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Index

Symbols

0i, j, i j zero matrix 120, 215A, associative algebra 217219A see conjugate transposeAff(n,R) see Lie group, affineBmL, generated Lie algebra 40, 42, 45, 51,

64B 40, 45, 47, 51, 57Bm, matrix list 10, 172Cr 240C see smoothC see real-analyticF,Fn,Fnn 215G see groupGL(n,R) 19, 4453GL+(n,R) 19, 4554H see quaternions

I, radical see ideal, polynomialIn, identity matrix 215,, imaginary part 215LI see input, locally integrableMn see manifoldPd, positive dminors property 93P(t; u), discrete transition matrix 130PC see input, piecewise continuousPK see input, piecewise constantPA, minimum polynomial of A 219Q, rationals 215R

n see punctured space

Rn+, positive orthant 17, 166

R see controllability matrixRn+, nonnegative reals 215, real part 7, 215S see semigroupS see time-shiftSL(n,Z) 19, 199

SU(n) see group, special unitaryS11, unit circle 129S, sphere 217Symm(n) 7S see semigroup, concatenationU see input, historyV0 61X(t; u), transition matrix 14, 147Y see output history(A; t), integral of etA 6d(A; t) 6LyA see operator, Lyapunov, constraint set 9, transition-matrix group 19, Kronecker sum 31, 222pA, characteristic polynomial 4(A, j), special sum 163, 222, composition of mappings 231det, determinant 216| , such that 4, n2 30, 38, 221+

x see successor-related 188x see generic rankP, principal polynomial 6270

see shufflespec(A), eigenvalues of A 5, concatenation 17Symm(n) 132, 216 see semidirect product see Kronecker producttr, trace 216a, vector field 36, 238 empty set 22(u + iv) see representationmax(A), real spectrum bound 27adA see adjoint

D.L. Elliott, Bilinear Control Systems, Applied Mathematical Sciences 169, 273DOI 10.1007/978-1-4020-9613-6_14, c Springer Science+Business Media B.V. 2009

274 Index

adg, adjoint representation 229aff(n,R) see Lie algebra, affine, 121U, boundary 22U, closure of U 22(X,Y) see CartanKilling formdminor 216i, j, Kronecker delta 215, transition-matrix group 3351f t see trajectoryf, Jacobian matrix 188, 232, 233, 237, matrix-to-vector 30, 31, 38, 43, 221, sharp (vector-to-matrix) 55, sharp (vector-to-matrix) 44, 221g, derived subalgebra 230ij,k see structure tensorg, Lie algebra 3582, 225238H0 see homogeneous rational functions

U, interior of U 22, set intersection 239

null(X), nullity of X see nullityn2 69|U, restricted to U 234 see similaritysl(2,R) 37, 69, 70sl(k,C) 254sl(k,H) 257sl(n,R), special linear 64, 81, 257so(n) see Lie algebra, orthogonalspec(A), spectrum of A 4, 7, 31, 217

, disjoint union 235, union 232

A

Abels relation 5, 18absolute stability 179accessibility 22, 100116, 118

strong 22, 23, 100, 102, 103action of a group 34, 122, 223, 239, 240,

242, 244, 253, 258adjoint, Ad 38, 39, 62transitive 90, 244, 252, 253

ad-condition 100, 102121multi-input 103weak 102, 103, 106, 110, 152

adjoint 37, 39, 101108, 121, 192, 193, 229adjoint representation see representationaffine variety 53, 6061, 67, 224, 248250

invariant 63, 64, 68, 78singular 61, 64, 65

algebraassociative see associative algebraLie see Lie algebra

algebraic geometry, real 65, 249algorithmAlgTree 81LieTree 43, 64, 82, 90, 92, 96Buchberger 61Euclid, GCD 62

alpha representation see representationarc 234, 236, 237, 245arc-wise connected 234associative algebra 5, 81, 216

Abelian 218conjugate 217generators 218

atlas 44, 46, 47, 232attainable set 21, 22, 23, 32, 47, 67,

90100, 116, 136, 139, 141, 179181,212

B

biaffine systems 12, 140, 143, 151, 153,161, 162, 170, 175

controllability 119124discrete-time 140, 156stabilization 119

bilinear (2-D) systems 13bilinear system

approximation property 162inhomogeneous see control system,

biaffinerotated semistable 89

boost converter, DCDC 173Borel subalgebra 230Brownian motion see Wiener process

C

CampbellBakerHausdorff see Theoremcanonical form

Jordan 5, 7, 38, 219real 220

Carleman approximation 164CartanKilling form 87, 178, 230category 161causal 161, 203, 206CCK1 see coordinates, first canonicalCCK2 see coordinates, second canonicalcentralizer, z 220, 231, 253character 62, 240charts, coordinate 44, 4850, 232, 234, 238charts, global 72chemical reactions 185ChenFliess series 201, 204205compartmental analysis 170

Index 275

competition, VerhulstPearl model 172completed system 45complex structure 254composition 231composition of maps 233composition property 20concatenation 17, 150cone 248

invariant 96conjugacy see similarityconjugate transpose 177, 216, 217, 219Continuation Lemma 22, 106, 132, 139,

142control 9, 10

feedback 10, 32open-loop 10, 11switching 9, 199

control set, invariant see trapcontrol system

biaffine see biaffine systemsbilinear 10linear 810, 116linearized 12matrix 55, 56, 179nonlinear 21positive 166stabilizable 83switching 13, 172179time-invariant 10variable structure 12, 32, 174

control system: causal 10controllability 21, 85, 89122

discrete-time 129139of bilinear system 2224of linear system 20of symmetric systems 34on positive orthant 168

controllability matrix 21, 32, 134136,155

controllable see controllabilitycontrollable pair 21, 123controlled switching 172convex cone 166

open 166convex set 21coordinate transformation 49

linear 36coordinates

first canonical 48, 7274manifold 232second canonical 50, 7375, 185

coset space 239Cremona group 191

D

delay operator z 155dendrite, Rall model 171dense subgroup 46derivation 234, 237derived series 74, 230diffeomorphism 50, 72, 139, 187, 188, 190,

192, 232of manifolds 233

difference equation 3difference equations 12, 22, 28, see

discrete-time systems, 146, 147, 150differential 231differential generator 212diffusions 210213direct product 243discrete-time systems 12, 127142

affine 6linear 31, 154156

discretization 28Euler 29, 141midpoint 29sampled-data 28, 118

domainnegative 104positive 104

drift term 11, 91dynamical polysystem 13, 14, 190, 237dynamical system 2, 236

affine 6discrete-time 3linear 24positive 166

dynamics 3linear 3

E

economics 166eigenvalues 217embedding, regular 233endpoint map 236enzyme kinetics 171equilibrium 24

stable 7unstable 7

equivalence, C 188euclidean motions 182Euler angles 74Euler operator 56events, probabilistic 207exponent

Floquet 27

276 Index

Lyapunov 27, 28, 178, 209exponential polynomial 5

F

feedback control 11, 83field

complex, C 215quaternion, H 251rationals, Q 215real, R 215

finite escape time 3, 115, 236Flatten 221flow 28, 235, 236formal power series

rational 204, 206frames on Rn 24, 72

G

gauge function 25, 26, 89, 93, 112, 128,139, 169

GCD, greatest common divisor 62generators 218generic 58, 153

rank 58, 60, 61, 64, 68good at 57Grbner basis see ideal, polynomialgroup 16, 238

Abelian 241algebraic 53analytic 240complex Lie 45diagonal 46fundamental, 1 91general linear 19Lie 3666, 240matrix 1920, 24matrix Lie 45, 242simple see simplesolvable 119special linear 46special orthogonal 46special unitary 77topological 239transition matrices 33transitive 34unipotent 74upper triangular 46

H

Hausdorff space 232, 243Heisenberg algebra 225

helicoid surface 23homogeneous function 56, 111homogeneous space 71, 78, 243homomorphism 19

group 239semigroup 16

Hurwitz property see matrixhybrid systems 176hypersurface system 99

I

idealLie algebra 71, 226zero-time 92, 93

ideal, polynomial 61, 247basis 61Grbner basis 59, 61, 64, 65, 92radical 61, 249real radical 249

identification of system parameters 154independence, stochastic 207, 209, 210,

213index

polynomial 15indistinguishable 150inner product 252input 9

concatenated 17generalized 203history 10, 13, 28locally integrable, LI 18, 201piecewise constant 9, 33, 35, 50, 148piecewise continuous 9piecewise continuous, PC 14, 18, 162stochastic 211switching 13universal 150, 152

input history 138inputoutput

mapping 154, 156, 158, 160, 162relation 144, 154

intertwining equation 188, 238invariant

function 56linear function 57polynomial 55, 57, 62relative 56set 26subset 24, 34subspace 9, 57variety 64

involutive see real-involutiveirreducible 220, 253

Index 277

isomorphismgroup 239Lie algebra 226, 238, 257Lie group 240

isotropysubalgebra 71, 82subgroup 71, 78

J

Jacobi axioms 35, 225Jacobian Conjecture 191Jordan canonical see canonical formJordan decomposition 194JQ system 108, 109111JurdjevicQuinn systems see JQ system

K

Kalman criterioncontrollability 21, 120, 179observability 143, 179

Killing form (X,Y) see CartanKillingform

Kolmogorovaxioms for probability 207backward equation 200, 209backward operator see differential

generatorKronecker delta 6, 215Kronecker power 222Kronecker product 128, 131, 220223, 252Kronecker sum 31

L

LARC matrix 58, 6062, 65LARC, Lie algebra rank condition 34, 58,

102, 121, 168LaSalles invariance principle 26, 108,

113left-invariant see vector fieldlexicographic (lex) order 247Lie algebra 33, 225231

Abelian 225affine 120center of 38, 231, 254compact 253derived 56, 230direct sum 226exceptional 256free 40, 227generated 4043, 192, 226generators 40

homomorphism 226isomorphic 36nilpotent 74, 76, 230of a Lie group 240, 241orthogonal 37, 57, 60orthonormal basis 47rank condition see LARCreductive 81, 231, 253semisimple see semisimplesolvable 73, 75, 81, 88, 185, 230special linear 36transitive 251258

Lie bracket 3435, 225vector field 237

Lie derivative 25Lie group see group, LieLie monomials 40, 227

left normalized 41, 42standard 227

Lie saturate 116Lie subgroup 242Lie wedge 119Lie, S. 33LieWord 43, 80local rank condition 139locally integrable, LI 9, 20, 146, 201203Lyapunov equation 8, 26, 105

discrete-time 128Lyapunov function 25, 26, 175

quadratic 107, 178Lyapunov spectrum 27Lyapunovs direct method 7, 104, 131

M

machine (in a category) 161manifold 26, 36, 4447, 55, 70, 72, 166,

182, 211, 232234mass action 171Mathematica 65

scripts 43, 59, 68, 104, 124, 125matrix 215217

analytic function 46augmented 65cyclic 21elementary 36exponential 430Hankel 155161Hermitian 216Hurwitz 7, 8, 29, 85Metzler 167neutral 97107nilpotent 128, 194, 218nonnegative 17

278 Index

orthogonal 216Pauli 77pencil 84permutation-reducible 117positive 167SchurCohn 129skew-Hermitian 77, 216skew-symmetric 97, 216strongly regular 95, 96, 117, 168symmetric 216totally positive 95transition 1424transition (discrete) 130133triangular 5, 17, 31, 88, 218, 221, 230unipotent 218unitary 216, 219

matrix control system 14, 33, 35, 51, 67matrix Lie algebra 227

conjugacy 36, 226matrix logarithm 6, 48minor 61, 63, 92, 216

positive 93, 95principal 7, 216

module, V(Mn) 234monoid 16, 94multi-index 192, 247

N

negative definite 7, 25, 26, 105neutral see matrixnil radical 230nilpotent Lie algebra see Lie algebranon-resonance 189norm

Frobenius 47, 217, 221operator 3, 217uniform 213vector 217

nullity of adA 38nullspace 38, 57, 134, 215numerical methods see discretizations

O

observability 147of bilinear systems 146, 148, 156

observability Gram matrix 151observability matrix, O(C; A) 143observable dynamical system 143, 147observable pair 143, 147octonions 257operator

adjoint see adjoint

Euler 27Lyapunov 8, 221Sylvester 8, 221

optimal control 180181orbit 236

of group 3457, 243of semigroup 84zero-time 92, 93

order, lexicographic 193, 204, 216output feedback 10output history 147output mapping 9output tracking 182

P

parameter estimation 166Partition 44path planning 182, 236Peano-Baker series 15, 162Philip Hall basis PH(r) 227polynomial

characteristic 85, 87, 217Hurwitz 85minimum 5, 219

polysystem see dynamical polysystempositive definite 7, 25, 61, 65, 114, 128,

177positive systems 165172

discrete 133prehomogeneous vector space 78probability space 207problems 16, 32, 67, 72, 79, 81, 82, 99, 100,

107, 109, 113, 136, 178, 198, 258product integral 14proper rational function 155pullback formula 38, 51, 101punctured space 22

transitivity on 24

Q

Q-ball 106quantum mechanics 184185quasicommutativity 123quaternions 55, 251

pure 252, 258

R

radially unbounded 26range 8, 142, 215rank

Lie 5767

Index 279

of a matrix 37, 43, 57of mapping 233

rank-one controls 134continuous-time 179

reachability matrix, R 157160reachable 21, see span-reachablereachable set see attainable setreal-analytic 40, 44, 49, 50, 55, 56,

232240real-involutive 36, 40, 42, 51realization

minimal 156of inputoutput relation 143, 153, 154,

162realization, group 78, 241representation, group 77, 240, 243, 253

adjoint 38, 55, 62representation, Lie algebra 6870, 227,

243, 251257 (alpha) 44, 67, 77, 251 (beta) 251adjoint 38, 229faithful 37, 227, 231, 253, 255

representation, semigroup 18resonance 7, 128, 219right-invariant see vector field

S

sans-serif type 2Schurs Lemma 220, 253SchurCohn 178semi-flow 3, 235, 236semidefinite 25, 26, 27, 148, 149semidirect product 120, 182, 243semidirect sum 243semigroup 16, 18, 40

concatenation 17, 18, 201203discrete parameter 128, 130discrete-time 132Lie 116matrix S 16topological 201transition matrices, S 18, 84with Pd property 94

semiorbit 236semisimple 77, 80, 82, 231, 254shuffle 145, 206sigma-algebra 207sign pattern 7signature 37, 57, 132similarity 216, 217, 223, 229

invariants 87simple 253256

algebra of Lie group 242group 239Lie algebra 231

Singular 65sliding mode 174175small-controllable 104smooth 232solvable see Lie algebraspan-reachable 154, 156, 157, 160spectrum 193, 217

in unit disc 129Lyapunov 209

stabilityasymptotic 25, 26, 27basin of 26, 83, 107, 123by feedback 32exponential 7globally asymptotic 26neutral 7UES, uniform exponential stability

176, 178uniform under switching 176

stability subgroup 71stabilization 83

by constant feedback 8589by state feedback 107113practical 113115

stabilizer see isotropystate space 2step function 15stochastic bilinear systems 166, 208213strong accessibility see accessibilitystructure tensor 229subalgebra

associative 217, 219isotropy 244, 258Lie 36, 226244

subgrouparc-wise connected 245closed 47, 53, 71connected 20dense 46isotropy 244Lie 242normal 92, 239one-parameter 19

submanifold 233subsemigroup 16subspace

u-observable 150unobservable 150

successor 3, 129, 137switching

autonomous 174

280 Index

controlled 176hybrid 176

Sylvester criterion 7Sylvester equation 8, 222symmetric bilinear systems 11, 3358system composition

cascade 145parallel 144product 145

T

tangent bundle 235tangent space 235Theorem

Ado 230AscoliArzela 202BJKS 122, 140Boothby A 64Boothby B 253CairnsGhys 199CampbellBakerHausdorff 3948Cartan solvability 231CayleyHamilton 218Correspondence 244EnestrmKakeya 129existence of log 219Generic Transitivity 79GrasselliIsidori 150Hrmander 212Hilberts Basis 248Inverse Function 234JurdjevicSallet 122, 140Kuranishi 80Lie, solvable algebras 88, 230M. Hall 227Malcev 53Morrey 234Poincar 188Rank 232Rectification 188Schur 194, 219StoneWeierstrass 206Whitney 233Whitney Embedding 234Yamabe 245

time-set 2, 9, 236time-shift 9time-variant bilinear system 11topology

group 239matrix norm 217of subgroup 241of tangent bundle 235

product 213uniform convergence 202, 206weak 202, 203Zariski 250

trajectory 56, 235concatenation 237controlled 21

transition mappingreal-analytic 199

transition mapping 2, 3, 4, 14, 24on a manifold 236real-analytic 187

transition matrix see matrix, transitiontransitivity 90, 239, 255258

Lie algebra 54, 66Lie group 5482on Rn 23weak 58, 69, 79

translation 45, 48, 49, 53, 240, 241trap 92, 133, 137, 169triangular see matrixtriangularization, simultaneous 88Trotter product formula 244

U

UES, uniform exponential stability 178unit, of semigroup 16, 18

V

variety see affine varietyvector bundle 28vector field 25

C 56, 57, 205, 234, 235affine 12, 238complete 236invariant 241left-invariant 240linear 45, 238right-invariant 44, 50, 70, 116, 240, 241,

244Volterra series 161163

discrete 163

W

weakly transitive see transitivity, 58Wiener process 210, 213

Z

Zariski closure 78, 250

ReferencesIndex

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