Multi-Time Euler-Lagrange Dynamics

Transcription

1 Proceedngs of the 7th WSEAS Internatonal Conference on Systems Theory and Scentfc Computaton, Athens, Greece, August 24-26, Mult-Tme Euler-Lagrange Dynamcs CONSTANTIN UDRISTE Unversty Poltehnca of Bucharest Department of Mathematcs Splaul Indpendente 313 ROMANIA IONEL TEVY Unversty Poltehnca of Bucharest Department of Mathematcs Splaul Indpendente 313 ROMANIA Abstract: Ths paper ntroduces new types of Euler-Lagrange PDEs requred by optmal control problems wth performance crtera nvolvng curvlnear or multple ntegrals subject to evolutons of multdmensonal-flow type. Partcularly, the ant-trace mult-tme Euler-Lagrange PDEs are strongly connected to the mult-tme maxmum prncple. Secton 1 comments the lmtatons of classcal mult-varable varatonal calculus. Sectons 2-3 refer to varatonal calculus wth gradent varatons and curvlnear or multple ntegral functonals. Secton 4 s dedcated to the study of the propertes of mult-tme Euler-Lagrange operator affne changng of the Lagrangan, ant-trace mult-tme Euler-Lagrange PDEs and new conservaton laws. Secton 5 formulates an applcaton to mult-tme rheonomc dynamcs. Secton 6 underlnes the mportance of the ant-trace mult-tme Euler-Lagrange PDEs. Key Words: gradent varatons, mult-tme Euler-Lagrange PDEs, mult-tme maxmum prncple. Typng manuscrpts, LATEX 1 Overvew of classcal multvarable varatonal calculus The foundatons of varatonal calculus have been bult usng the classc Lagrange varaton of an admssble functon, but ths determnes some lmtatons that are not sutable for mult-tme control theory. The most mportant lmtaton comes from the fact that the classcal mult-varable varatonal calculus cannot be appled drectly to create a mult-tme maxmum prncple. In fact, the functonals gven as multple ntegrals, subject to general varaton functons produce mult-varable Euler-Lagrange or Hamlton PDEs contanng a trace total dvergence, whch s not convenent for the conservaton of the Hamltonan. Indeed, the Hamltonan s not a frst ntegral for the mult-varable Hamlton PDEs, even n the autonomous case. Our mult-tme control theory successfully overcomes the prevous lmtatons [3]-[8]. Ths theory requres m-needle-shaped varatons and complete ntegrablty condtons as core ssues. Addng new deas n varatonal calculus va the gradent varatons n curvlnear and multple ntegrals and the ant-trace Euler-Lagrange or Hamlton PDEs, we have justfed a mult-tme maxmum prncple whch s smlar to the Pontryagun maxmum prncple. The prevous two types of functonal varatons WSEAS Transactons on Mathematcs,.,. 2007, can be consdered as celebrtes of the optmzaton theory, although the use of classcal varatons s hardly compatble wth ampltude constrants, whle m-needle-shaped varatons are barely used n smooth optmzaton problems. 2 Curvlnear ntegral functonal and gradent varatons Let x, 1,..., n denote the feld varables on the target space R n, let t α, α 1,..., m be the multtme varables on the source space R m, and let x α x be the partal veloctes. In ths context, the jet tα bundle of order one s the manfold J 1 R m, R n {t α, x, x α}. Assume we are gven a smooth completely ntegrable 1-form L L β xt, x tdt β, β, 1,..., m, t R m, called autonomous Lagrangan 1-form. The Lagrangan 1-form s determned by the Lagrange covector feld L β xt, x t. The complete ntegrablty condtons are L β x x t λ L β x x t λ L λ x x t β L λ x x t β. Let be an arbtrary pecewse C 1 curve jonng the ponts 0 and t 0 n R m, and let be a par-

2 Proceedngs of the 7th WSEAS Internatonal Conference on Systems Theory and Scentfc Computaton, Athens, Greece, August 24-26, allelepped represented as the closed nterval 0 t t 0, fxed by the dagonal opposte ponts 00,..., 0 and t 0 t 1 0,..., tm 0 n Rm. We fx the mult-tme t 0 R m, and two ponts x 0, x 1 R n and we ntroduce a new problem of the calculus of varatons askng to fnd an m-sheet x : R n that mnmzes the functonal curvlnear ntegral Jx L β xt, x tdt β, α, β 1,..., m, among all functons x satsfyng the condtons x0 x 0, xt 0 x 1, usng C 1 gradent varaton functons constraned by boundary condtons. Fundamental queston: how can we characterze that functon x whch s the soluton of the prevous varatonal problem? Theorem 1 mult-tme non-homogeneous Euler-Lagrange PDEs. If the m-sheet x mnmzes the functonal Jx n the prevous sense, then x s a soluton of the mult-tme non-homogeneous Euler-Lagrange PDEs L β x L β t x c β, 1,..., n, β, 1,..., m. E L 1 Here we have a system of nm second order PDEs wth n unknown functons x. Theorem 1 shows that f we can solve the E L 1 PDEs system, then the mnmzer of the functonal J α assumng that t exsts wll be among the solutons. Proof. Step 1. Select a smooth gradent varaton y α : R nm, yα0 0, yαt 0 0 wth the prmtve y : R n, y 0 0, y t 0 0. We add the condtons complete ntegrablty condtons of 1-forms L and of gradent varatons L β x y λ L β y x t λ L λ x y β L λ y x t β, y β t y t β. CI 1 Let us defne the parameter ɛ ɛ 1,..., ɛ m and J α ɛ J α x ɛ β y β, for ɛ R m and we wrte x x,.e., we omt the superscrpt. We remark that the perturbed functon x ɛ β y β takes the same values as x at the dagonal ponts endponts 0 and t 0. Snce x s a mnmzer, we can wrte Jɛ Jx J0. In ths way, the functon Jɛ has a mnmum at the pont ɛ 0, and consequently ths must be a crtcal pont,.e., J 0 0. ɛβ Step 2. We compute the partal dervatves J ɛ β of the functon and we wrte Jɛ 0 J ɛ β 0 L σ xt ɛ λ y λ t, x t t ɛλ y λ t t dt σ Lσ x xt, x ty βt L σ x xt, x t y β t t dt σ. To process ths formula we use the condtons for varatons, wrtng 0 Lσ x y β L σ y x t β dt σ Lβ x y σ L β y x t σ dt σ Lβ x δ σ L β t σ x ydt σ Lβ t σ y dt σ. x Step 3. Snce the 1-forms Lβ x δ σ L β t σ x dt σ must be pullbacks of some 1-forms da β, we can evaluate the frst curvlnear ntegral usng the formula t y da y β t da β y t da β. One obtans y t da β 0, for all varatons y wth y 0 0, y t 0 0, yα0 0, yαt 0 0 satsfyng CI 1 and for all curves n the curvlnear ntegrals. Consequently, Lβ t x δ σ L β t σ x 0. Therefore the E L 1 PDEs hold for all mult-tmes t. Remark. There are two other ways to process the prevous formula, but fnally they are nconvenent because of the complete ntegrablty condtons:

3 Proceedngs of the 7th WSEAS Internatonal Conference on Systems Theory and Scentfc Computaton, Athens, Greece, August 24-26, the creaton of a dvergence operator, used usually n the varatonal calculus for multple ntegrals, Lσ x L σ t x yβdt σ Lσ t x yβ dt σ ; 2 the ntroducton a procedure suggested by our mult-tme maxmum prncple theory, based on the dea y β t 0 y t β, namely Lσ x δ β y L σ y β x t dt σ Lσ x δ β y L σ y x t β dt σ Lσ x δ β L σ t β ydt σ t β Lσ x y x dt σ. 3 Multple ntegral functonal and gradent varatons Assume we are gven a smooth Lagrangan Lxt, x t, t R m. We fx the mult-tme t 0 R m, the parallelepped R m wth the dagonal opposte ponts 00,..., 0 and t 0 t 1 0,..., tm 0, and two ponts x 0, x 1 R n.. We ntroduce a new problem of the calculus of varatons askng to fnd an m-sheet x : R n that mnmzes the functonal multple ntegral Jx Lxt, x tdt 1...dt m, among all functons x satsfyng the condtons x0 x 0, xt 0 x 1, usng gradent varaton functons satsfyng not only usual boundary condtons but also the complete ntegrablty condtons. Fundamental queston: how can we characterze that functon x whch s the soluton of the prevous varatonal problem? Theorem 2 mult-tme non-homogeneous Euler-Lagrange PDEs. If the m-sheet x mnmzes the functonal Jx n the prevous sense, then x s a soluton of the mult-tme non-homogeneous Euler-Lagrange PDEs L x L t x c, 1,..., n, 1,..., m. E L 2 Here we have a system of n second order PDEs wth n unknown functons x. Theorem 2 shows that f we can solve the E L 2 PDEs system, then the mnmzer of the functonal J assumng t exsts wll be among the solutons. Proof. Step 1. Select a smooth gradent varaton y α : R nm, y α Ω0,t0 0 wth the prmtve y : R n, y Ω0,t0 0. We add the complete ntegrablty condtons y β t y t β. CI 2 Defne the parameter ɛ ɛ 1,..., ɛ m and Jɛ Jx ɛ β y β, for ɛ R m and we wrte x x,.e., we omt the superscrpt. We remark that the perturbed functon x ɛ β y β takes the same values as x at the boundary of. Snce x s a mnmzer, we can wrte Jɛ Jx J0. In ths way, the functon Jɛ has a mnmum at the pont ɛ 0, and consequently ths must be a crtcal pont,.e., J 0 0. ɛβ Step 2. We compute the partal dervatves J ɛ β of the functon Jɛ L and we put J 0 0 or ɛβ 0 xt ɛ λ y λ t, x t t ɛ λ y λ t t dt 1...dt m, L x xt, x ty βt L x xt, x t y β t t dt 1...dt m.

4 Proceedngs of the 7th WSEAS Internatonal Conference on Systems Theory and Scentfc Computaton, Athens, Greece, August 24-26, To process ths formula we use the condtons CI 2 for the varatons, wrtng 0 L x y β L x L x δ β t β y t β L t β x L x y dt 1...dt m y dt 1...dt m dt 1...dt m. The last ntegral s evaluated by the Fubn formula. The frst ntegral can be modfed va Gauss formula snce t y B β y t B β y B β L x δ β L t β x. Step 3. It remans y t L x δ β t B β L t β x dt 1...dt m. Ths equalty holds for all dfferentable varatons y wth y Ω0,t0 0. Therefore L t x δ β L t β x 0 and consequently the mult-tme E L 2 PDEs hold for t. 4 Propertes of mult-tme Euler- Lagrange operator 4.1 Affne changng of the Lagrangan Suppose that Lt, xt, x t, t R m s a smooth Lagrangan. The followng Lemma s well-known. Lemma 3. The Euler-Lagrange dervatve ELL s ndependent of the partal acceleratons x αβ f and only f the Lagrangan L s an affne functon n veloctes,.e., Lt, xt, x t W t, xt A α t, xtx α. Let us consder the homogeneous Euler-Lagrange PDEs system L x L t x 0, 1,..., n, 1,..., m 1 together wth the non-homogeneous Euler-Lagrange PDEs system L x L t x c, 1,..., n, 1,..., m. The non-homogeneous Euler-Lagrange PDEs system becomes homogeneous f the Lagrangan L s replaced by ˆLt, xt, x t Lt, xt, x t W t, xt where A x j A α t, xtx α, A j x x A j t B x j c j. A partcular soluton s W 0, A α 1 m c t α. Combnng the prevous remarks wth some formulas n Sectons 2-3, we obtan the followng Theorem 4. The Euler-Lagrange operator EL has the property ELˆL, h ELL, h, where h stands for standard varatons. Therefore, the non-homogeneous Euler-Lagrange PDEs system whch orgnally was obtaned from L usng gradent varatons s a homogeneous system for the modfed Lagrangan ˆL and the standard varatons. From another pont of vew, the nonhomogeneous Euler-Lagrange PDEs system s a controlled Lagrangan system for a constant control [1]. Open problem. What s the sense of affne changng of classcal Lagrangans? What s the sense of homographc changng of classcal Lagrangans? When the gradent varatons are adequate? 4.2 Ant-trace mult-tme Euler-Lagrange PDEs The statements n Sectons 2-3 suggest to ntroduce the ant-trace mult-tme Euler-Lagrange PDEs L x δ β L t β x 0, as generalzatons of classcal homogeneous Euler- Lagrange PDEs. But, whle the classcal homogeneous Euler-Lagrange equatons are nvarant wth respect to changes of varables t, x t, x, the ant-trace mult-tme Euler-Lagrange PDEs are nvarant only wth respect to t, x t α a α β tβ b α, x.

5 Proceedngs of the 7th WSEAS Internatonal Conference on Systems Theory and Scentfc Computaton, Athens, Greece, August 24-26, Let us consder the ant-trace procedure as an algebrac operator appled to Euler-Lagrange operators. Lookng for the nverse operator, we dscovered two nverses: one algebrac and one dfferental. Theorem 5. 1 The trace followed by a fbre scalng s an algebrac nverse of the ant-trace procedure. 2 The dvergence s a dfferental nverse of the ant-trace procedure. Proof. 1 If we start wth ant-trace mult-tme Euler-Lagrange PDEs, by the trace after β, and by scalng the partal veloctes, n the sense of changng the Lagrangan after the rule Lt, xt, x t L 1 t, xt, mx t, we obtan the classcal Euler-Lagrange PDEs assocated to the new Lagrangan L 1. 2 Applyng the dvergence operator to ant-trace mult-tme Euler-Lagrange PDEs,.e., t L x δ β L t β x 0, we fnd L x L t x c. Consequently the dvergence and the ant-trace procedure are related as the prmtve wth the dervatve. 4.3 New conservaton law n case of multple ntegrals We start wth the autonomous Lagrangan Lxt, x t and the assocated homogeneous Euler-Lagrange PDEs 1. Gven the m-sheet x, let us ntroduce the mult-momentum p p α by the relatons p α L t xt, x t. Suppose that these x α nm equatons defne nm functons x x x, p. Sometmes, the varables x and p are called canoncal varables. We defne two tensor felds: ant-trace Euler-Lagrange tensor feld, A α β x, x L x x, x δβ α L t β x x, x, α energy-momentum tensor feld T α β x, p pα x β x, p Lx, pδα β. These tensor felds represent conservaton laws for Euler-Lagrange PDEs 1 snce t α Aα β 0, t α T β α 0 along the solutons of 1. Whle the second law s well-known, the frst s new and t appears from the ant-trace dea. 4.4 New conservaton laws and ant-trace PDEs n case of curvlnear ntegrals When we use path ndependent curvlnear ntegral functonals, we have smlar propertes. But, a smooth Lagrangan Lxt, x t, t R m produces two smooth completely ntegrable 1-forms: - the dfferental dl L x dx L x dx of components L x, L x, wth respect to the bass dx, dx ; - the restrcton of dl to xt, x t,.e., the pullback dl xt,xt of components L x x t β L x x t β dt β, L β xt, x t L x xt, x t x t β t L x xt, x t x t β t, wth respect to the bass dt β. In ths case, we must underlne that we have two dfferent ant-trace Euler- Lagrange tensor felds A αβ x, x λ L α x x, x λδ β B βα x, x λ L β x x, x λδ α snce the followng Theorem s true. Theorem 6. The relatons L α t β x L β t α x 0 L α t β x, x λ x L α t β x, x λ, x L α t β L β x t α δ x αδ λ β δλ βδα L t λ x t A αβ x, x λ t B βα x, x λ hold true. The frst relaton can be wrtten as a conservaton law t λ L α x δλ β L β x δλ α 0

6 Proceedngs of the 7th WSEAS Internatonal Conference on Systems Theory and Scentfc Computaton, Athens, Greece, August 24-26, or as L α x δλ β L β x δλ α t α δλ β t β δλ α L x. It mples the thrd relaton between ant-trace multtme Euler-Lagrange operators. Furthermore, the tensor felds A αβ x, x λ, B βα x, x λ represent equvalent conservaton laws for the homogeneous Euler- Lagrange PDEs L α x t L α x 0. 2 Indeed, ther dfference s a tensor of curl type and t A αβ x, x λ 0, t B βα x, x λ 0 along the solutons of 2. Startng from the Euler-Lagrange PDEs 2, we ntroduce: 1 the frst knd of ant-trace mult-tme Euler- Lagrange PDEs L α x x, x λδ β L α t β x, x λ 0; x 2 the second knd of ant-trace mult-tme Euler- Lagrange PDEs L β x x, x λδ α L α t β x, x λ 0. x Both ant-trace procedures have the same dfferental nverse dvergence, but dfferent algebrac nverses the frst, trace followed by a fbre scalng; the second, the trace. Consequently, applyng the dvergence, both mply the same controlled Euler-Lagrange PDEs system L α x t L α x c α. That s why, the solutons of ant-trace mult-tme Euler-Lagrange PDEs are among the solutons of the controlled Euler-Lagrange PDEs. 5 Applcaton n Mult-Tme Rheonomc Dynamcs Our theory has applcatons n Relatvstc and Mult- Tme Rheonomc Dynamcs; the electromagnetc feld E E, H H, 1, 2, 3 determnes the densty of electromagnetc deformaton energy L 1 E 2 δ jδ αβ E j t α t β H H j t α t β, where t 1 x, t 2 y, t 3 z, t 4 t. The extremals of L under gradent varatons are descrbed by the controlled wave PDEs E 2 E t 2 6 Concluson b, H 2 H t 2 c, 1, 2, 3. The present pont of vew regardng the mult-tme Euler-Lagrange PDEs has the key deas n the unon between [3]-[8] and [1], [2]. Acceptng that the evoluton s m-dmensonal, all the results confrm the possblty of passng from the sngle-tme Pontryagun s maxmum prncple to a mult-tme maxmum prncple [4]-[8]. Furthermore, the man results belong to PDEs-constraned optmal control theory. Acknowledgements: Partally supported by Grant CNCSIS 86/ 2007 and by 15-th Italan- Romanan Executve Programme of S&T Cooperaton for , Unversty Poltehnca of Bucharest. References: [1] D. E. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden, C. A. Woolse, The euvalence of controlled Lagrangan and controlled Hamltonan systems, ESAIM: Control, Optmsaton and Calculus of Varatons, , [2] D. F. M. Torres, On the Noether theorem for optmal control, European Journal of Control EJC, 8, , pp [3] C. Udrşte, Geodesc moton n a gyroscopc feld of forces, Tensor, N. S., 66, , [4] C. Udrşte, Mult-tme maxmum prncple, short communcaton at Internatonal Congress of Mathematcans, Madrd, August 22-30, [5] C. Udrşte, Maxwell geometrc dynamcs, manuscrpt, [6] C. Udrşte, M. Ferrara, Mult-tme optmal economc growth, manuscrpt, [7] C. Udrşte, A. M. Teleman, Hamltonan approaches of feld theory, IJMMS, , pp [8] C. Udrşte, I. Ţevy, Mult-Tme Euler-Lagrange- Hamlton Theory, WSEAS Transactons on Mathematcs, 6, ,