Vietnam J Math (20I4| 42:409-419 DOI 10 10O7/SI0OI3-0I3-0O48-9
Higher-Order Variational Problems of Herglotz l^pe
Simao P.S. Santos • Natalia Martins • Delfim F.M. Torres
Received 13 June 2013 / Accepted- 24 September 2013 / Published onhne: 5 November 2013
© Vietnam Academy of Science and Technology (VAST) and Spnnger Science+Business Media Singapore 2013
Abstract We obtain a generalized Euler-Lagrange differential equation and transversality optimahty conditions for Herglotz-type higher-order variational problems. Illustrative ex- amples of the new results are given.
Keywords Euler-Lagrange differential equations Namral boundary condiuons - Generahzed calculus of variations
Mathematics Subject Classification (2010) 34H05 • 49K15
1 Introduction
The generalized variational calculus proposed by Herglotz [7, 8] deals with an initial value problem
t(t} = L(t,x{t).i(t).z(t)), te[a.h]. (I) z{a) = y, yeR. (2) and consists in deterrmning trajectories x that extremize (minimize or maximize) the value
z(b). Observe that (I) represents a family of differential equations, for each function x a different diiferential equation arises. Therefore. ; depends on A, a fact that can be made
Dedicated 10 Boris Mordukhovich on the occasion of his 65ih Birthday S.P.S Santos N. Marims DFM Torres (IS])
CIDMA-Cenier for Research and Development in Madiemaiics and Applications.
Department of Mathematics. University of Aveiro. 3810-193 Aveiro. Portugal e-mail: [email protected]
S.PS Santos e-mail: [email protected]
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S P.S. Santos el al explicit by writing z(t,x(t),x(t)) or z[x; r], but for brevity and convenience of notation it is usual to write simply z{t). The problem reduces to the classical fundamental problem ofthe calculus of variations (see, e.g , (11) if the Lagrangian L does not depend on the variable z:
z(t) = L{t.x{t),i(t)). tela.b].
z(a) = y. yeR, then we obtain the classical variational problem
z(b)= I L{t,x{t),x(t))dt—*exlr
Herglotz proved that a necessary condition for a trajectory x to be an exlremizer of the generalized variational problem z(b) -^ extr subject to (l)-(2) is given by
-^{t,x{t),x{t),z(t))-— — {t,x(t),x{t),z(t)) dL, ,dL,
-\-—(t,x(t),x(t),z(t))-^{t,x(t),x{t).z(t))=0. (4) t e [a,fr]. This equation is known as the generalized Euler-Lagrange equation [6, 7, 12].
Observe that for the classical problem of the calculus of variations (3) one has | ^ = 0, and the differential equation (4) reduces to the classical Euler-Lagrange equation:
dL, , d dL, .
The variational problem of Herglotz was the basis of the Ph D. thesis [2] The main goal of this thesis, doneundersupervisionofRonaldB.Guenther, was to generalize the well known Noether's theorems (see, e.g., [ 13]) to problems of Herglotz type [3-5]. As reported in [3,4], unlike the classical variational principle, the variational principle of Herglotz gives a vana- tional description of nonconservative processes, even when the Lagrangian is autonomous.
For the importance to include nonconservativism in the calculus of variations, we refer the reader to the recent book [9].
In this paper we generalize Herglotz's problem by considering the following higher-order variational problem.
Problem (P) Determine the trajectories x e C^([a, b], R) that extremize the value of the functional zlx; b],
z(b) —> extr, where z satisfies the differential equation
Z(t) = t{t,x(t),x{t),...,x^''\t).z(t)), te[a.b], (<;j
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Higher-Order Variational Problems of Herglotz Type subject to the initial condition
where y is a fixed real number. The Lagrangian L is assumed to satisfy the following hy- potheses'
(HI) Z . i s a C ' ( M " + ^ E ) function;
(H2) functions / M- J^(t,x(t),i{t) x'">((),z(f)) and t \-^ ^(t,x(t).x(t) x'-"'(t),z{t)), j =0,...,n, are differentiable up to order n for any admissible tra- jectory X.
Clearly, Problem (P) generalizes the classical higher-order variational problem: if the Lagrangian L is independent of z, then
z(t)=Li',x{t),i(t) x'">(t)). teia.bl z(a) = y, yeR, which implies that the problem under consideration is
z{b) j L(t,x{t),x(').---,x^"Ht))dt-
L{t.x.x Jt:'"') = L{t.x.x,...,x'^"') -\
b-a'
The paper is organized as follows In Sect 2 we recall the necessary results from the classical calculus of variations. Our results are then proved in Sect. 3. in Sect 3 1 we obtain the generalized Euler-Lagrange equation for Problem (P) in the class of functions X S C^{[a,b],W) satisfying given boundary conditions
x(a)^ao,.... j:'''""(fl) =ff„_i, x(b)^fif,,...,x^"-'\b) = fi,.i,
where ao, ...,ar„_i, and ^o. •••<Pn-\ are given real numbers The transversality conditions (or natural boundary conditions) for Problem (P) are obtained in Sect. 3.2. We end with Sect. 4, presenting some illustrative examples of application of the new results.
The results of the paper are trivially generalized for the case of vector functions x . {a, b] —* R'", m e N, but for simplicity of presentation we restrict ourselves to the scalar case. Along the text, we use the standard conventions A:"" = ^ =X and Yl,k=\ "^i^) ~ 0 whenever j — 0.
2 Preliminary Results
We recall some results of the classical calculus of variations that are useful in the sequel.
Definition 1 We say that TJ e C^"([a,fe],lR) is an admissible variation for Problem (P) subject to boundary conditions (7) if, and only if, ri(a) = q{b) — ••• = j)*"""(a) —
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Lemma 1 (Higher-orderfundamentallemmaof the calculus of variations—cf. [10]) Let fo, . , / „ e C ( [ a , ^ ] . K ) . / /
/ (E/.{')')'"(OJrff=0
for all admissible variations ij of Problem (P) subject to boundary conditions (7), then
X](-l)'/,"'(0 = 0, t€[a,b].
a 2 (Higher-order integration by parts formulas—cf [11]) LetnEN,a,beR.a<b, and f.ge C"({a. b], R). The following n equalities hold
j"f{t)g^-\t)dt = \j2(-'i)'f^''(0g^'~'~''m + ( - " ' / f'"(')g(t)dt.
= i.
3 Main Results
For simplicity of notation, we introduce the operator (-, •)„, n € N, defined by {x.z}„(t)-={'.Mt),x(t) x'"\t).z{t)).
3.1 Generalized Euler-Lagrange Equation
The following resuh gives a necessary condidon of Euler-Lagrange type for an admissible function JT to be an exlremizer of the functional z[x;b], where z is defined by (5), (6) and (7).
Theorem 1 (Generalized higher-order Euler-Lagrange equation) Ifx is a solution to Prob- lem (P) subject to the boundaiy conditions {!), then x satisfies the generalized Euler- Lagrange equation
wliereUt):=e-'-i<^"""".
Proof Suppose that x is a solution of (/•) subject to (7), and let ij E C^([a.b]. R) be an admissible variation such that f;""{«) — 0- Let € be an arbitrary real number. Define f : UI.
fc] ^ a by
CM • = : r z l - ' + *''•'!
ae |j_(i
= -rz{l.x(0 + eri(l),i(l) + eii(l) ;r'"i(>) + ei(""(f))
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Obviously, <(a) = 0. Since jr is a minimizer (resp., maximizer), we have z(*.Jr(W + e>((fc),i(!>)+fii(6) x""(b) + €ri""(b))
>(resp. <)z(b,xib),i(b-i x''\b)).
Hence, /;{b) — ^z[x + ei]: fc]|6=o — 0. Because
i(t) = ^^z{l.x(.t)-l-fr,U),H,)-Hi,U). ..x<"'(l)-Hil'"'(l))\
dtde li d
we conclude that
-X^C£^{^,s)„(0'?'*'(o') + ^(^.z).a)?(o.
Thus, t, satisfies a first order linear differential equation whose solution is found according to
_y + / > y ^ ( 2 «=> e - ^ > < ^ " " ' y ( t ) - y ( « ) - y e-/"'^'""'^(2(r)t/r Therefore,
e-r.-ik".='-«>-\(,)-;(a)==fe-i:'"- ' ^ •"•"" ( j ] - j ^ ( . . , z ) . ( r ) , " l ( t ) | r f t .
Denoting i(») — e " ' ' S ' " a-'"*", we get
In particular, for f = fc, we have
Since C (') = 0, f e I", fc), the left-hand side of the previous equation vanishes and we get 0 = f V i ( r ) - ^ ( . v , ; ) . ( r ) , ' " ( t W r .
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414 SPS Santos eial Using the higher-order fundamental lemma of the calculus of vanations (Lemma 1), we obtain the generalized Euler-Lagrange equation
= 0. ( e [ a , f o ] .
proving the intended result.
In order lo simplify expressions, and in agreement with Theorem 1, from now on we use the notation k(t) ;—£-/= 3F<-"-='"<*>''^. If n ^ l, then die differential equation of Prob- lem (P) reduces to j(;)—Z.(r,;i:(f), i ( f ) , ; ( / ) ) , which defines the functional z of Herglotz's variational principle. This pnnciple is a particular case of our Theorem 1 and is given in Corollary L
Corollary 1 (See [7, 8]) Let z be a solution of z(t) = L{t,x{t),x{t),z(t)), t e [a.b], sub- ject to the boundary conditions z(a) = y, x(a) — a and x(b) = ^, where y, a, and ^ are given real numbers. Ifx is an extremizer of functional z[x;b], then x satisfies the differential equation
SL , . dL , , B L ,
— {t,x(t),x(t),z{t))^':^Jt,x(t),x{t),z(t)) — {t.x{t).k(t),z{t)) - — — {t,x{t),x(t),z{t))^(i, t^\a,b-\.
Our Euler-Lagrange equation (8) is also a generalization ofthe classical Euler-Lagrange equation for higher-order variational problems.
Corollary 2 (See, e.g., [1]) Suppose that x is a .lolution of Problem (P) subject to (7), and that the Lagrangian L is independent of z- Then x satisfies the classical higher-order Euler-Lagrange differential equation
V - di { dL , , , A
3.2 Generalized Natural Boundary Conditions
We now consider the case when the values of x(a) A:*"~"(a), x{b), . . . , jr'''""(&), are not necessarily specified.
Theorem 2 (Generalized natural boundary conditions) Suppose that x isa solution to Prob- lem (P). Then x satisfies the generalized Euler-Lagrange equation (8). Moreover I. Ifx*'^'*(b), A e (0, . . . , n — 1), is free, then the natural boundary condition
holds.
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Higher-Order Vanational Problems of Herglotz Type
2. Ifx*''\a), k e{0,.. .n — 1), is free, then the natural boundary condition
holds.
Proof Suppose that J; is a solution to Problem (P) Let TJ e C-"([a.b\.R) and define die function ? jusi like in the proof of Theorem I. From the arbitrariness of TJ, and using similar argumentsas the ones in the proof of Theorem I, we conclude that.T satisfies the generalized Euler-Lagrange equation (8). We now prove (10) (the proof of (11) follows exactly the same arguments). Suppose that x^'''{b), it e (0 « — 1|, is free. Define the function C(') •=
^z[;c-t-6fj,(]U=o- LelJ:={j€{0,...,n-l] :x^J\a) is given}. For any j G (0 H - i j , if y e J. then )j'"(i) — 0; otherwise, we restrict ourselves to those functions >] such that j)'J'(a) = 0. For convenience, we also suppose that ij''"((() = 0. Using the same arguments as the ones used in the proof of Theorem 1, we find that f satisfies the first order linear differential equation
3L dL 3L , , n') = ^{x.z)At)n(i) + ^{x.z},.(m(t) + --- + ^^Ax.z}„(t)7j^'"{t)
dx dx djc""
dL
whose solution is found by
k(t)at)-;{<i)=f Y,MT)-^{x.z)AT)r)^'\T)dT.
Again, since ^(t) = 0, for t e {a, h], we get
dxO) and, therefore,
5 i ; M r ) T % ( ^ . z)„(r)r)^'\T)dz = 0
j k{z) — (x.z)„{T)ij{r)dT-\-}2j
Using the higher-order integration by parts formula (Lemma 2) in the second parcel we get r X ( r ) ^ ( A : , . - ) „ ( r ) T 7 ( r ) d T + X ^ ( M r ) ^ ( ^ . ^ ) „ ( r ) i j ' ^ - " ( r )
-^{-ly j'(^Mr)-^{x.z)„{T)y r,(T)dT\=0,
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which is equivalent to
f)dz
th^'
/
' " / dL \'-''-^X:(-ly(«r)lt^(,,^).(t))"',/-'-|(r)j =0.
Using the generalized Euler-Lagrange equation (8) into the last equation we get
+ l ; ( - i y ( i ( r ) 5 ^ ( . : . z ) „ ( r ) ) " ' , " ^ ' - ' > ( r ) j = 0
and since rj(a) = rj{a) — •• — rj'"~"{a) = 0, we conclude thaiThis equation is equivalent to
Let / : ^ {f € | 0 , . . . , « - I } : x'^'Hb) is given}. Note that k^ I. For any ( e | 0 , . . . , « - I}, if I e I, then jj'''(fo) = 0; otherwise, for; ?t k, we restrict ourselves to those functions TJ such that/j'"(fo) = 0 . From the arbitrariness of J j ' ' ' ( t ) , h follows that
This concludes the proof. U Remark 1 If .c is a solution to Problem (P) without any of the 2n boundary conditions (7)
then JT satisfies the generalized higher-order Euler-Lagrange equation (8) and n transver- sality conditions (10) and n transversality conditions (11). In general, for each boundary condition missing in (7), there is a corresponding natural boundary condition, as given bv Theorem 2.
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Higher-Order Variational Problems of Herglotz Type
Next we remark thai our generalized transversahty conditions (10) and (11) are general- izations of the classical transversality conditions for higher-order variational problems (cf }{/'• =0,k = 0,....n-], withi^* given as in [13, Sect. 5]).
Corollary 3 Suppose that x is a solution of Problem (P) with L independent of z- Then x satisfies the classical Euler-Lagrange equation (9). Moreover,
1. Ifx'^(b),k&{0 « — I}, is free, then the natural boundary condition
|'-"'-'S^(d!fe)('-<« ^•"<«)-''
holds.
2. Ifx'''^(a) IS free, ks |0 n-l], then the natural boundary condition
^ , I dJ-^ ( dL \ , ,„, ,
4 Illustrative Examples
We illustrate the usefulness of our results with some examples that are not covered by previ- ous available results in the hterature Let us consider the particular case of Theorem I with n=2.
Corollary 4 Let z be a solution of z{t) = L(t.x(t),x(t),x.(t),z{t)), t e [a,b\. subject to the boundary conditions z(a) = y, x(a) — ao, x{a) — ai, x(b) = ft, and x(b) = Pi. where y.ao.ai. Po.and PI.are given realnumbers.Ifx is an exlremizer offiinctioiial z[x: h\, then X satisfies the differential equation
dL dL dL d 'dL -r-{x.z)2(t) + —(x.z)2{t)^{x,z.)2(r)--'^{x,z)i(t)
ax dz ox dt dx (dL \-3L dL d dL
+ \^{x,z)2(t)] ^{x,z)2(')-2—{x.z)2(t)~^{X,z)2{t)
\dz / dx dz dt dx {d dL \dL d- dL
-{-r^{^-^)2(0)^{x,zh(t)+-^~{x.zh(n^0. ts{a,bl (12)
\dt dz / ox dt- dx where (x.z)2(t) = {t,x(t),x{0,x(t).z(l)}.
We now apply Corollary 4 to concrete situations.
Example I Let us consider the following Herglotz's higher-order variational problem:
;(1) —»• min,
z(t)=x\t)-i-z-(t), r e [ 0 , l ] , ^(0) = - . (13)
;r(0) = 0, i ( 0 ) = I. ^(1) - 1, x ( l ) ^ 1.
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For this problem, the necessary optimahty condition (12) asserts that
x'-'Ht) - 4z{t)x^'Ht) + {4zHl) - 2 z ( 0 ) ^ ' " ( ' ) - 0 . (14) Solving the system fonned by (14) and z(t) - x^{t) + z'(t), subject to the given boundary conditions, gives the extremal
for which 2(1) — t.
Example 2 Consider problem (13) widi z(0) = zo free. We show that such problem is not well defined. Indeed, if a solution exists, we obtain the optimahty system
'x'-^Ht) - 4j(r)A:<3' (0 -I- {4zHt) - 2z(t))x'-^^ (t) - 0, z(t)^XHt)+zHt)
subject lo x(0) ^ 0 and i ( 0 ) ^ x ( l ) = x ( l ) ^ t. It follows that x(t)^t. E(0 = — ^ ,
1 -zot
and we conclude dial the problem has no solution: the infimum is —co obtained when z o - » l + .
Example 3 Consider now the following problem:
z(l) —> nun,
z(r) = i 2 ( i ) - F z ( t ) , r e [ 0 , l ] , z(0) = i , (15) 1(0) = 0 , i ( 0 ) = l, xa) = l, i ( l ) = 0.
For problem (15), the necessary optimahty condition (12) asserts that
i ™ ( I ) - 2 j < " ( 0 + J : ™ ( t ) = 0 , (16) Solving the system fonned by (16) and i(t) = x^{t) + z(r), subject to the given boundary
conditions, gives the extremal
, (1 - r ) e ' + ' + (2i - l ) c ' + (e - 3 ) « - e-t-1
* ' ' ' " e 2 - 3 e + l
, , V.\ + t^)e'*^- 2(2(2 j _ , ^ 2)e'+l +(il^ + il + 5)e' + «< - 6«3 + 10,2 _ 2« - „i
^<" = f . 2 - 3 . - H ) 2 - •
Our last example shows the usefulness of Theorem 2.
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Higher-Order Variational Problems of Herglotz Type
Example 4 W e n o w c o n s i d e r p r o b l e m ( 1 5 ) w i t h i ( l ) free. I n t h i s c a s e , s o l v i n g
( ; r ' ' " ( r ) - 2 j r < ' i ( 0 -I-x<->(f) = 0 .
\z(t)=x^{t) + z(f)
s u b j e c t t o t h e b o u n d a r y c o n d i t i o n s z(0) = 1, j r ( 0 ) = 0 , i ( 0 ) ^ 1, j r ( l ) = 1, a n d t h e n a t u r a l b o u n d a r y c o n d i t i o n ( 1 0 ) for « — 2 a n d i = I, w h i c h m t h e p r e s e n t s i m a r i o n s i m p l i f i e s to x ( l ) = 0 , g i v e s d i e e x t r e m a l
x{t)^t. z(t) = e'.
for w h i c h i ( l ) ^ l a n d j ( l ) - t < 2 , 7 2 .
Acknowledgements This work was supported by FEDER funds through COMPETE-Operational Pro- gramme Factors of Competitiveness ("Programa Operacional Faetores de CompetiUvidade") and by Por- higuese funds through die Center for Research and Development in Malhematics and Applications (Univer- sity of Aveiro) and the Portuguese Foundation for Scienee and Technology fFCT-Funda^ao para a Cien- cia e a Teenologia"), within project PEsi-CA1ATAJ14l06/2011 with COMPETE number FCOMP-OI-0124- FEDER-022690 Tories was also supported by the FCT project PTDC/EEl-AUT/1450/2012, co-financed by FEDER under POFC-QREN with COMPETE reference FCOMP-01-0I24-FEDER-028894 The authors are grateful to two anonymous referees for their valuable comments and helpful suggestions.
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