput y( )ofthesystem, we consider y(t) = C S x s=b (1.1.5) t the output end s = b. This setting is lrgely motivted by the urge for mthemticl models of networked dynmicl systems involving the interction of components tht operte t distinct time scles. In prticulr we im t pplictions from civil engineering, where pumps, vlves nd other ctutors in networked dynmicl systems interct. Such modeling pproch will be discussed in more detil in Section

12 1 Introduction nd preliminries In the following, we will study three specific problems for hybrid dynmicl system with modes governed by (1.1.3) (1.1.5): 1. The direct problem, i. e., well-posedness of the system on n rbitrry finite time horizon [0, T] when switching signl is given by specifying finite sequence of switching times in {(τ 0,...,τ K ) R K+1 : 0 = τ 0 τ 1 τ K = T} (1.1.6) with n ssignment of mode q k Q on ech intervl [τ k 1, τ k ]. 2. Open loop optiml switching, i. e., existence nd chrcteriztion of switching signls s bove when these re subject to optimiztion in order to minimize given cost of the form J = J (x, (τ 1, q 1 q 2 ), (τ 2, q 2 q 3 ),...), (1.1.7) involving the solution x(, ) of the system. 3. Switching by feedbck, i. e., well-posedness of the dynmicl system with modes governed by (1.1.3) (1.1.5) nd switching is ccomplished by rules of the form If one is in the mode q t time t, then: switching q q is permitted (only) if y(t) C(q q ), stying in mode q is permitted (only) if y(t) A(q). (1.1.8) where, for ech q Q, the sets A(q), {C(q q ) : q q} cover the spce in which the output y( ) tkes its vlues. These specific problems re motivted by the concept of supervisory control. Switching mong cndidte controllers orchestrted by highlevel decision mker is clled supervisor. When the controller selection is performed in discrete fshion, one is no longer forced to construct continuously prmeterized fmilies of controllers (which my be difficult tsk, especilly when using dvnced controllers) nd performing optiml prmeter estimtion. The switching pproch rther llows for hndling process models tht re nonlinerly prmeterized over nonconvex sets nd voids potentil loss of stbilizbility of the estimted model. These re well-known difficulties in continuous dptive 4

13 1.1 Hybrid systems governed by semiliner hyperbolic PDEs control. Moreover, supervisory control by switching provides greter flexibility in pplictions, where one often wishes to utilize existing control structures suitbly orchestrted. For exmple, the opertion of compressor in gs network is crried out by continuous controller specificlly designed to run the compressor with compromise of mximum performnce nd longevity whenever it is switched ON. The supervisory control tsk consists of switching the compressor ON nd OFF. Severl other exmples illustrting the dvntges of switching supervisorycontrol in the contextof ODEs cn be found in [36] nd the references therein. Exmples for distributed prmeter systems esily come into mind. For instnce, open loop optiml control s well s feedbck control in terms of switching rules ply n importnt role in so clled supervisory control nd dt cquisition systems (short: SCADA systems) with lrge modeling uncertinties. Typiclly such SCADA systems involve centrlized devices which monitor nd control lrge plnts spred out over res tht rnge up to continents, for exmple in the cse of pipelines, nd most control ctions re performed utomticlly bsed on sensor mesurements, see for instnce [11]. Clerly, the bove problems tht we wish to study re non-stndrd in view of clssicl mthemticl system nd control theory, even in n ODE setting. Nevertheless we nticipte tht they present interesting theoreticl chllenges s well s being importnt for mny rel world problems. Becuse of their non-stndrd nture, they lso bring in nonstndrd difficulties not ppering for instnce in clssicl control theory. First nd foremost, for control in mixed discrete nd continuous fshion, we will see tht it is importnt to consider the possibility of this leding to the Zeno phenomenon 1 which is ssocited with the ccumultion points of discrete events or switching times. This phenomenon is best illustrted by n exmple. Exmple 1.1.1: (Thomson s Lmp [57]) There re certin reding lmps tht hve button in the bse. If the lmp is off nd you press the button the lmp goes on, nd if the lmp is on nd you press the button the lmp goes off. So if the lmp ws originlly off, nd you 1 The nme refersto the prdoxin Achilles nd the Tortoise of the Greekphilosopher Zeno of Ele. 5

14 1 Introduction nd preliminries pressed the button n odd number of times, the lmp is on, nd if you pressed the button n even number of times the lmp is off. Suppose now tht the lmp is off, nd I succeed in pressing the button n infinite number of times, perhps mking one jb in one minute, nother jb in the next hlf minute, nd so on, ccording to Russell s recipe. After I hve completed the whole infinite sequence of jbs, i.e. t the end of the two minutes, is the lmp on or off? One cn now rgue tht it is impossible to nswer this question: It cnnot be on, becuse one did not ever turn it on without t once turning it off. On the other hnd, it cnnot be off, becuse if one did in the first plce turn it on, nd therefter one never turned it off without t once turning it on. But the lmp must be either on or off. This is contrdiction. The Zeno phenomenon is one of the min technicl difficulties in obtining globl existence results, even for hybrid dynmicl systems governed by ODEs. In the context of switching control, prcticl solution to void the Zeno phenomenon is to implement dwell-time, i.e., mechnism tht forbids ny further switchings for fixed mount of time fter switch occurred. The drwbck of dwell-time is tht, in cse of open loop optiml switching, there re no gurntees tht the performnce of the system in the chosen mode will not deteriorte to n uncceptble level before the next switch is permitted. Similrly, for the cse of switching feedbck control, it my become impossible for the controller to rect to system filures during tht dwell time intervl. Therefore, we will not consider dwell-time nd must, for sound theory, py ttention to the voidnce of the Zeno phenomenon. A significnt difference to the theory of hybrid dynmicl systems governed by ODEs resides in the fct tht we must, in ddition to the evolution in time, be strongly concerned with the regulrity of solutions in the coupled spce vrible. Moreover, due to the distributed nture of the dynmics, we lso see the necessity of including n output mp in the nlysis which hs prtil stte informtion only. In n ODE setting, typiclly, full stte feedbck is ssumed. Therefore, n pproprite theory beyond the ODE perspective pying prticulr ttention to these forementioned spects is needed. 6

15 1.2 Motivtion: Multiscle modeling for networked dynmicl systems L R L R L L L R L R L R R Figure 1.2: Exmple of grph modeling networked dynmicl system. The direction of the rcs coincide with the prmeteriztion long the edge, but not necessrily with the direction of flow long the edge. The grey vertices re sensor nodes, the dimond shped vertices represent nodes with ctutors. 1.2 Motivtion: Multiscle modeling for networked dynmicl systems Consider network of pipes (or wires, rods, strings, bems, etc.) represented by grph G = (E, V) with edges E = {e j } j=1,...,m, ech edge e j correspondingto pipe prmeterized by n intervl [ (j), b (j) ], nd with vertices V = {v i }, i = 1,...,n where pipes my be interconnected. Assume tht the network is consistent in the sense tht the grph is biprtite, i. e., 2-colorble for instnce with lbels left nd right, directed (coinciding with the direction from left to right nd the positive direction of the prmeteriztion of ech edge), nd wekly connected (i. e., replcing ll of its directed edges with undirected ones, then for ech vertex there exist pth to ech other vertex). Note tht these ssumption re not restrictive for pplictions such s gs, sewer or trffic networks, becuse one my, for instnce, introduce uxiliry vertices to mke grph biprtite while modeling the sme physicl system. Now suppose tht the networked system under considertion involves ctutors nd sensors plced t selected vertices. This is ubiquitous for exmple in SCADA systems mentioned bove. 7

16 1 Introduction nd preliminries Assuming tht the sensors nd ctutors re non-collocted in such network, this implies tht the set of vertices consists of three disjoint subsets V = V S V A V N, where V S contins ll vertices with sensors, V A contins ll vertices with ctutors, V N contins ll vertices with neither sensors nor ctutors. To ech edge e j, j = 1,...,m we ssocite dynmicl system t x (j) + Ã (j) s x (j) = f (j) (x) (1.2.1) for x(t, s) = (x 1 (t, s), x 2 (t, s),...,x N (t, s)) R N in the region (j) < s < b (j), t > 0. The choice of this prticulr dynmic is strongly motivted by pplictions, noting tht such systems model vriety of complex physicl systems in networks rnging from the vibrting string or vibrting bem [50] over conterflow processes in chemicl engineering [51] up to dynmicl flows in communiction or logistic res [33]. Further, (1.2.1) cn be obtined by pproprite lineriztion of non-liner blnce equtions round (stedy-stte) solutions modeling gs flow in pipelines by mens of the isotherml Euler gs equtions [8], wter flow in open chnnels by mens of the St. Vennt equtions [35] or trffic flow on highwys by mens of the Lighthill-Whithm-Richrds model [21]. Common scling It is useful to prmeterize ll edges e j, j = 1,...,N with single prmeter s over common intervl [, b]. For the j-th edge, this entils prmeter chnge s b(j) (j) b s nd the dynmicl system (1.2.2) corresponding to tht edge becomes with t x (j) + A (j) s x (j) = f (j) (x), s (, b) (1.2.2) A (j) = b(j) (j) Ã (j), f (j) (t, s, x (j) ) = f(t, b(j) (j) b b s, x(j) ) (1.2.3) 8

17 1.2 Motivtion: Multiscle modeling for networked dynmicl systems Nodl conditions To ech vertex v i, i = 1,...,n with connected incoming pipes δ v i E nd outgoing pipes δ + v i E, we ssocite nodl conditions. Typiclly, these consist of conditions for continuity of quntity x κ t node, given s x (k) κ (t, b (k) ) = x (k ) κ (t, (k ) ) for ll k δ v i, k δ + v i (1.2.4) for some κ {1,...,N}. blnce eqution for quntity x κ t node, given s ω ik x (k) κ (t, b (k) ) ω ik x k δ v i k δ + v i (k ) κ (t, (k ) ) = ϕ (i) κ (t) (1.2.5) forsome κ {1,...,N} withppropriteweights ω ik [0,1] such tht k δ v i ω ik = 1 nd k δ + v i ω ik = 1 nd prescribed vlues ϕ (i) = (ϕ κ (i) 1,...,ϕ κ (i) ) modeling sources/demnds. n (i) Using tht the equtions (1.2.4) nd (1.2.5) re liner in x κ nd x κ, respectively, ny combintion of these with κ κ imply two liner eqution systems for ech edge e j C (j) L x(k) (t, (k) ) = u (j) L, C(j) R x(k) (t, b (k) ) = u (j) R (1.2.6) with coefficient mtrices C (j) L/R involving the respective ω ik nd righthnd-side u (j) L/R involving the respective ϕ(i). Sensors Physics commonly suggests tht the solution of distributed prmeter systems cn only be observed prtilly, e. g., temperture cn often only be mesured t (prts of) the boundry of physicl system. Here, we tke ccount of tht by ssuming tht some components of the solution cn be observedt certin nodesin the grph, i.e., t sensor nodes. It is not restrictive to ssume tht ll sensor nodes re lbeled with right by 9

18 1 Introduction nd preliminries introducing uxiliry nodes if needed. So for ech edge e (k), we djoin n output eqution of the form y (k) (t) = κ σ κ (k) x κ (k) (t, b) (1.2.7) with the convention tht non-zero outputs re only obtined from those edges connected to sensor node v i V S, i.e., { (k) σ κ R if k δ v i, v i S, (1.2.8) σ (k) κ = 0 else. A typicl exmple would be the projection of the solution t the endpoint of pipe onto its component ssocited with pressure in correspondence with physicl mesurement. With coefficient mtrices C (k) S the liner eqution system Actutors = dig(σ κ (k) ), we my write (1.2.7) s y (k) (t) = C (k) S x(k) (t, b). (1.2.9) For ech v i in the subset of vertices with ctutors V A, we ssume tht ω ik = ω q ik, nd ϕ(i) = ϕ (i),q (1.2.10) with q Q (i) for some finite set Q (i). We will tke q s n dditionl stte of the networked dynmicl system, with the physicl interprettion tht q indictes the mode of n inlet, outlet, vlve, pump, etc., e.g., Q (i) {ON,OFF}. Exmple 1.2.1: Suppose tht t some vertex v i, i 1,...,n, compressordynmicllyincresesthepressure p i (correspondingtosomequntity x κ for κ {1,...,N})byginof β i,wherethegin β i isgoverned by n ODE d dt β i(t) = π i (β i (t), p i (t)), (1.2.11) coupled with the dynmics on the network by setting ϕ (i) κ (t) = { βi (t) if ON 0 if OFF (1.2.12) 10

19 1.2 Motivtion: Multiscle modeling for networked dynmicl systems in (1.2.5) nd which my undergo gin resetwith every modl switch ON OFF or OFF ON t switching time τ k, such s β(τ k ) = 0. (1.2.13) Note tht dditionl nodl dynmics s in Exmple requires creful nlysis of ech respective compressor model (1.2.11), (1.2.13). But t this point, we do not wnt to focus on developing relistic (descriptive) model of compressor rther thn investigting the theoreticl implictions of modeling like involving logicl decisions such s ON nd OFF t the nodes. In prticulr we wish to study fundmentl wellposedness properties of forementioned bstrct systems in (optiml) open loop or closed loop opertion. So we continue writing (1.2.12) s ϕ (i) κ = ϕ (i),q κ with ϕ (i),on κ (t) = β i (t), ϕ (i),off κ (t) = 0, (1.2.14) effectively ssuming ϕ (i),on/off κ (t) s given. As with the sensors, it is not restrictive to ssume tht the boundry ctutors re locted t nodes lbeled with left by introducing uxiliry nodes if needed, so the left boundry conditions (1.2.6) for ech edge e j becomes C (j),q L x (k) (t, (k) ) = u (j),q L. (1.2.15) We will be llowing tht, long with the nodl conditions, the convection nd rection term my depend on discrete prmeter q, i.e., A(t, s) = A q (t, s) nd f(t, s, x) = f q (t, s, x) (1.2.16) in (1.2.2). This permits the modeling of numerous complex physicl systems rrely considered in the literture so fr. Exmples 1.2.2: Consider fmily of trffic flow models on highwy ssocited with mode q Q = {1,2,3,4} ech corresponding to prescribed recommended speed of 60, 80, 100 or 120 km/h on dynmicl speed limit signs flected by different dvection function A q. The control tsk consists of prescribing the optiml mode in order to void congestion. For n ppliction in the context of civil engineering, see [49]. 11

20 1 Introduction nd preliminries In chemicl engineering, rective trnsport my involve mterils tht instntneously switch between dormnt nd ctive, e. g., depending on the vilbility of nother mteril concentrtion. Obviously, this cn be modeled by different rection terms f q. For n ppliction in the context of bio-remedition, see [54]. If the sign of the eigenvlues in A q depend on q, it might lso be necessry for the well-posedness of the boundry conditions to include dependency on C (j) R generl become nd u(j) R on q, so the right boundry conditions in C (j),q R x (k) (t, b (k) ) = u (j),q R. (1.2.17) It is cler tht the implementtion of switching the stte of ny such ctutors, e. g., opening vlve or inlet, turning on pump, compressor, etc., will ctully consume some time. However, in mny pplictions, the time scle in which these control elements operte is significntly finer thn the time scle used for modeling the trnsporttion process within the network. Therefore we consider these timescles distinct nd pproximte the dynmics on the fine scle s effectively instntnous. It is now precisely the interction of the discrete nd continuous dynmics resulting from this multi-scle model tht we wish to study. Reformultion to single system in mtrix form The systems (1.2.2) cn be rrnged to single system in mtrix form in x = t x + A q s x = f q (x) (1.2.18) ( x (1) x (2) x (n) ) with the following block structures à (1),q 0 f (1) (t, s, x (1),q ) A q à (2),q =..., f (2) (t, s, x (2),q ) fq (t, s, x) =.. 0 à (n),q f (n) (t, s, x (n),q ) The boundry conditions in the vrible x become C q L x = uq L, Cq R x = uq R 12

21 1.2 Motivtion: Multiscle modeling for networked dynmicl systems with C q L = nd C (1),q L 0 C (2),q L... 0 C (n),q L d q L = d (1),q L d (2),q L. d (n),q L, Cq R =, dq R = The sensor output in the vrible x becomes C (1),q d (1),q R d (2),q R. d (n),q R R 0 C (2),q R... 0 C (n),q R., y(t) = C S x(t, b) with C S = C (1) S 0 C (2) S... 0 C (n) S. Without loss of generlity we my tke the discrete prmeter q s globlforthegrph(rtherthnindividul q (i) forechnodendedge) by introducing sufficiently mny uxiliry modes in Q. Thus we hve derived quite generl hybrid model in the form of (1.1.3) (1.1.5) for brod clss of networked dynmicl system with multi scle chrcter. Moreover, in ddition to the pplictions lredy mentioned, the bove model provides systemtic theoreticl foundtion for studying the coupling of different models in single network, for exmple in order to increse the computtionl efficency for the simultion of lrge gs networks s proposed in [7]. 13

22 1 Introduction nd preliminries 1.3 Solution concepts: Brod solutions nd hybrid time evolution Both the prtil differentil eqution(1.1.3) nd the hybrid nture of the modelrequireclrifictionofwhtismentbysolutionofthesystem. For the PDE, we will consider generlized solutions defined in brod sense, following the pttern in [13] to be restricted by the requirement tht they nd their relevnt derivtives represent ideliztions by limiting processes. The hybrid time evolution introduced below is bsed on similr principles. Brod solutions for systems of hyperbolic prtil differentil equtions Consider semiliner hyperbolic system of n equtions in two independent vribles x + A(t, s) x = f(t, s, x) (1.3.1) t s where x tkes vlues inr n, A is n n n mtrix for ll t, s nd f tkes vlues inr n for ll t, s, x. Assuming tht the system is strictly hyperbolic, i.e., ech A(t, s) hs n rel distinct eigenvlues λ 1,...,λ n, one cn select bses of left nd right eigenvectors {l 1,...,l n } nd {r 1,...,r n } such tht l i A = λ i l i, Ar i = λ i r i (1.3.2) t every point (t, s). These eigenvectors cn be normlized ccording to r i 1, l j r i = { 1 if i = j 0 if i j. (1.3.3) Observe tht (1.3.2) nd (1.3.3) imply x = n (l i x)r i, A x = i=1 n λ i (l i x)r i (1.3.4) for every x R n. By clssicl solution of the hyperbolic system (1.3.1) we men continuously differentible function x(, ) which stisfies(1.3.1) for ll (t, s) in the domin of considertion. But one cn lso interpret the equtions i=1 14

23 1.3 Solution concepts: Brod solutions nd hybrid time evolution in broder sense nd consider weker solutions tht do not need to be continuously differentible everywhere s commonly ccepted for the theory of conservtion or blnce lws. We will follow clssicl methods of prtil differentil equtions [13] nd will interprete the semiliner hyperbolic system in the sense of brod solutions, mening tht the solution components with componentwise right-hnd-side x i (t, s) = l i x(t, s) (1.3.5) f i = l i f + [ t l i + λ i s l i] x, x = stisfy the (non-liner) ordinry differentil eqution n x i r i, (1.3.6) d dt x i(t, s i (t; τ, σ)) = f i (t, s i (t; τ, σ), x(t, s i (t; τ, σ))) (1.3.7) long the integrl curves s i ( ; τ, σ) in the vector field v i = (1, λ i ) going through (τ, σ) in the domin of considertion. This notion of solution is motivted by the observtion tht, for sufficient regulr dt nd by using the chin rule, (1.3.7) nd the definition of s i ( ; τ, σ) together imply d dt x i(t, s i (t; τ, σ)) = t x i(t, s i (t; τ, σ)) + d dt s i(t; τ, σ)x i (t, s i (t; τ, σ)) i=1 (1.3.8) = t x i(t, s i (t; τ, σ)) + λ i (t, s)x i (t, s i (t; τ, σ)) (1.3.9) = f i (t, s i (t; τ, σ), x(t, s i (t; τ, σ))). (1.3.10) Moreover, this notion of solution is known s physiclly meningful nd it is esy to see tht ny clssicl solution is lso solution in this brod sense. From (1.3.7), we then see for instnce tht discontinuities, if involved in the solution, cn only propgte long the chrcteristic curves s i ( ; τ, σ). Though the theory developed in this thesis will strongly build on the notion of brod solutions, t this point, we note tht there re vrious other wys to define generlized solutions for the system (1.3.1) nd we list some of them without climing completeness: 15

24 1 Introduction nd preliminries Limit solutions, mening tht there exists sequence of clssicl solutions x ν to (1.3.1) with boundry dt u ν stisfying u ν u in L 1 loc s ν (1.3.11) nd the sequence of solutions stisfying see, e.g., [13]. x ν x in L 1 loc s ν, (1.3.12) Distributionl solutions, mening tht for every test function ϕ C 1 c one hs x ϕ + A(t, s)x ϕ f(x, t, s)ϕdx dt = 0, (1.3.13) t s see, e.g., [37]. Mild solutions, in the sense tht x stisfies the integrl eqution x(t) = U(t, τ)x + t τ U(t, ξ)f(ξ, x(ξ))dξ, t τ (1.3.14) where U(t, τ) is continuous liner evolution process {U(t, τ) : t τ} such tht for every τ R nd x in the domin of the (unbounded) liner opertor A(t)x(s) := A(t, s) x(s) (1.3.15) s the function x(t) = U(t, τ)x is the determined solution of the opertor eqution dx = A(t)x, (1.3.16) dt stisfying x(τ) = x, see, e.g., [47]. Wek solutions in the sense tht the solution is piecewise smooth nd stisfies ( ξds + Λξdt = g(t, s, ξ) + Λ ) (t, s)ξ ds dt (1.3.17) s Ω Ω for ny domin Ω [, b] {t 0} with piecewise Lipschitz boundry Ω, where Λ = dig(λ 1,...,λ n ), g = (g 1,...,g n ) nd ξ = (x 1,...,x n ) with λ i, g i nd x i defined s before, see [20]. 16

25 1.3 Solution concepts: Brod solutions nd hybrid time evolution Some of these generlized solutions though obtined by different concepts my coincide nd it my be of interest to investigte this since some concepts hve dvntges over others. Time evolution for hybrid dynmicl systems A conceptul difficulty of hybrid dynmicl systems is the forml tretment of time evolution. As lredy indicted in the sketch of the solution concept (1.1.1), time evolution is broken up into sequence of closed intervls which my even be reduced to single points. Therefore n pproch widely considered in the literture of hybrid dynmicl systems is the forml tretment clled hybrid time evolution usingsetoftimeevents, see[60]or[53,38,24,25]withsimilr concepts. A time event consists on n event time t Rtogetherwith multiplicity m(t) nd cn be denoted by sequence (t 0, t 1, t 2,...,t m(t) ) (1.3.18) specifying the sequentilly ordered discrete trnsition times t the sme continuous time instnt t R(the event time). A time eventwith multiplicity equlto 1is just given by pir (t 0, t 1 ) with the interprettion of denoting the time instnts just before nd just fter the event hs tken plce. If the multiplicity of the time event is lrger thn one ( multiple time event) then there re some intermedite time instnts (ll t the sme event time t) ordering the sequence of discrete trnsitions tking plce t t. So sequence of time events with event times (t 1 < t 2 < t 3 < ) implies sequence of discrete trnsition times or switching times (τ k ) k N = (t 1 1,...,tm(t 1) 1, t 1 2,..., t m(t 2) 2, t 1 3,...,tm(t 3) 3,...) (1.3.19) with mode q(t) = q k (nd continuous stte x(t), output y(t), etc.) defined for t in the interswitching intervls [τ k, τ k+1 ] for two consecutive switching times τ k, τ k+1 in (1.3.19). The embedding of this hybrid time evolution in norml time intervl T Rcn then relized by n ugmenttion T = {(t,0) : t T} {(t j k, j) : 0 < j m(t k), k = 1,...,K}, (1.3.20) 17

26 1 Introduction nd preliminries so we hve T R Z. Solutions of hybrid dynmicl systems re then functions defined on this prtilly ordered structure T. A difficulty with this hybrid time evolution nd ugmenttion for set of solutions is tht the relevnt T will vry with the solution. The time evolution is closely connected with the pproprite notion of switching signl. Therefore, we include t this point the following bsic definitions we will work with subsequently. Definition 1.3.1: On finite time horizon [0, T] for some T > 0 nd finite set of modes Q switching signl is determined by finite sequence of pirs (q k, t k ) K k=1 Q R +, K Nsuch tht K k=1 t k = T. Here, q k denotes the ctive mode nd t k the length of the interswitching intervl in time where this mode is ctive. This defines switching times τ k = k κ=1 t κ wheretheswitchingsignldiscontinuouslychngesfrom q k to q k+1, in the following denoted by q k q k+1. We denote the collection of ll such switching signls by S([0, T]; Q). We note the following remrks concerning the bove definition. Remrks 1.3.1: 1. Finitenessofthesequences (q k, t k ) K k=1 hsclosetieswiththezeno phenomenon. Similrly s with Zeno s prdoxes, we hve seen in Exmple tht n infinite series of switches in finite period of time my result in contrdiction. Therefore we exclude tht possibility in the definition of switching signl, but point out tht this only voids the Zeno phenomenon when the switchingsignlis-prioriknown. Forournlysisoftheproblemwhen the switching signl s control is given implicitly by optimlity or feedbck rules, we must mke sure tht the -priori unknown switching signl is then in ccordnce with this definition. We note tht there re physicl systems whose modeling ideliztion involves switching with infinitely mny switches on finite time horizons which re not covered by the Definition 1.3.1, for instnce in modeling buzzer. Solutions of such chttering systems cn be defined in the sense of Filippov [19], replcing differentil eqution d dtx(t) = f(x) with discontinuous righthnd side 18

27 1.3 Solution concepts: Brod solutions nd hybrid time evolution f( ) by differentil inclusion d x(t) F(x), (1.3.21) dt where F(x) is defined to be the smllest convex closed set contining ll limit vlues of the function f( ) t x. The Zeno phenomenon then shows up s the possibility of non-unique solutions of (1.3.21), cf. prgrph 10.4 in [19]. Indeed, Filippov hypothesizes in Theorem in [19] the non-zenoness of the system s sufficient condition for unique solutions, but without providing conditions how to verify this -priori. The theory developed here provides such conditions for the system (1.1.3) (1.1.5) in cse of optiml switching nd switching by feedbck, but does not cover such chttering systems unless one uses some verging (homogeniztion) to redefine this behvior s single mode. As possible extension of the theory developed here, we note tht the notion of hybrid time evolution s well s the notion of switching signl re ble to cover some situtions in which switching times ccumulte by llowing m(t), K N { }. It is then nturl to require tht for K = m(t) =, the corresponding sequence of modes q 1, q 2, q 3,... becomes sttionry nd tht lim k x(t k ) exists in n pproprite sense. 2. The definition of switching signls llows t k = 0 nd, consequently, switching signl my hve cscded switches q k q k+1 q k+l (short: q k q k+l ) (1.3.22) t the sme switching time τ k = τ k+1 = = τ k+l for some L N. The llownce of such cscded switches coincides with multiple time event in hybrid time evolution nd hs close ties with desired limit properties for sequences of switching signls s we will see in Definition below. As with the hybrid time evolution, the difficulty with this retention of these 0-length interswitching intervls is tht switching signl s such is not function of time with embeddings of the switching times τ k in [0, T] nd the modes Q in [1, Q ] s subsets ofrrther thn being function defined on ugmented time intervls, cf. (1.3.20). 19

28 1 Introduction nd preliminries Somewht busing nottion, we will nevertheless write q( ) for switching signls (q k, t k ) K k=1 S([0, T]; Q) when the interprettion of q( ) s sequence is cler from the context. For the considertions of well-posedness nd optiml switching in n pproprite sense for hybrid dynmicl system, we lso need topology for S([0, T]; Q). The multi-scle perspective of hybrid dynmicl systems suggests the following notion for the convergence of switching signls. Definition 1.3.2: Let q ν ( ) = [ (q k, t k ) K ν k=1] be sequence of switching signls in S([0, T]; Q). We sy tht [ (q k, t k ) K ν k=1] converges to limit switching signl q ( ) = [ (q k, t k ) K k=1] in S([0, T]; Q), if for lrge ν K = K ν t ν k t k, k = 1,...,K (1.3.23) q ν k = q k, k = 1,...,K. We do denote this convergence by q ν ( ) q ( ) in S([0, T]; Q). For switching signl q( ) = (q k, t k ) K k=1 S([0, T]; Q) one cn define projections π 0 to l (R) nd π to l (R) by nd π 0 (q( )) = (τ 1,...,τ K,0,0,0,...), τ k = k t κ (1.3.24) κ=1 π (q( )) = (q 1,...,q K,0,0,0,...). (1.3.25) It is then esy to see tht the topology defined implicity in Definition is induced by the distnce defined, for two switching signls q 1 ( ), q 2 ( ) S([0, T]; Q), by d S (q 1 ( ), q 2 ( )) = π 0 (q 1 ( )) π 0 (q 2 ( )) + π (q 1 ( )) π (q 2 ( )). (1.3.26) Moreover, we see tht ny sufficiently smll neighborhood of ny q( ) S([0, T]; Q) in this topology contins only switching signls with the sme number of switching times. For n illustrtion of this sense of convergence see Figure 1.3 on pge

29 1.4 The sclr eqution nd piecewise continuous brod solutions 1.4 The sclr eqution nd piecewise continuous brod solutions Consider the constitutionl sclr eqution t x(t, s) + λq (t, s) s x(t, s) = fq (t, s, x(t, s)) (1.4.1) for s (, b), t > 0, nd the initil-boundry condition x(0, s) = x(s), x(t, ) = u q (t) (1.4.2) for s (, b) nd t 0, respectively, with given dt x(s) nd u q (t) prescribed on the gnomon 2 G := {(t, s) : t 0, s = 0} {(t, s) : t = 0, s b}. (1.4.3) As output of the sclr system bove, we consider y(t) = x(t, b). For well-posedness of (1.4.1) (1.4.2), suppose tht for ll q Q the given functions λ q (, ) re continuous in t nd continuously differentible in s nd tht there re bounds 0 < λ < λ such tht λ λ q (t, s) λ. (Thus,bycompctness,thereexistsLipschitz-constnt L λs suchtht λ q (, s) λ q (, s ) L λs s s for ll s, s [, b].) Moreover, ssume tht for ll q Q the given functions f q (,, ) re continuous, tht there existsconstnt C f suchtht f q (,, x) C f (1+x) nd thtthereexists Lipschitz-constnt L fx such tht f q (,, x) f q (,, x ) L fx x x for ll x, x R. As motivted bove, we will consider solutions of the hyperbolic equtions in brod sense. The interprettion of (1.4.1) (1.4.2) using the method of chrcteristics simplifies in the sclr cse to chrcteristic curves being obtined s solutions of the ordinry differentil equtions d dt s(t) = λq (t, s(t)). (1.4.4) 2 Geometriclly, gnomon is the L-shped piece of prllelogrm remining when similr prllelogrm is excised from its corner. We re modifying this usge to consider together the bottom nd left side of the infinite rectngle [0, ) [, b], visulizing flow slefttoright longnedgessocitedwith theintervl [,b], sothis gnomonprecisely contins the input dt for trnsport long such n edge modeled by(1.4.1) when λ q (, ) > 0. 21

30 1 Introduction nd preliminries Setting ˆx = x(t, s(t)), (1.4.1) becomes the ODE d dtˆx(t) = fq (t, s(t), ˆx(t)). (1.4.5) The brod solution x(, ) of (1.4.1) (1.4.2) is thus given by the fmily of ODEs (1.4.5) with initil dt x(s), u q (t) on the gnomon G. The following exmple shows tht we must ccept the introduction nd propgtion of discontinuities in the solution of the system(1.4.1) (1.4.2) when the mode q discontinuously chnges. Exmple 1.4.1: Consider the trnsport dynmics (1.4.1) tken simply s t x(t, s) + x(t, s) = 0 (1.4.6) s nd with constnt initil dt x 1. In first mode q = 1 suppose we use x(t,0) = u 1 (t) 1, so the solution remins x 1. But suppose we woulddiscontinuouslyswitchfrom mode q = 1tonewmode q = 2t switching time t = τ with keeping the dynmics fixed but now using input dt x(t,0) = u 2 (t) 2. Then jump discontinuity is introduced t the input boundry nd propgtes(long the chrcteristic s = t τ) into the edge. Moreover, both for the boundry conditions(1.1.4) rising from nodl coupling if we consider trnsport on network nd for our results on feedbck switching control bsed on sensor observtions t nodes, we must be strongly concerned with the regulrity of boundry trces for the brod solution: Exmple 1.4.2: Continuing Exmple 1.4.1, suppose we would observe the solution t the output node corresponding to s = 1. We would see constnt vlue y(t) = x(t,1) 1 for t < τ + 1, constnt vlue y(t) = x(t,1) 2 for t > τ +1 nd jump discontinuity t t = τ +1. Now consider the following non-stndrd spce of piecewise continuous functions. Definition 1.4.1: A piecewise continuous function g( ) on n intervl [, b] tking vlues in R is constructed by specifying finitely mny prtition points = p 0 p 1 p K = b, (1.4.7) 22

31 1.4 The sclr eqution nd piecewise continuous brod solutions k N, nd on ech prtition subintervl [p k 1, p k ], k {1,...,K} ssigning continuous function g k C([0,1];R). For degenerte 0-length subintervls we impose the restriction tht the ssigned g k ( ) must be constnt. As function on [, b], we hve ( ) s pk 1 g(s) = g k, for s (p k 1, p k ) (1.4.8) p k p k 1 nd g(s) = [g k 1 (1), g k (0)] if sis oneof theprtition points p k. Wemy even hve g(s) = [g k 1 (1), g k,...,g k+l 1, g k+l (0)] (1.4.9) t colesced prtition points p k =... = p k+l, noting tht such multiple ssignment is not set but preserves the sequence order. The set of ll such piecewise continuous functions on [, b] will be denoted by C pw ([, b]). Furthermore, suppose we topologize the set C pw ([, b]) of piecewise continuous functions by the convergence of sequences similr to those of switching signls S([0, T]; Q) in Definition Definition 1.4.2: Let {g ν ( )} be sequence in C pw ([, b]). We sy tht {g ν ( )} converges to some {g ( )} C pw ([, b]), if for lrge ν K = K ν p ν k p k, k = 1,...,K g ν k g k uniformly in C([0,1]), k = 1,...,K. (1.4.10) We do denote this convergence by g ν ( ) g ( ) in C pw ([, b]). Now observe tht for given q( ) S([0, T]; Q), the ssumption on λ q (, ) nd f q (,, )ensurebythepicrd-lindelöf theoremtheexistence nd uniqueness of solutions to the equtions (1.4.4) nd (1.4.5) when q is replced by q(t). Moreover, using tht these solutions depend continuously on the initil dt, the resulting coordinte trnsformtion (t, s) (t, s(t)) is homeomorphism between the domin [0, T] [, b] nd the corresponding portion of the gnomon G. Hence, if the boundry dt u q ( ) is continuous for ll q Q, we hve u q( ) C pw ([0, T]) nd we obtin unique brod solution on [0, T] with x(t, ) C pw ([, b]) for ny initil dt x( ) C pw ([, b]). By exchnging t nd s, we lso see 23

32 1 Introduction nd preliminries tht the output y( ) = x(,1) is in C pw ([0, T]). Finlly, stndrd wellposedness theory for ODEs shows tht the solution x(t, ) C pw ([, b]) (ndtheoutput y( ) C pw ([0, T])) dependcontinuously(stopologized in Definition nd 1.4.2) on x( ), u q ( ) nd q( ). In ddition to tht we hve the following compctness result. Lemm The subspce S K ([0, T]; Q) S([0, T]; Q) corresponding to bound on K in Definition is sequentilly compct s topologized in Definition Proof. Let q ν ( ) be sequence in S K ([0, T]; Q). Then, with bound on the number of switching points τ k, we my extrct subsequence (reindexed by ν), such tht K ν = K ν = K K (1.4.11) for ll ν, ν. With K fixed, switching sequence is equivlent to point in the compct set {(τ 0,...,τ K ) R K+1 : 0 = τ 0 τ 1 τ K = T} Q K. (1.4.12) Moreover, we remrk tht the definition of piecewise continuous functions in Definition nd the topology in Definition is here primrily intended for the description of the sptil regulrity of brod solutions for the sclr trnsport eqution (1.4.1) nd the temporl regulrity of the system s output y( ) in time. But it bers nlogy to ttempts in developing topologicl structures for hybrid ODE trjectories, mostly used in the context of stbility nlysis: Similrly s with S([0, T]; Q), one cn lso metrize C pw ([, b]) by defining, for g( ) C pw ([, b]), the projection π to l (R) by π (g( )) = (p 0,...,p K,0,0,0,...) (1.4.13) nd, for g 1 ( ), g 2 ( ) C pw ([, b]), the distnce d Cpw (g 1 ( ), g 2 ( )) = π (g 1 ( )) π (g 2 ( )) + mx k 1 g1 k ( ) g2 k ( ). (1.4.14) For pirs [q( ), g( )] S([0, T]; Q) C pw ([0, T]) s solutions of hybrid dynmicl systems with modes governed by ODEs the distnce given 24

33 1.5 Systems of equtions nd BV setting by the sum of (1.3.26) nd (1.4.14) is known s the Tvernini metric [56]. Morover, this metric induces the Skorohod J 1 topology, defined, for càdlàg functions f 1, f 2 D([0, T],R), by d S-J1 (f 1, f 2 ) = inf λ Λ mx{ f1 λ f 2, f 1 e }, (1.4.15) where Λ is the group of ll strictly order preserving homeomorphisms of [0, T] nd e is the identity, see [25]. The J 1 -topology is corser thn the compct open topology nd finer thn the Lebesque topologies. For exmple mong rel functions on the unit intervls, the chrcteristic function of [0, 1 2 ) is δ-close to the indictor function of [0, δ) in the Skorohod J 1 distnce, but in the uniform metric the distnce between these functions is one. On the other hnd, the chrcteristic function of [0, δ) is δ-close to zero in ny of the Lebesgue metrics, but the Skorohod J 1 distnce between them is one. WelsoremrkthttheTverninimetric(ndtheSkorohodJ 1 metric) re known to be incomplete, see [25] (nd [10]). Nevertheless, we hve developed, with these tools, well-posedness theory for the sclr eqution (1.4.1) (1.4.2), cn consider open loop optiml switching for quite generl cost functions s well s closed loop switching by feedbck with mild ssumption on theprtitions A q, C(q q ) of the observtion spce indeed lso for sclr networked system. For detils, see [27]. But generliztion of the results in [27] for the constitutionl sclr eqution to (fully coupled) systems of equtions cnnot be obtined by direct use of Bnch s fixed point theorem in n incomplete vector vlued spce C pw ([, b];r n ). This is one reson why we investigte different pproch for the mtrix cse s discussed in the next Section. 1.5 Systems of equtions nd BV setting Motivted by pplictions, we wish theory for switching (fully coupled) systems of hyperbolic prtil differentil equtions, i. e., the mtrix cse (1.1.3) (1.1.5). For tht cse, continuous dependency on input dt becomes delicte for systemswith n 2, even when the equtions re just coupledt the 25