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"" {TEXT 230 15 "Evelyne Hubert " }}{PARA 218 "" 0 "" {TEXT 231 12 "INRIA Sophia" }}{PARA 219 "" 0 "" {TEXT 232 23 "Evelyne.Hubert@inria.fr" }}{PARA 220 "" 0 "" {TEXT 233 43 "http:// www-sop.inria.fr/cafe/Evelyne.Hubert" }}{PARA 221 "" 0 "" {TEXT 234 20 "(revised March 2004)" }}{PARA 222 "" 0 "" }{PARA 216 "" 0 "" {TEXT 235 4 "This" }{TEXT 236 359 " worksheet illustrates several appl ications of an a triangulation-decomposition algorithm for systems of \+ differential equations. The main point is to find equivalent sytems to the given one that are minimal in some sense. This minimality depends on the ranking we choose. With different ranking we can determine cer tain properties of our system. For instance:" }}{PARA 216 "" 0 "" {TEXT 236 1 " " }}{PARA 216 "" 0 "" {TEXT 236 1 " " }{TEXT 236 129 "- \+ are there purely algebraic equations satisfied by the solutions of our system? Find a representation of all those constraints. " }{TEXT 236 26 "\n See Constraint Sytems" }}{PARA 216 "" 0 "" {TEXT 236 117 " - are there equations in only one or two of the unknown functions? Fin d a representation for all those equations. " }{TEXT 236 105 "\n S ee for instance Solving differential systems, Control theory, Symmetry by pattern matttern matching" }}{PARA 216 "" 0 "" {TEXT 236 141 " - a re there ordinary differential equations satisfied by the solutions of a system of partial differential equations? Find those equations. " } {TEXT 236 58 "\n See Determining equations of symmetry infinetesima ls." }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 236 110 "We give exa mples taken from the litterature of control theory, numerical analysi s and ode solver litterature." }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 235 37 "This maple worksheet is available at " }{HYPERLNK 237 "h ttp://www-sop.inria.fr/cafe/Evelyne.Hubert/webdiffalg/applications/app lications.mws" 4 "www-sop.inria.fr/cafe/Evelyne.Hubert/webdiffalg/appl ications/applications.mws" "" }{TEXT 235 1 " " }}}{EXCHG {PARA 216 "> \+ " 0 "" {MPLTEXT 1 238 14 "with(diffalg);" }}{PARA 223 "" 1 "" {TEXT 239 70 "Warning, the protected name version has been redefined and unp rotected" }{TEXT 239 1 "\n" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7:I3Ros enfeld_GroebnerG6\"I-delta_leaderGF%I1delta_polynomialGF%I'denoteGF%I, derivativesGF%I2differential_ringGF%I.differentiateGF%I*equationsGF%I5 essential_componentsGF%I0field_extensionGF%I(greaterGF%I,inequationsGF %I(initialGF%I3initial_conditionsGF%I)is_primeGF%I'leaderGF%I6power_se ries_solutionGF%I7preparation_polynomialGF%I.print_rankingGF%I%rankGF% I'reduceGF%I.rewrite_rulesGF%I)separantGF%I(versionG6$I(_syslibGF%F>" }{TEXT 240 1 " " }}}{SECT 0 {PARA 225 "" 0 "" {TEXT 241 37 "Solving or dinary differential systems" }}{SECT 0 {PARA 226 "" 0 "" {TEXT 242 18 "Singular solutions" }}{EXCHG {PARA 216 "" 0 "" {TEXT 235 58 "We consi der the non linear ordinary differential equation:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 94 "chazy := (diff(y(t),t,t)+y(t)^3*diff( y(t),t))^2-(y(t)*diff(y(t),t))^2*(4*diff(y(t),t)+y(t)^4);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%&chazyG,&*$),&-%%diffG6$-%\"yG6#%\"tG-%\" $G6$F/\"\"#\"\"\"*&)F,\"\"$F4-F*6$F,F/F4F4F3F4F4*()F,F3F4)F8F3F4,&F8\" \"%*$)F,F>F4F4F4!\"\"" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 72 "We encode the differential indeterminates and the deriva tions in a table" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 68 "R : = differential_ring(ranking=[y], derivations=[t], notation=diff);" }} {PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"RG%)ODE_ringG" }{TEXT 240 1 " " } }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 36 "G := Rosenfeld_Groebne r([chazy], R);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7%%0character isableGF&F&" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 122 "The characteristic decomposition has 3 components. We get their c haracteristic sets one after the other, or in one stance." }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 42 "The first represents the general solut ion:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 16 "equations(G[1]) ;" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7#,(*$)-%%diffG6$-%\"yG6#%\"tG-% \"$G6$F-\"\"#F1\"\"\"F2**F1F2F'F2)F*\"\"$F2-F(6$F*F-F2F2*(\"\"%F2)F*F1 F2)F6F5F2!\"\"" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 128 "The second component reflects two singular solutions, both th e general solution of a first order ordinary differential equation." } }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 32 "equations(G[2]); map(f actor, %);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7#,&*&)-%\"yG6#%\"tG\" \"%\"\"\"-%%diffG6$F'F*F,F,*&F+F,)F-\"\"#F,F," }{TEXT 240 1 " " }} {PARA 224 "" 1 "" {XPPMATH 20 "6#7#*&-%%diffG6$-%\"yG6#%\"tGF+\"\"\",& F%\"\"%*$)F(F.F,F,F," }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 122 "The last component is also a singular solutions. It is \+ in fact a particular case of one of the previous singular solution." } }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 16 "equations(G[3]);" }} {PARA 224 "" 1 "" {XPPMATH 20 "6#7#-%\"yG6#%\"tG" }{TEXT 240 1 " " }}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 13 "equations(G);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7%7#,(*$)-%%diffG6$-%\"yG6#%\"tG-%\"$G6$F. \"\"#F2\"\"\"F3**F2F3F(F3)F+\"\"$F3-F)6$F+F.F3F3*(\"\"%F3)F+F2F3)F7F6F 3!\"\"7#,&*&)F+F:F3F7F3F3*&F:F3)F7F2F3F37#F+" }{TEXT 240 1 " " }}} {EXCHG {PARA 216 "" 0 "" {TEXT 235 82 "We shall also use the practical following so called jet notation for using diffalg" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 27 "map(denote, chazy, jet, R);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#,(*$)&%\"yG6$%\"tGF)\"\"#\"\"\"F+**F*F+F&F+ )F'\"\"$F+&F'6#F)F+F+*(\"\"%F+)F'F*F+)F/F.F+!\"\"" }{TEXT 240 1 " " }} }}{SECT 1 {PARA 227 "" 0 "" {TEXT 243 114 "To perform a chain resoluti on of a system of ordinary differential equations one shall use an eli mination ranking." }}{EXCHG {PARA 227 "" 0 "" {TEXT 244 63 "\nConsider the differential system defined by the following set " }{TEXT 245 1 " S" }{TEXT 244 54 " of differential polynomials in the unknown function s " }{TEXT 245 14 "x(t),y(t),z(t)" }{TEXT 244 1 "." }}{PARA 216 "" 0 " " {XPPEDIT 18 0 "diff(x(t),t)=x(t)*(x(t)+y(t)+z(t))\n" "6#/-%%diffG6$- %\"xG6#%\"tGF**&F'\"\"\",(F'F,-%\"yGF)F,-%\"zGF)F,F," }{TEXT 246 1 " " }{XPPEDIT 18 0 "diff(y(t),t)=-y(t)*(x(t)+y(t)+z(t))\n" "6#/-%%diffG6$ -%\"yG6#%\"tGF*,$*&F'\"\"\",(-%\"xGF)F-F'F--%\"zGF)F-F-!\"\"" }{TEXT 246 2 "\n " }{XPPEDIT 18 0 "diff(z(t),t)=-z(t)*(x(t)+y(t)+z(t))" "6#/- %%diffG6$-%\"zG6#%\"tGF*,$*&F'\"\"\",(-%\"xGF)F--%\"yGF)F-F'F-F-!\"\"" }{TEXT 246 2 "\n " }}{PARA 216 "" 0 "" {TEXT 247 1 "\n" }{TEXT 247 1 "\n" }}{PARA 227 "" 0 "" {TEXT 244 35 "Using an elimination ranking wh ere " }{TEXT 245 14 "z(t)>y(t)>x(t)" }{TEXT 244 89 " we will obtain di fferential characteristic sets containing a differential polynomial in " }{TEXT 245 4 "x(t)" }{TEXT 244 46 " alone, a differential polynomia l determining " }{TEXT 245 4 "y(t)" }{TEXT 244 13 " in terms of " } {TEXT 245 4 "x(t)" }{TEXT 244 51 " and finally a differential polynomi al determining " }{TEXT 245 4 "z(t)" }{TEXT 244 13 " in terms of " } {TEXT 245 4 "y(t)" }{TEXT 244 5 " and " }{TEXT 245 4 "x(t)" }}{PARA 227 "" 0 "" {TEXT 244 88 "To see more clearly the dependencies of thes e differential polynomials, one can use the " }{TEXT 245 13 "rewrite_r ules" }{TEXT 244 41 " command, which is an alternative to the " } {TEXT 245 9 "equations" }{TEXT 244 9 " command." }}}{EXCHG {PARA 216 " " 0 "" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 113 "S:= \{diff(x(t ),t)-x(t)*(x(t)+y(t)+z(t)), diff(y(t),t)+y(t)*(x(t)+y(t)+z(t)), diff(z (t),t)+z(t)*(x(t)+y(t)+z(t))\};" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% \"SG<%,&-%%diffG6$-%\"xG6#%\"tGF-\"\"\"*&F*F.,(F*F.-%\"yGF,F.-%\"zGF,F .F.!\"\",&-F(6$F1F-F.*&F1F.F0F.F.,&-F(6$F3F-F.*&F3F.F0F.F." }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 14 "with(diffal g):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 73 "DR := differenti al_ring(ranking=[z,y,x], derivations=[t], notation=diff):" }{MPLTEXT 1 238 31 "\nG :=Rosenfeld_Groebner(H, DR);" }{MPLTEXT 1 238 26 "\nT := rewrite_rules(G[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7#%0ch aracterisableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% \"TG7&/-%\"zG6#%\"tG,$*&,(-%%diffG6$-%\"xGF)F*!\"\"*$)F1\"\"#\"\"\"F7* &F1F7-%\"yGF)F7F7F7F1F3F3/-F/6$F9F*,$*&*&F9F7F.F7F7F1F3F3/-F/6$F1-%\"$ G6$F*F6,$*&F1F7F.F7F60F1\"\"!" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 18 "dsolve(T[3],x(t));" }{MPLTEXT 1 238 28 "\nd solve(eval(T[2],%), y(t));" }{MPLTEXT 1 238 29 "\nnormal( eval(T[1],[% ,%%]) );" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#/-%\"xG6#%\"tG*&-%$tanG6# *&,&F'\"\"\"%$_C2GF.F.%$_C1G!\"\"F.F0F1" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&%$_C3G\"\"\"-%$tanG6#*&,&F'F*%$ _C2GF*F*%$_C1G!\"\"F2" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#/-%\"zG6#%\"tG,$*&,&!\"\"\"\"\"*&%$_C3GF,%$_C1GF,F,F,*&F/F,-%$ta nG6#*&,&F'F,%$_C2GF,F,F/F+F,F+F+" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 248 26 "The singular case x(t)=0 :" }}{PARA 216 "" 0 "" {TEXT 235 23 "IIt is actually in the " }{TEXT 249 10 "adherence " }{TEXT 235 55 "of the general case, but it can be computes separately. " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 43 "G0 :=Rosenfeld_Groe bner(\{op(H), x(t)\}, DR);" }{MPLTEXT 1 238 19 "\nrewrite_rules(G0);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#G0G7#%0characterisableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7#7&/-%\"zG6#%\"tG,$*&,&- %%diffG6$-%\"yGF(F)\"\"\"*$)F0\"\"#F2F2F2F0!\"\"F6/-F.6$F0-%\"$G6$F)F5 ,$*&*$)F-F5F2F2F0F6F5/-%\"xGF(\"\"!0F0FD" }{TEXT 240 1 " " }}}{PARA 216 "" 0 "" {TEXT 235 1 " " }{TEXT 248 85 "\nThe system to be solved c ertainly does not need to be in the standard explicit form." }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 66 "S := [diff( x(t),t)-x(t)*(x(t)+y(t)),diff(y(t),t)+y(t)*(x(t)+y(t))," }{MPLTEXT 1 238 52 "\ndiff(x(t),t)^2+ diff(y(t),t)^2+ diff(z(t),t)^2-1 ];" }} {PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG7%,&-%%diffG6$-%\"xG6#%\"tGF-\" \"\"*&F*F.,&F*F.-%\"yGF,F.F.!\"\",&-F(6$F1F-F.*&F1F.F0F.F.,**$)F'\"\"# F.F.*$)F5F;F.F.*$)-F(6$-%\"zGF,F-F;F.F.F.F3" }{TEXT 240 1 " " }}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 72 "R := differential_ring(r anking=[z,y,x], derivations=[t], notation=diff):" }{MPLTEXT 1 238 30 " \nG :=Rosenfeld_Groebner(S, R);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% \"GG7#%0characterisableG" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 21 "rewrite_rules(G[1]); " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7'/*$)-%%diffG6$-%\"zG6#%\"tGF-\"\"#\"\"\",$*&,**&)-%\" xGF,\"\"%F/)-F(6$F5F-F.F/F.*$F4F/!\"\"*$)F9F7F/F/*(F.F/)F9\"\"$F/)F5F. F/F " 0 "" {MPLTEXT 1 238 20 "rewrite_rules(G[2]);" }}{PARA 228 "" 1 "" {TEXT 250 33 "Error, invalid subscript selector" }{TEXT 250 1 "\n" }}}{PARA 227 "" 0 "" {TEXT 244 100 "We can perform a chain resolution to both c omponents of the result. Note that the singular solution " }{TEXT 245 6 "x(t)=0" }{TEXT 244 115 " has to be discarded in the resolution of t he first component, as indicated by the differential polynomials of it s " }{TEXT 245 11 "inequations" }{TEXT 244 37 ". The correct complete \+ solution with " }{TEXT 245 6 "x(t)=0" }{TEXT 244 34 " is given by the \+ second component." }}{PARA 216 "" 0 "" }}}{SECT 1 {PARA 225 "" 0 "" {TEXT 241 17 "Contraint systems" }}{PARA 216 "" 0 "" {TEXT 235 223 "Th e problem here is to reduce any system of differentio-algebraic equati ons (DAE) to a differential system of index zero or index 1, that is \+ more accessible to generic numerical codes. The problem can be stated \+ as follow. " }}{PARA 229 "" 0 "" {TEXT 251 65 "How to determine the hi dden constraints of a differential system " }{XPPEDIT 18 0 "F(t, Y, Di ff(Y,t)) = 0;" "6#/-%\"FG6%%\"tG%\"YG-%%DiffG6$F(F'\"\"!" }{TEXT 252 1 " " }{TEXT 251 8 " where " }{XPPEDIT 18 0 "F[Diff(Y,t)] = 0;" "6#/& %\"FG6#-%%DiffG6$%\"YG%\"tG\"\"!" }{TEXT 252 1 " " }{TEXT 251 11 " uni formly?" }}{PARA 216 "" 0 "" {TEXT 235 117 "We choose an orderly ranki ng. We illustrate this with the pendulum and then double pendulum in c artesian coordinates." }{TEXT 235 1 "\n" }}{SECT 1 {PARA 226 "" 0 "" {TEXT 242 12 "The pendulum" }}{PARA 227 "" 0 "" {TEXT 244 100 "Conside r the set of differential polynomials describing the motion of a pendu lum, i.e. a point mass " }{TEXT 245 1 "m" }{TEXT 244 39 " suspended at a massless rod of length " }{TEXT 245 1 "l" }{TEXT 244 32 " under the influence of gravity " }{TEXT 245 1 "g" }{TEXT 244 27 ", in Cartesian coordinates " }{TEXT 245 5 "(x,y)" }{TEXT 244 119 ". The Lagrangian f ormulation leads to two second order differential equations together w ith an algebraic constraint. " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 253 1 " " }{TEXT 244 56 " is the Lagrangian multiplier. It is \+ a function of time." }}{PARA 227 "" 0 "" {XPPEDIT 18 0 "m*y[t,t]=-lamb da*y-g" "6#/*&%\"mG\"\"\"&%\"yG6$%\"tGF*F&,&*&%'lambdaGF&F(F&!\"\"%\"g GF." }{TEXT 253 1 " " }}{PARA 227 "" 0 "" {XPPEDIT 18 0 "m*x[t,t]=-lam bda*x" "6#/*&%\"mG\"\"\"&%\"xG6$%\"tGF*F&,$*&%'lambdaGF&F(F&!\"\"" } {TEXT 253 1 " " }}{PARA 227 "" 0 "" {XPPEDIT 18 0 "x^2+y^2=l^2" "6#/,& *$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"lGF'" }{TEXT 253 1 " " }}{PARA 227 " " 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 71 "P := [m*y[t,t]+l ambda[]*y[]+g, m*x[t,t]+lambda[]*x[], x[]^2+y[]^2-l^2];" }}{PARA 224 " " 1 "" {XPPMATH 20 "6#>%\"PG7%,(*&%\"mG\"\"\"&%\"yG6$%\"tGF-F)F)*&&%'l ambdaG6\"F)&F+F1F)F)%\"gGF),&*&F(F)&%\"xGF,F)F)*&F/F)&F7F1F)F),(*$)F9 \"\"#F)F)*$)F2F=F)F)*$)%\"lGF=F)!\"\"" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 54 "K := field_extension(transcende ntal_elements=[m,l,g]):" }}}{PARA 227 "" 0 "" }{PARA 227 "" 0 "" {TEXT 244 49 "We find the lowest order system of equations for " } {XPPEDIT 18 0 "x,y " "6$%\"xG%\"yG" }{TEXT 253 1 " " }{TEXT 244 5 " an d " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 253 1 " " }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 87 "RO := differential_ring(ranking =[[lambda,x,y]], derivations=[t], field_of_constants=K):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 31 "GO:= Rosenfeld_Groebner(P, RO); " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#GOG7$%0characterisableGF&" } {TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 18 "rewri te_rules(GO);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7$7'/&%'lambdaG6#%\" tG,$*&*&&%\"yGF(\"\"\"%\"gGF/F/*$)%\"lG\"\"#F/!\"\"!\"$/*$)F-F4F/*&,** &)&F.6\"\"\"$F/F0F/F5*(F>F/F0F/F2F/F/*(&F'F?F/F2F/)F>F4F/F5*&FCF/)F3\" \"%F/F/F/*&F2F/%\"mGF/F5/*$)&%\"xGF?F4F/,&*$FDF/F5F1F/0FM\"\"!0F-FR7&/ FC,$*&*&F>F/F0F/F/F1F5F5/FMFR/FPF10F>FR" }{TEXT 240 1 " " }}}{PARA 227 "" 0 "" {TEXT 244 38 "The second component in the output of " } {TEXT 245 18 "Rosenfeld_Groebner" }{TEXT 244 254 " corresponds to the \+ equilibria of the pendulum. From the first component of the output we \+ see that the actual motion is described by two first order differentia l equations together with a constraint: the solution depends on only t wo arbitrary constants. " }}{PARA 227 "" 0 "" }{PARA 227 "" 0 "" {TEXT 244 58 "If we wish to obtain the differential system satisfied b y " }{TEXT 245 1 "x" }{TEXT 244 5 " and " }{TEXT 245 1 "y" }{TEXT 244 27 " alone one shall eliminate " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT 253 1 " " }{TEXT 244 1 ":" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 87 "RM := differential_ring(ranking=[lambda,[x,y]], der ivations=[t], field_of_constants=K):" }{MPLTEXT 1 238 32 "\nGM:= Rosen feld_Groebner(P, RM);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#GMG7$%0ch aracterisableGF&" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 21 "rewrite_rules(GM[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7'/&%'lambdaG6\",$*&,(*(%\"mG\"\"\")&%\"yG6#%\"tG\"\"#F -)%\"lGF3F-!\"\"*&)&F0F'\"\"$F-%\"gGF-F6*(F9F-F;F-F4F-F-F-*&F4F-,&*$)F 9F3F-F6*$F4F-F-F-F6F6/&F06$F2F2,$*&,**&)F9\"\"%F-F;F-F-**F3F-F@F-F;F-F 4F-F6**F9F-F,F-F.F-F4F-F-*&)F5FJF-F;F-F-F-*(F4F-F,F-F>F-F6F6/*$)&%\"xG F'F3F-F>0FS\"\"!0,&F?F-FAF6FV" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 21 "rewrite_rules(GM[2]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7&/&%'lambdaG6\",$*&*&&%\"yGF'\"\"\"%\"gGF-F-*$)%\"lG\" \"#F-!\"\"F3/&%\"xGF'\"\"!/*$)F+F2F-F/0F+F7" }{TEXT 240 1 " " }}} {PARA 227 "" 0 "" {TEXT 244 78 "Note here that the motion is given by \+ a second order differential equation in " }{TEXT 245 4 "y(t)" }{TEXT 244 64 ". Then the Lagrange multiplier is given explicitely in terms o f " }{TEXT 245 4 "y(t)" }{TEXT 244 26 " and its first derivative." }} {PARA 216 "" 0 "" }}{SECT 0 {PARA 226 "" 0 "" {TEXT 242 19 "The double pendulum" }}{PARA 216 "" 0 "" {TEXT 235 122 "The upper point mass is located by its Cartesian coordinates (x,y) relative to the point wher e the upper rod is attached." }}{PARA 216 "" 0 "" {TEXT 235 281 "(X,Y) are the cartesian coordiates of the lower point mass taken relatively to the upper point mass. The Lagrangian formulation leads to the fol lowing equations where L and l are the Lagrange multipliers. They cons ist of 4 second order differential equations plus two constraints." }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 59 "PP := [ -x[t,t] +2*x*l - 2*X*L, -y[t,t] +2*y*l - 2*Y*L - 1," }{MPLTEXT 1 238 60 "\n -(X [t,t]+x[t,t]) +2*X*L, -(Y[t,t]+y[t,t]) +2*Y*L-1," }{MPLTEXT 1 238 34 " \n -1 + x^2+y^2,-1+X^2+Y^2]:" }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 167 "We wish to know the minimal order differential equation s satisfied by x,y,X,Y. We use a ranking eliminating the Lagrange mult ipliers and an orderly ranking on x,y,X,Y." }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 14 "with(diffalg);" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#7<%3Rosenfeld_GroebnerG%-delta_leaderG%1delta_polynomialG%'denot eG%,derivativesG%2differential_ringG%.differentiateG%*equationsG%5esse ntial_componentsG%0field_extensionG%(greaterG%,inequationsG%(initialG% 3initial_conditionsG%)is_primeG%'leaderG%6power_series_solutionG%7prep aration_polynomialG%.print_rankingG%%rankG%'reduceG%(reducedG%-reduced _formG%.rewrite_rulesG%)separantG%(versionG" }{TEXT 254 1 " " }}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 72 "R := differential_ring(r anking=[[L, l],[ X, Y, x, y]], derivations=[t]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"RG%)ODE_ringG" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 34 "_Env_diffalg_char := \"Kalkbrener\":" }{MPLTEXT 1 238 32 "\nG := Rosenfeld_Groebner(PP,R); " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7$%0characterisableGF&" }{TEXT 240 1 " " }} }{EXCHG {PARA 216 "" 0 "" {TEXT 235 46 "The second component represent the equilibria:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 20 "rew rite_rules(G[2]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7*/%\"LG,$*&\"\" \"F(%\"YG!\"\"#F(\"\"#/%\"lG*&F(F(%\"yGF*/%\"XG\"\"!/*$)F)F,F(F(/%\"xG F3/*$)F0F,F(F(0F0F30F)F3" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 41 "The first component describes the motion:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 11 "rank(G[1]);" }}{PARA 224 "" 1 " " {XPPMATH 20 "6#7(%\"LG%\"lG&%\"YG6$%\"tGF)&%\"yGF(*$)%\"XG\"\"#\"\" \"*$)%\"xGF/F0" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 63 " map(collect, rewrite_rules(G[1]) ,[y[t,t],Y[t],y[t ]], factor);" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#7-/%\"LG,&*&&%\"yG6$% \"tGF+\"\"\"%\"YG!\"\"#F,\"\"#*&*&F/F,,&&F-F*F,F,F,F,F,F-F.F,/%\"lG,&* &F(F,F)F.F,*&*&F/F,,&F0F,F4F,F,F,F)F.F,/F4,**&*(%\"XGF,,&*&F-F,F)F,F,* &%\"xGF,F@F,F,F,F(F,F,*&FDF,,&*$)F@F0F,F,*$)F-F0F,F,F,F.F.*&*&F-F,)&F- 6#F+F0F,F,FGF.F.*&**F-F,F@F,,&*$)F)F0F,F,*$)FDF0F,F,F,)&F)FOF0F,F,*&)F D\"\"$F,FFF,F.F.*&FGF,FFF.F./F(,(*&**FDF,FFF,,&*&F)F,F@F,F,*&FDF,F-F,F .F,FMF,F,*&FHF,,,*&FVF,FHF,F,*,F0F,F)F,F@F,FDF,F-F,F.*(F0F,FVF,FJF,F,* (F0F,FTF,FHF,F,*&FJF,FTF,F,F,F.F.*&*(FRF,,(*&F)F,FHF,F0*&F)F,FJF,F,*(F @F,FDF,F-F,F.F,FWF,F,*&FVF,F_oF,F.F.*&*&FDF,,(*(F)F,F@F,F-F,F,*&FDF,FH F,F.*(F0F,FDF,FJF,F.F,F,F_oF.F,/FG,$*&,&F-F,F,F.F,,&F-F,F,F,F,F./FU,$* &,&F)F,F,F.F,,&F)F,F,F,F,F.0F)\"\"!0F-F]q0F_oF]q0FDF]q0*&F@F,FFF,F]q" }{TEXT 254 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 142 "The motion is thus described by two coupled second order differential equations \+ in y and Y. The solution depends thus on 4 initial conditions." }} {PARA 216 "" 0 "" {TEXT 235 76 "The Lagrange multipliers are determine d by the solutions of these equations:" }}}}}{SECT 1 {PARA 225 "" 0 "" {TEXT 241 14 "Lie symmetries" }}{SECT 1 {PARA 226 "" 0 "" {TEXT 242 48 "Determining the symmetries by \"pattern matching\"" }}{PARA 216 "" 0 "" {TEXT 235 77 "These kinds of methods were investigated by E. Che b Terrab. The reference is " }{TEXT 249 72 "P.Olver, Applications of L ie Groups to Differential Equations (Springer)" }{TEXT 235 1 "." } {TEXT 235 28 "\nWe look for conditions on " }{XPPEDIT 18 0 "F(x,y)" " 6#-%\"FG6$%\"xG%\"yG" }{TEXT 246 1 " " }{TEXT 235 54 " under which the differential equation of first order " }{XPPEDIT 18 0 "diff(y(x),x) = F(x,y(x));" "6#/-%%diffG6$-%\"yG6#%\"xGF*-%\"FG6$F*F'" }{TEXT 246 1 " " }}{PARA 216 "" 0 "" {TEXT 235 41 "has a symmetry group with infinet esimals " }{XPPEDIT 18 0 " [a*x-y,x+a*y" "6#7$,&*&%\"aG\"\"\"%\"xGF'F' %\"yG!\"\",&F(F'*&F&F'F)F'F'" }{TEXT 246 1 " " }{TEXT 235 88 ", a an a rbitrary constant. If so the equation above can be transformed to a q uadrature " }{XPPEDIT 18 0 "diff(v(u),u)=G(u)" "6#/-%%diffG6$-%\"vG6#% \"uGF*-%\"GGF)" }{TEXT 246 1 " " }{TEXT 235 31 " with the change of va riables: " }{XPPEDIT 18 0 "u=1/2*ln(x^2+y^2)-a, v=arctan(y/x)" "6$/%\" uG,&*(\"\"\"F'\"\"#!\"\"-%#lnG6#,&*$%\"xGF(F'*$%\"yGF(F'F'F'%\"aGF)/% \"vG-%'arctanG6#*&F1F'F/F)" }{TEXT 246 1 " " }{TEXT 235 79 " .This can thus be used to create new classification and method of resolution. " }{TEXT 235 1 "\n" }}{PARA 216 "" 0 "" {TEXT 235 61 "Such infinitesima ls are defined by the differential system: " }{XPPEDIT 18 0 "xi[y]=-1 , x*xi[x]-xi=y, eta[x]=1, y*eta[y]-eta=-x, eta[y]=xi[x]" "6'/&%#xi G6#%\"yG,$\"\"\"!\"\"/,&*&%\"xGF)&F%6#F.F)F)F%F*F'/&%$etaGF0F)/,&*&F'F )&F3F&F)F)F3F*,$F.F*/F7F/" }{TEXT 246 1 " " }}{PARA 216 "" 0 "" {TEXT 235 26 "We look for conditions on " }{XPPEDIT 18 0 "F(x,y)" "6#-%\"FG6 $%\"xG%\"yG" }{TEXT 246 1 " " }{TEXT 235 1 "." }}{PARA 216 "" 0 "" {TEXT 235 1 " " }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 72 "pde := DEtools[odepde](diff(y(x),x)=F(x,y(x)),y(x), [xi(x,y),eta(x,y)]);" }} {PARA 224 "" 1 "" {XPPMATH 20 "6#>%$pdeG,,-%%diffG6$-%$etaG6$%\"xG%\"y GF,\"\"\"*&,&-F'6$F)F-F.-F'6$-%#xiGF+F,!\"\"F.-%\"FGF+F.F.*&-F'6$F5F-F .)F8\"\"#F.F7*&F5F.-F'6$F8F,F.F7*&F)F.-F'6$F8F-F.F7" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 82 "Rtemp:= differential_ ring(ranking=[[xi,eta],F], derivations=[x,y], notation=diff):" }}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 92 "S := [xi[y]+1, x*xi[x]-x i-y, eta[x]-1, y*eta[y]-eta+x, eta[y]-xi[x], denote(pde,jet,Rtemp)];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG7(,&&%#xiG6#%\"yG\"\"\"F+F+,( *&%\"xGF+&F(6#F.F+F+F(!\"\"F*F1,&&%$etaGF0F+F+F1,(*&F*F+&F4F)F+F+F4F1F .F+,&F7F+F/F1,.F3F+*&%\"FGF+F7F+F+*&F;F+F/F+F1*&F'F+)F;\"\"#F+F1*&F(F+ &F;F0F+F1*&F4F+&F;F)F+F1" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 64 "R := differential_ring(ranking=[[xi,eta],F], der ivations=[x,y]):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 43 "G : = Rosenfeld_Groebner(S,R); map(rank, G);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7$%0characterisableGF&" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7$7&%#xiG%$etaG&%\"FG6$%\"xGF*&F(6$F*%\"yG7 '&F&6#F*&F&6#F-F%&F(F0&F(F2" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 27 "There are 2 possibilities. " }}{PARA 216 "" 0 "" {TEXT 235 18 "In the first case," }{XPPEDIT 18 0 " F(x,y)" "6#-%\"FG6$ %\"xG%\"yG" }{TEXT 246 1 " " }{TEXT 235 59 " must be a common zero of \+ the two differential polynomials:" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 40 "equations(G[1])[3]; ;equations(G[1])[4];" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#,^p**)&%\"FG6#%\"yG\"\"#\"\"\")F)\"\"$F+F'F +&F'6#%\"xGF+\"\"%*&F&F+F)F+!\"\"*&F.F+F0F+F+*&,6*()F)F*F+F%F+)F0F*F+F ***F*F+F8F+F&F+F0F+F3*$F9F+F+*&F%F+)F)F1F+F+*,F*F+F8F+F&F+F0F+)F'F*F+F 3*(F*F+F?F+F9F+F+*(F*F+F&F+)F0F-F+F3*&F%F+)F0F1F+F+**F*F+F&F+FBF+F?F+F 3*&F9F+)F'F1F+F+F+&F'6$F0F0F+F+*&)F&F-F+F,F+F3*.F*F+F.F+F9F+F)F+&F'6$F )F)F+F?F+F3**F*F+F.F+F0F+F?F+F+*(F.F+FBF+F%F+F+**F.F+F8F+F%F+F0F+F+*() F.F*F+F=F+FMF+F3**F*F+)F'F-F+F8F+F%F+F+*(FGF+F)F+F&F+F3*(FGF+F0F+F.F+F +*(FKF+F9F+F)F+F3*(FSF+FDF+FMF+F3*,F*F+F?F+F0F+FSF+F)F+F+*,F*F+FSF+F9F +F8F+FMF+F3*,F*F+F?F+F,F+FMF+F.F+F3*,F*F+F&F+F9F+F.F+F?F+F3*,F*F+F?F+F 0F+F%F+F)F+F+**F&F+F9F+FSF+F)F+F3*.F1F+F%F+F9F+F)F+F'F+F.F+F+**F*F+FSF +F9F+F'F+F3**F*F+F&F+F9F+F.F+F3**F*F+FUF+F9F+FSF+F3*&)F.F-F+FBF+F+*,F* F+FMF+F)F+F.F+F9F+F3*,F*F+F&F+F8F+F.F+F?F+F3*(F&F+F,F+FSF+F3*,F1F+F&F+ FBF+FSF+F'F+F+**F*F+F&F+F)F+F?F+F3**F*F+F%F+F8F+F'F+F+*(F_oF+F8F+F0F+F +**F*F+FMF+F8F+F?F+F3**F*F+FMF+F,F+F.F+F3**F*F+FSF+F0F+F)F+F+*&FMF+F8F +F3**F*F+F%F+F)F+F0F+F+**F*F+F&F+F8F+F.F+F3*(FGF+F8F+FMF+F3*.F1F+F&F+F 8F+FSF+F0F+F'F+F+" }{TEXT 254 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6 #,8*&,**&&%\"FG6#%\"yG\"\"\")F*\"\"#F+F+%\"xG!\"\"*&F'F+)F.F-F+F+*&)F( F-F+F.F+F/F+&F(6$F.F*F+F+*,F-F+&F(6#F.F+F.F+F(F+F'F+F+*&)F'F-F+F.F+F+F 'F/*(&F(6$F*F*F+F7F+F1F+F/*&)F7F-F+F.F+F+*&F'F+F3F+F/*&F " 0 "" {MPLTEXT 1 238 26 "rewrite_rules(G[1])[1..2 ];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7$/%#xiG,$*&,**&&%\"FG6#%\"yG\" \"\")F-\"\"#F.F.%\"xG!\"\"*&F*F.)F1F0F.F.*&)F+F0F.F1F.F2F.,&*&&F+6#F1F .F1F.F.*&F*F.F-F.F.F2F2/%$etaG,$*&,*F-F2*&F6F.F-F.F2*&F9F.F4F.F2*&F9F. F/F.F2F.F7F2F2" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 71 "The second case corresponds in fact to the cases where the sol ution to " }{XPPEDIT 18 0 "diff(y(x),x)=F(x,y)" "6#/-%%diffG6$-%\"yG6# %\"xGF*-%\"FG6$F*F(" }{TEXT 246 1 " " }{TEXT 235 39 " is given by the \+ change of coordinates:" }{TEXT 235 2 "\n " }{XPPEDIT 18 0 "1/2*ln(x^2+ y^2)+a*arctan(y/x)=C" "6#/,&*(\"\"\"F&\"\"#!\"\"-%#lnG6#,&*$%\"xGF'F&* $%\"yGF'F&F&F&*&%\"aGF&-%'arctanG6#*&F0F&F.F(F&F&%\"CG" }{TEXT 246 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 20 "rewrite_rules(G[2] );" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7'/&%$etaG6#%\"xG\"\"\"/&F&6#% \"yG,$*&,&F&!\"\"F(F)F)F-F1F1/%#xiG,$*&,(*&F(F)F&F)F1*$)F(\"\"#F)F)*$) F-F:F)F)F)F-F1F1/&%\"FGF',$*&,&F-F)*&)F?F:F)F-F)F)F),&F;F)F8F)F1F1/&F? F,,$*&,&F(F1*&FDF)F(F)F1F)FEF1F1" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 29 "Practically: let us be given " }{XPPEDIT 18 0 "F(x,y)" "6#-%\"FG6$%\"xG%\"yG" }{TEXT 246 1 " " }{TEXT 235 5 ". If \+ " }{XPPEDIT 18 0 "F(x,y) " "6#-%\"FG6$%\"xG%\"yG" }{TEXT 246 1 " " } {TEXT 235 103 "satisfies one of the differential systems output, we wi ll be able to reduce the differential equations " }{XPPEDIT 18 0 "diff (y(x),x) = F(x,y)" "6#/-%%diffG6$-%\"yG6#%\"xGF*-%\"FG6$F*F(" }{TEXT 246 1 " " }{TEXT 235 59 " to a quadrature with the change of variable s given above." }}}}{PARA 216 "" 0 "" }{SECT 0 {PARA 226 "" 0 "" {TEXT 242 34 "Reducing the determining equations" }{TEXT 242 1 "\n" }} {PARA 216 "" 0 "" {TEXT 235 699 "This is a successfull application of \+ differential elimination that has been brought by E.Mansfield, G. Reid , F. Schwartz and others. When making the symmetry analysis of a diffe rential equations, the infinitesimals are defined by an overdetermined system of PDEs. The reduction of such system often leads to closed fo rm solutions. A proper differential elimination is anyway vital to de termine if there does exist any symmetries. For classical symmetry ana lysis the system determing the infinetesimas is linear (as the one bel ow) but when one tries non-classical (or dynamical) symmetries, the s ytem satified by the infinetesimals is non-linear (see work by P. Clar kson, E. Mansfield and G. Reid)." }}{PARA 216 "" 0 "" }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 14 "with(liesymm):" }}{PARA 223 "" 1 "" {TEXT 239 43 "Warning, the name reduce has been redefined" }{TEXT 239 1 "\n" }}{PARA 223 "" 1 "" {TEXT 239 68 "Warning, the protected name c lose has been redefined and unprotected" }{TEXT 239 1 "\n" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 69 "Burgers_pde := Diff(u(t,x),t)=D iff(u(t,x),x,x)-u(t,x)*Diff(u(t,x),x);" }{MPLTEXT 1 238 1 "\n" }} {PARA 224 "" 1 "" {XPPMATH 20 "6#>%,Burgers_pdeG/-%%DiffG6$-%\"uG6$%\" tG%\"xGF,,&-F'6$F)-%\"$G6$F-\"\"#\"\"\"*&F)F5-F'6$F)F-F5!\"\"" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 57 "symm_eq := \+ determine(Burgers_pde, `V`, u(t,x), [ux, ut]):" }{MPLTEXT 1 238 84 "\n Q := differential_ring(ranking=[lex[V3,V2,V1]],derivations=[t,x,u], no tation=Diff):" }{MPLTEXT 1 238 83 "\nS := subs(V1=xi,V2=tau,V3=phi,map (denote, map(x->rhs(x)-lhs(x),symm_eq), jet, Q));" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG<+,&&%$tauG6#%\"uG!\"\"&%#xiG6$%\"xGF*F+,$&F-6#F /F+,$&F-6$F*F*F+,$&F-F)F+,$&F(F5F+,(&F(F2\"\"#&F-6$F/F/\"\"\"&F-6#%\"t GF+,(&%$phiGF>F?*&F*F?&FEF2F?F+&FEFAF+,(&F(F.F<&FEF5F+*(FF?*&F " 0 "" {MPLTEXT 1 238 70 "R := differential_ring(r anking=[lex[phi,tau,xi]],derivations=[t,x,u]):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 27 "G:=Rosenfeld_Groebner(S,R);" }{MPLTEXT 1 238 21 "\nrewrite_rules(G[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% \"GG7#%0characterisableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7,/&%$phiG6#%\"tG,$*&%\"uG\"\"\"&F&6#%\"xGF,!\"\"/&%$ta uGF',&F&F,*&F+F,&F&6#F+F,F0/&%#xiGF',$F6!\"#/&F&6$F/F/\"\"!/&F&6$F/F+F @/&F3F.,$F6F0/&F:F.F@/&F&6$F+F+F@/&F3F7F@/&F:F7F@" }{TEXT 240 1 " " }} }{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 70 "R := differential_ring( ranking=[lex[phi,tau,xi]],derivations=[u,x,t]):" }}}{EXCHG {PARA 216 " > " 0 "" {MPLTEXT 1 238 27 "G:=Rosenfeld_Groebner(S,R);" }{MPLTEXT 1 238 21 "\nrewrite_rules(G[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% \"GG7#%0characterisableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7+/&%$phiG6#%\"uG,$&%#xiG6#%\"tG#!\"\"\"\"#/&%$tauGF'\" \"!/&F+F'F4/&F&6#%\"xG,$*&&F&F,\"\"\"F(F/F//&F3F9,$F*#F>F0/&F+F9F4/&F& 6$F-F-F4/&F+FG,$FF(F>FBF&F>" }{TEXT 240 1 " " }}} {PARA 216 "" 0 "" }}}{SECT 0 {PARA 225 "" 0 "" {TEXT 241 63 " Identifi ability and observability in non-linear control system" }}{PARA 216 "" 0 "" {TEXT 235 109 "A control system is usually first modelled as a \+ system of the following form, called the \"state-space\" form:" }} {PARA 216 "" 0 "" {TEXT 235 37 " " }{XPPEDIT 18 0 "diff(X,t) = F(X, U, P);" "6#/-%%diffG6$%\"XG%\"tG-%\" FG6%F'%\"UG%\"PG" }{TEXT 246 1 " " }}{PARA 216 "" 0 "" {TEXT 235 42 " \+ " }{XPPEDIT 18 0 "Y = G(X, U, P);" "6#/%\"YG-%\"GG6%%\"XG%\"UG%\"PG" }{TEXT 246 1 " " }}{PARA 216 " " 0 "" {TEXT 235 23 "where U is a vector of " }{TEXT 249 16 "input var iables " }{TEXT 235 92 "(the commands or things that can be controled \+ by the operator), P is a vector of parameters," }}{PARA 216 "" 0 "" {TEXT 235 17 "X is a vector of " }{TEXT 249 15 "sate variables " } {TEXT 235 79 "(things that cannot be observed or measured directly) wh ile Y is the vector of " }}{PARA 216 "" 0 "" {TEXT 235 38 "output vari ables that can be observed." }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 235 18 "The problems are: " }}{PARA 216 "" 0 "" {TEXT 235 20 " \+ - can we compute " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 246 1 " " } {TEXT 235 15 " directly from " }{XPPEDIT 18 0 "U" "6#%\"UG" }{TEXT 246 1 " " }{TEXT 235 50 "? i.e. what is the differential system relat ing " }{XPPEDIT 18 0 "U" "6#%\"UG" }{TEXT 246 1 " " }{TEXT 235 5 " t o " }{XPPEDIT 18 0 "Y" "6#%\"YG" }{TEXT 246 1 " " }{TEXT 235 15 ", ca lled the i" }{TEXT 249 11 "nput-output" }{TEXT 235 18 " representati on." }}{PARA 216 "" 0 "" {TEXT 235 24 " - are the parameters " } {XPPEDIT 18 0 "identifiable" "6#%-identifiableG" }{TEXT 246 1 " " } {TEXT 235 26 " i.e can be computed from " }{XPPEDIT 18 0 "U" "6#%\"UG" }{TEXT 246 1 " " }{TEXT 235 5 " and " }{XPPEDIT 18 0 "Y" "6#%\"YG" } {TEXT 246 1 " " }{TEXT 235 1 "." }}{PARA 216 "" 0 "" {TEXT 235 19 " \+ - is the system " }{TEXT 249 10 "observable" }{TEXT 235 59 " i.e. can \+ the values of X can be deduced from the value of " }{TEXT 249 1 "U" } {TEXT 235 5 " and " }{TEXT 249 2 "Y " }{TEXT 235 36 "and their deriv atives at any time?" }}{PARA 216 "" 0 "" {TEXT 235 1 " " }}{SECT 1 {PARA 226 "" 0 "" {TEXT 242 14 "Simple example" }}{EXCHG {PARA 216 "> \+ " 0 "" {MPLTEXT 1 238 66 "S := [x[1][t]-x[1]-lambda*u, x[2][t]-x[2]*(1 -x[1]),y-lambda*x[1]];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG7%,(& &%\"xG6#\"\"\"6#%\"tGF+F(!\"\"*&%'lambdaGF+%\"uGF+F.,&&&F)6#\"\"#F,F+* &F4F+,&F+F+F(F.F+F.,&%\"yGF+*&F0F+F(F+F." }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 18 "X := [x[1], x[2]];" }{MPLTEXT 1 238 10 "\nY := [y];" }{MPLTEXT 1 238 15 "\nP := [lambda];" } {MPLTEXT 1 238 10 "\nU := [u];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% \"XG7$&%\"xG6#\"\"\"&F'6#\"\"#" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"YG7#%\"yG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"PG7#%'lambdaG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"UG7#%\"uG" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 71 "R := differential_ring(ranking=[P,X,Y,U],pa rameters=P,derivations=[t]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"R G%)ODE_ringG" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 29 "C := Rosenfeld_Groebner(S,R);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"CG7$%0characterisableGF&" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 20 "rewrite_rules(C[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7*/%'lambdaG,$*&,&*&&%\"xG6#\"\"\"F-&%\"yG6#%\"tGF -!\"\"*&F/F-F*F-F-F-*&%\"uGF-F/F-F2F2/&&F+6#\"\"#F0,&F8F-*&F8F-F*F-F2/ *$)F*F:F-*&*&)F/F:F-F5F-F-,&F.F-F/F2F2/&F/6$F1F1,$*&,(*&&F5F0F-F/F-F-* &F5F-F.F-F2*&FKF-F.F-F2F-F5F2F20F/\"\"!0F5FO0F*FO0FCFO" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 20 "rewrite_rules(C[2] );" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7&/%'lambdaG\"\"!/&&%\"xG6#\"\" #6#%\"tGF)/&F*6#\"\"\"F&/%\"yGF&" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "" 0 "" {TEXT 235 12 "Generically " }{XPPEDIT 18 0 "lambda" "6#%'l ambdaG" }{TEXT 246 1 " " }{TEXT 235 29 " is locally identifiable and " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT 246 1 " " }{TEXT 235 29 " is locally observable, but " }{XPPEDIT 18 0 "x[2]" "6#&%\"xG6#\" \"#" }{TEXT 246 1 " " }{TEXT 235 8 " is not." }}}}{SECT 0 {PARA 226 "" 0 "" {TEXT 242 26 "An non identifiable model " }}{PARA 216 "" 0 "" {TEXT 235 11 "taken from " }}{PARA 216 "" 0 "" {TEXT 249 92 "F. Ollivi er, A. Sedoglavic Algorithmes efficaces pour tester l'identifiabilite \+ locale (2002)" }}{PARA 216 "" 0 "" {TEXT 235 14 "and originally" }} {PARA 231 "" 0 "" {TEXT 255 128 "S. Vajda, K.Godfrey, H. Rabitz , Simi larity Tranformation Approach to identifiability Analysis of Nonlinear Compartmental Models" }{TEXT 255 32 "\nMathematical Bioscience (1989) ." }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 88 "S := [x[1][t]-p[1]* x[1]^2-p[2]*x[1]*x[2]-u, x[2][t]-p[3]*x[1]^2+p[4]*x[1]*x[2], y-x[1]];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG7%,*&&%\"xG6#\"\"\"6#%\"tGF+ *&&%\"pGF*F+)F(\"\"#F+!\"\"*(&F06#F2F+F(F+&F)F6F+F3%\"uGF3,(&F7F,F+*&& F06#\"\"$F+F1F+F3*(&F06#\"\"%F+F(F+F7F+F+,&%\"yGF+F(F3" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 58 "P := [seq(p[i],i=1 ..4)]; X := [x[1],x[2]]; Y:=[y]; U:=[u];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"PG7&&%\"pG6#\"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" } {TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"XG7$&%\"xG6#\" \"\"&F'6#\"\"#" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% \"YG7#%\"yG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"U G7#%\"uG" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 74 "R0 := differential_ring(ranking=[Y,X,U,P], derivations=[t], pa rameters=P):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 32 "G0 := R osenfeld_Groebner(S, R0);" }{MPLTEXT 1 238 22 "\nrewrite_rules(G0[1]); " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#G0G7#%0characterisableG" } {TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7%/%\"yG&%\"xG6#\" \"\"/&F&6#%\"tG,(*&&%\"pGF(F))F&\"\"#F)F)*(&F16#F3F)F&F)&F'F6F)F)%\"uG F)/&F7F,,&*&&F16#\"\"$F)F2F)F)*(&F16#\"\"%F)F&F)F7F)!\"\"" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 73 "R := differenti al_ring(ranking=[P,X,Y,U], derivations=[t], parameters=P):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 13 "Time:=time():" }{MPLTEXT 1 238 31 "\nG := Rosenfeld_Groebner(S, R);" }{MPLTEXT 1 238 19 "\nTime:=time ()-Time;" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7#%0characterisable G" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%%TimeG$\"&T \\)!\"$" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 13 "map(rank, G);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7#7)&%\"pG6# \"\"\"&F&6#\"\"#&F&6#\"\"$&F&6#\"\"%&&%\"xGF*6#%\"tG&F4F'&%\"yG6'F6F6F 6F6F6" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 29 "map(indets, equations(G[1]));" }}{PARA 230 "" 1 "" {XPPMATH 20 "6# 7)<,%\"yG%\"uG&F&6$%\"tGF)&F&6%F)F)F)&F%F(&F&6#F)&F%F+&F%6&F)F)F)F)&F% F.&%\"pG6#\"\"\"<-F%F&F'F*F,F-F/F0F2&F46#\"\"#&%\"xGF9<-F%F&F'F*F,F-F/ F0F2F;&F46#\"\"$<,F%F&F'F*F,F-F/F0F2&F46#\"\"%<-F%F&F'F*F,F-F/F0F2F;&F ;F.<$F%&F " 0 "" {MPLTEXT 1 238 32 "for g in rewrite_rules (G[1]) do " }{MPLTEXT 1 238 41 "\ncollect(g, [y[t,t,t,t]],factor); end do;" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#/&%\"pG6#\"\"\",&*(,**&%\"yGF '&%\"uG6#%\"tGF'!\"\"*&\"\"#F')&F,F/F3F'F1*(F3F'F5F'F.F'F'*&F,F'&F,6$F 0F0F'F'F',2**F3F'F,F'&F.F9F'F4F'F1*()F,F3F'F-F'&F,6%F0F0F0F'F1*(F>F'F8 F'FF'FDF'F5F'F.F'F'*,F3F'F>F'F 5F')F.F3F'F?F'F1*,\"#SF'F,F'F8F'FHF'F.F'F1*,\"#=F'F>F'F8F'F-F'F4F'F'*, F3F'FOF'F8F'F?F'F.F'F'*,\"#?F'F,F'F-F'FHF'F.F'F'**FOF'F-F'F?F'F.F'F'*, 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F,F,F,&F26&F*F*F*F*F,F6F,F2F6,fo*(FHF,FhpF,F]qF,F6*,F>F,FHF,F?F,F1F,FJ F,F6*,FTF,FBF,FCF,F1F,F5F,F,*,FTF,FBF,F1F,F4F,FJF,F6*,\"#UF,F2F,F?F,F= F,F5F,F6*,\"#?F,FBF,F?F,FIF,FDF,F,*,F(F,FHF,F?F,FJF,F5F,F,*,FjoF,F2F,F IF,F=F,F5F,F,*,F(F,FHF,FIF,FJF,F5F,F,*,F[oF,F2F,FDF,F4F,F?F,F,*&FZF,Fg nF,F,*(F3F,FHF,FWF,F,*(F/F,FfoF,F4F,F6**F]oF,F2F,F?F,FfoF,F,*(FZF,F?F, FMF,F,**F[oF,FBF,FCF,FDF,F6**\"#5F,FHF,FCF,FIF,F6**FjoF,F2F,FIF,FfoF,F 6*(FZF,FIF,FMF,F6**F3F,FHF,FOF,F?F,F,**FTF,FBF,FOF,FDF,F6**F(F,FHF,FDF ,FMF,F6**F:F,FBF,F=F,FJF,F6**F(F,FZF,FJF,FFF,F6**FTF,FBF,FCF,F4F,F,**F (F,FBF,F=F,FFF,F,*.F[oF,FBF,FIF,F1F,F5F,F?F,F6*,F(F,FHF,F1F,F5F,FMF,F, *,FcrF,FBF,FDF,F5F,FJF,F,*,F(F,FBF,FDF,F5F,FFF,F6*,F3F,FHF,FFF,F1F,F?F ,F,**FHF,FFF,F1F,FIF,F6*,FTF,FHF,FFF,F?F,F5F,F6*(F8F,F0F,F5F,F,*&F/F,F 9F,F6*&FZF,F`pF,F,**FHF,FIF,F1F,FJF,F6F6" }{TEXT 254 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#/&%\"xG6#\"\"\"%\"yG" }{TEXT 240 1 " " }} {PARA 230 "" 1 "" {XPPMATH 20 "6#/&%\"yG6'%\"tGF'F'F'F',(**F%\"\"\",&* (\"\"#F*&F%6#F'F*%\"u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oF*F8F*F1*,F-F*FYF*F.F*F0F*FjnF*F**,FTF*FYF*F>F*F;F*F 0F*F***FenF*F:F*FEF*F6F*F**&FVF*)F6F-F*F**(FVF*F>F*FjnF*F1*(FVF*F3F*Fj nF*F1*(FMF*FSF*FPF*F1*(FMF*)F.FXF*F0F*F*F*F4F1FGF*F1*(,fv*.\"#gF*F0F*F AF*FEF*F:F*F3F*F1*.F`pF*F0F*F%F*F3F*F\\pF*F>F*F***\"#[F*F%F*F_oF*)F.Fe nF*F1**\"#7F*FPF*FYF*)F>FTF*F1**F-F*)F%FXF*F0F*)F;FFF*F***F-F*)FjnF-F* F8F*FipF*F***FMF*FjnF*F:F*FdpF*F**(FcpF*F3F*)F.FOF*F1**FcpF*FPF*F>F*Fd pF*F1**FcpF*F0F*F>F*)F.\"\"(F*F***&F0FHF*F6F*F>F*)F%FenF*F**.F-F*FeqF* FipF*F3F*F.F*F>F*F**(FfqF*FjnF*)F;F-F*F1**FenF*FgoF*FAF*FipF*F1**FcpF* F6F*FbqF*F%F*F***\"#CF*FgoF*FYF*FSF*F1**FXF*FjnF*FipF*)F>FFF*F1*,\"#yF *FYF*F3F*F8F*F_rF*F1*,FarF*F0F*FYF*FgpF*F.F*F***FfqF*F6F*FjnF*F;F*F**, F-F*F.F*FipF*F;F*FgoF*F**,F-F*FeqF*F6F*F8F*FipF*F1*,\"$K\"F*F0F*F:F*F_ rF*FEF*F1*.FOF*F;F*FPF*FAF*F.F*FYF*F**.\"#aF*F0F*FYF*F_rF*F3F*F.F*F1*. 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FjrF*F6F*FYF*F3F*FEF*F>F*F1*.FhnF*FjnF*FYF*F>F*FEF*F0F*F*F*F%!\"%F4F1F *" }{TEXT 254 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#0,2**\"\"#\"\" \"%\"yGF'&%\"uG6$%\"tGF,F')&F(6#F,F&F'!\"\"*()F(F&F'&F*F/F'&F(6%F,F,F, F'F0*(F2F'&F(F+F'F)F'F'**F&F'F-F'F7F'F*F'F0**F&F'F(F'F*F')F7F&F'F0*,F& F'F(F'F3F'F7F'F.F'F'*,F&F'F(F'F*F'F4F'F.F'F'*(F&F'F3F')F.\"\"$F'F'\"\" !" }{TEXT 240 1 " " }}{PARA 230 "" 1 "" {XPPMATH 20 "6#0,fo*()%\"yG\" \"$\"\"\",**&F'F)&%\"uG6#%\"tGF)!\"\"*&\"\"#F))&F'F.F2F)F0*(F2F)F4F)F- F)F)*&F'F)&F'6$F/F/F)F)F)&F'6&F/F/F/F/F)F)*,F(F)F&F)F7F)F4F)&F'6%F/F/F /F)F)*,\"\"%F))F'F2F))F7F2F)F4F)F-F)F0*,F?F)F@F)F4F))F-F2F)F " 0 "" {MPLTEXT 1 238 80 "R1 := di fferential_ring(ranking=[P,X,p[23],Y,U], derivations=[t], parameters=P ):" }{MPLTEXT 1 238 14 "\nTime:=time():" }{MPLTEXT 1 238 94 "\nG1 := R osenfeld_Groebner([p[23]-p[2]*p[3],op(equations(G[1]))],inequations(G[ 1]), R1, G0[1]);" }{MPLTEXT 1 238 19 "\nTime:=time()-Time;" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#G1G7#%0characterisableG" }{TEXT 240 1 " \+ " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%%TimeG$\"&+)>!\"$" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 14 "map(rank, G1);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7#7*&%\"pG6#\"\"\"&F&6#\"\"#&F&6# \"\"$&F&6#\"\"%&&%\"xGF*6#%\"tG&F4F'&F&6#\"#B&%\"yG6'F6F6F6F6F6" } {TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 30 "map(i ndets, equations(G1[1]));" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#7*<,%\"y G%\"uG&F&6$%\"tGF)&F&6%F)F)F)&F%F(&F&6#F)&F%F+&F%6&F)F)F)F)&F%F.&%\"pG 6#\"\"\"<-F%F&F'F*F,F-F/F0F2&F46#\"\"#&%\"xGF9<-F%F&F'F*F,F-F/F0F2F;&F 46#\"\"$<,F%F&F'F*F,F-F/F0F2&F46#\"\"%<-F%F&F'F*F,F-F/F0F2F;&F;F.<$F%& F " 0 "" {MPLTEXT 1 238 33 "fo r g in rewrite_rules(G1[1]) do " }{MPLTEXT 1 238 41 "\ncollect(g, [y[t ,t,t,t]],factor); end do;" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#/&%\"pG6 #\"\"\",&*(,**&\"\"#F')&%\"yG6#%\"tGF,F'!\"\"*&F/F'&%\"uGF0F'F2*&F/F'& F/6$F1F1F'F'*(F,F'F.F'F5F'F'F',2*(&F5F8F'F7F')F/F,F'F'**F,F'F5F')F7F,F 'F/F'F2*(F4F'&F/6%F1F1F1F'F=F'F2*,F,F'F7F'F.F'F4F'F/F'F'*,F,F'F5F'FAF' F.F'F/F'F'**F,F'FF+F.F+F*F+F2**F'F+F5F+F.F+F7F+F2*(F'F+)F/\"\"$F+F4F+F+F2&F*6& F1F1F1F1F+F2**,do**\"\"%F+F?F+FAF+)F5F'F+F+**F'F+)F*FQF+FCF+F>F+F2**F' F+F?F+FJF+F>F+F+**\"\"'F+F?F+FJF+FCF+F2**F'F+)F*FKF+F.F+&F5FDF+F2**FQF +F?F+)F4F'F+F.F+F2**\"\"&F+FYF+FfnF+F7F+F+**\"#=F+F*F+F4F+)F/FQF+F2** \"#5F+FYF+FAF+F4F+F2**\"#7F+F?F+FAF+F.F+F2*(FTF+F7F+FZF+F+**\"#IF+F*F+ F7F+F[oF+F+*(FhnF+FYF+)F7FKF+F+*&FTF+)FCF'F+F+*&\"#;F+)F/FWF+F2*,FQF+F ?F+FAF+F/F+F5F+F+*,FKF+FYF+F7F+F/F+FCF+F2*(FhoF+F[oF+FRF+F2*(\"#KF+)F/ FhnF+F5F+F+*,FQF+F?F+F/F+FRF+FCF+F2*.F_oF+F?F+F4F+F/F+F5F+F7F+F2*,\"#U F+F*F+F7F+FJF+F5F+F2*,FQF+FYF+F>F+F7F+F5F+F2**FYF+F>F+F/F+F4F+F2*,FhnF +FYF+F>F+F/F+F7F+F+*,F'F+F?F+F.F+F5F+F>F+F2*,F]oF+F?F+F.F+F5F+FCF+F+*, F'F+FYF+F/F+F5F+FZF+F+*,F_oF+F*F+F.F+FRF+F7F+F+*,F'F+FYF+F4F+FCF+F5F+F +*,FjnF+F*F+F4F+FJF+F5F+F+*,F'F+FYF+F7F+FCF+F5F+F+*,\"#?F+F?F+F7F+F4F+ F.F+F+**FYF+F4F+F/F+FCF+F2*(FTF+F4F+FZF+F2*&FTF+)F>F'F+F+F+F*!\"#F:F2F 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" }{TEXT 240 1 " " }}}}}{SECT 0 {PARA 226 "" 0 "" {TEXT 242 39 "Identi fiability in models of bioscience" }}{PARA 216 "" 0 "" {TEXT 235 61 "T he following examples are extracted from the study cases of " }}{PARA 233 "" 0 "" {TEXT 257 187 "G. Margaria, E. Riccomagno, M. Chappell, H. Wynn, Differential Algebra Method for the study of the structural ide ntifiability of Rational Polynomial State-Space Model in Bioscience (2 001)" }}{SECT 0 {PARA 232 "" 0 "" {TEXT 256 65 "One compartment model \+ with non-liear Michaelis-Menten (example 5)" }}{PARA 216 "" 0 "" {TEXT 235 45 "The model is globally identifiable in general" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 37 "S := [(p[3]+x)*(x[t]+p[2]*x)+p[ 1]*x];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG7#,&*&,&&%\"pG6#\"\"$ \"\"\"%\"xGF-F-,&&F.6#%\"tGF-*&&F*6#\"\"#F-F.F-F-F-F-*&&F*6#F-F-F.F-F- " }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 21 "P \+ :=[p[1],p[2],p[3]];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"PG7%&%\"pG 6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> \+ " 0 "" {MPLTEXT 1 238 68 "R0 := differential_ring(ranking=[x,P],deriva tions=[t],parameters=P):" }{MPLTEXT 1 238 43 "\nG0 := Rosenfeld_Groebn er(S,\{p[3]+x\}, R0); " }{MPLTEXT 1 238 23 "\nrewrite_rules(G0[1]); " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#G0G7#%0characterisableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7$/&%\"xG6#%\"tG,$*&,(*(& %\"pG6#\"\"$\"\"\"&F.6#\"\"#F1F&F1F1*&F2F1)F&F4F1F1*&&F.6#F1F1F&F1F1F1 ,&F-F1F&F1!\"\"F;0F:\"\"!" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 68 "R := differential_ring(ranking=[P,x], derivatio ns=[t],parameters=P):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 85 "Time := time(): G := Rosenfeld_Groebner(S,\{x+p[3]\}, R, G0[1]); T ime := time()-Time();" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7#%0ch aracterisableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>% %TimeG$\"#\"*!\"$" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 11 "rank(G[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7&& %\"pG6#\"\"\"&F%6#\"\"#&F%6#\"\"$&%\"xG6&%\"tGF1F1F1" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 20 "rewrite_rules(G[1]); " }}{PARA 230 "" 1 "" {XPPMATH 20 "6#7*/&%\"pG6#\"\"\",$*&,**&&F&6#\" \"$F(&%\"xG6#%\"tGF(F(*(F-F(&F&6#\"\"#F(F1F(F(*&F1F(F0F(F(*&F5F()F1F7F (F(F(F1!\"\"F;/F5,$*&,(*(F1F(F-F(&F16$F3F3F(F(*&F:F(FAF(F(*&F-F()F0F7F (F;F(*&F:F(F0F(F;F;/F-,$*&,&*&)F1F/F()FAF7F(F;*(FLF(F0F(&F16%F3F3F3F(F (F(,**(F1F(FEF(FAF(!\"#*&F7F()F0\"\"%F(F(*&F:F(FMF(F;*(F:F(F0F(FOF(F(F ;F;/&F16&F3F3F3F3,$*&,0**F1F()F0F/F(FAF(FOF(\"#9*,F7F(F:F(F0F(FMF(FOF( F;**\"#7F(F1F(FEF()FAF/F(F;*(\"\"'F()F0\"\"&F(FOF(F;*(FaoF(FUF(FMF(F(* (F/F(F:F()FAFVF(F(**F/F(F:F(FEF()FOF7F(F;F(,&*(F:F(FEF(FAF(F7*(F7F(F1F (FUF(F;F;F;0F0\"\"!0F1F]p0,&*&F1F(FAF(F(*$FEF(F;F]p0FQF]p" }{TEXT 254 1 " " }}}{EXCHG {PARA 216 "> " 0 "" }}}{PARA 216 "" 0 "" }{SECT 0 {PARA 232 "" 0 "" {TEXT 256 36 "Microbial growth in a batch reactor " }}{PARA 216 "" 0 "" {TEXT 235 34 "The model is globally identifiable" }}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 96 "S := [ (p[2]+x[2])*(x[ 1][t]+p[3]*x[1])-p[1]*x[2]*x[1], (p[2]+x[2])*x[2][t]+p[4]*p[1]*x[2]*x[ 1]];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG7$,&*&,&&%\"pG6#\"\"#\" \"\"&%\"xGF+F-F-,&&&F/6#F-6#%\"tGF-*&&F*6#\"\"$F-F2F-F-F-F-*(&F*F3F-F. F-F2F-!\"\",&*&F(F-&F.F4F-F-**&F*6#\"\"%F-F;F-F.F-F2F-F-" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 25 "P := [seq(p[i], i=1..4)];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"PG7&&%\"pG6#\"\"\"& F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 80 "R0 := differential_ring(ranking=[[x[1],x[2] ], P], derivations=[t],parameters=P):" }{MPLTEXT 1 238 46 "\nG0 := Ros enfeld_Groebner(S,\{p[2]+x[2]\}, R0); " }{MPLTEXT 1 238 22 "\nrewrite_ rules(G0[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#G0G7#%0characteri sableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7%/&&%\"xG 6#\"\"\"6#%\"tG,$*&,(*(&%\"pG6#\"\"#F)&F16#\"\"$F)F&F)F)*(&F'F2F)F4F)F &F)F)*(&F1F(F)F8F)F&F)!\"\"F),&F0F)F8F)F;F;/&F8F*,$*&**&F16#\"\"%F)F:F )F8F)F&F)F)F " 0 "" {MPLTEXT 1 238 78 "R := differential_ring(ranking=[P,[x[1],x[2]]], derivations=[t],parameters=P):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 16 "Time := time(): " }{MPLTEXT 1 238 48 "\nG := Rosenf eld_Groebner(S,\{p[2]+x[2]\}, R, G0); " }{MPLTEXT 1 238 23 "\nTime := \+ time()-Time();" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7#%0character isableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%%TimeG$ \"$S\"!\"$" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 12 "map(rank,G);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7#7(&%\"pG6# \"\"\"&F&6#\"\"#&F&6#\"\"$&F&6#\"\"%&&%\"xGF'6%%\"tGF6F6&&F4F*F5" } {TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 20 "rewri te_rules(G[1]);" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#7./&%\"pG6#\"\"\", $*&,**&&F&6#\"\"#F(&&%\"xGF'6#%\"tGF(!\"\"*(F-F(&F&6#\"\"$F(F1F(F5*&&F 2F.F(F0F(F5*(F;F(F7F(F1F(F5F(*&F;F(F1F(F5F5/F-,$*&,&**&F&6#\"\"%F()F;F /F(F1F(&F16$F4F4F(F(*(FCF(FFF()F0F/F(F5F(,(**FCF(F;F(F1F(FGF(F(*(FCF(F ;F(FJF(F5*&)&F;F3F/F(F1F(F(F5F5/F7,$*&,&FPF(*&FCF(F0F(F(F(*&FCF(F1F(F5 F5/FC,$*&,&*&F1F(&F;FHF(F(*&F0F(FPF(F5F(,&*&F1F(FGF(F(*$FJF(F5F5F5/&F1 6%F4F4F4,$*&,,*(FOF(F1F(FGF(F/*(F/F(FOF(FJF(F5**FPF(F;F(F0F(FGF(F5**F9 F(FfnF(F;F(FJF(F(*,F9F(F1F(FfnF(F;F(FGF(F5F(*(FPF(F;F(F1F(F5F5/&F;F]o, $*&,,*&F0F()FPF9F(!\"#*,F9F(F0F(FPF(FfnF(F;F(F(**F/F(FOF(F1F(FfnF(F(** F9F(F1F()FfnF/F(F;F(F5*(FOF(F;F(FGF(F5F(FfoF5F50FK\"\"!0F;Fep0F1Fep0FP Fep0FhnFep0FCFep" }{TEXT 254 1 " " }}}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" }}{PARA 216 "" 0 "" }{SECT 0 {PARA 232 "" 0 "" {TEXT 256 46 "Immu nological model for mastisis in diary cows" }}{PARA 216 "" 0 "" {TEXT 235 90 "This system is globally identifiable! (not the conclusion of the paper above mentionned)" }}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" {TEXT 235 191 "We need to use the alternative characteristic decomposi tion algorithm based on Kalkbrener's algorithm. The one based on Groeb ner bases fails to complete the computation in a reasonnable time." }} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 60 "S := [x1[t]-p1*x1-p2*x1* x2, x2[t]-p3*x2*(1-p4*x2)+p5*x1*x2];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"SG7$,(&%#x1G6#%\"tG\"\"\"*&%#p1GF+F(F+!\"\"*(%#p2GF+F(F+%#x2GF+ F.,(&F1F)F+*(%#p3GF+F1F+,&F+F+*&%#p4GF+F1F+F.F+F.*(%#p5GF+F(F+F1F+F+" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 29 "P := [seq(p||(6-i), i=1..5)];" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"PG7' %#p5G%#p4G%#p3G%#p2G%#p1G" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 76 "R0 := differential_ring(ranking=[[x1,x2], P], d erivations=[t],parameters=P):" }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 36 "G0 := Rosenfeld_Groebner(S, \{\}, R0);" }{MPLTEXT 1 238 22 " \nrewrite_rules(G0[1]);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%#G0G7#%0 characterisableG" }{TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6# 7$/&%#x1G6#%\"tG,&*&%#p1G\"\"\"F&F,F,*(%#p2GF,F&F,%#x2GF,F,/&F/F',(*&% #p3GF,F/F,F,*(F4F,)F/\"\"#F,%#p4GF,!\"\"*(%#p5GF,F&F,F/F,F9" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 74 "R := differ ential_ring(ranking=[P,[x1,x2]], derivations=[t],parameters=P):" }}} {EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 32 "_Env_diffalg_char:=\"Kal kbrener\";" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%2_Env_diffalg_charGQ+ Kalkbrener6\"" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 16 "Time := time(): " }{MPLTEXT 1 238 36 "\nG := Rosenf eld_Groebner(S, R, G0); " }{MPLTEXT 1 238 23 "\nTime := time()-Time(); " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%\"GG7#%0characterisableG" } {TEXT 240 1 " " }}{PARA 224 "" 1 "" {XPPMATH 20 "6#>%%TimeG$\"$\\\"!\" $" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 13 "m ap(rank, G);" }}{PARA 224 "" 1 "" {XPPMATH 20 "6#7#7)%#p5G%#p4G%#p3G%# p2G%#p1G&%#x2G6&%\"tGF-F-F-&%#x1G6%F-F-F-" }{TEXT 240 1 " " }}}{EXCHG {PARA 216 "> " 0 "" {MPLTEXT 1 238 20 "rewrite_rules(G[1]);" }}{PARA 230 "" 1 "" {XPPMATH 20 "6#7//%#p5G,$*&,(&%#x2G6#%\"tG\"\"\"*&%#p3GF-F *F-!\"\"*(F/F-)F*\"\"#F-%#p4GF-F-F-*&F*F-%#x1GF-F0F0/F4,$*&,**(F6F-F*F -&F*6$F,F,F-F0*(&F6F+F-F*F-F)F-F-*(F?F-F2F-F/F-F0*&F6F-)F)F3F-F-F-,&** F6F-F2F-F/F-F)F-F0*()F*\"\"$F-F?F-F/F-F-F0F0/F/,$*&,4**F6F-F)F-F2F-&F* 6%F,F,F,F-F-*(F3F-F6F-)F)\"\"%F-F-*(F6F-F2F-)F " 0 "" }}}{PARA 216 "" 0 "" }{PARA 216 "" 0 "" }}}{PARA 234 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }