On variational formulations for functional differential equations

c 2007, Scientific Horizon

JOURNAL OF FUNCTION SPACES AND APPLICATIONS Volume 5, Number 1 (2007), 89-101

http://www.jfsa.net

On variational formulations for functional differential equations I. A. Kolesnikova, A. M. Popov and V. M. Savchin (Communicated by Vladimir Stepanov )

2000 Mathematics Subject Classification. 34Jxx, 34Kxx, 34A30. Keywords and phrases. Functional differential equations, inverse problem of the calculus of variations, conditions of potentiality, variational principles.

Abstract. Necessary and sufficient conditions for the existence of integral variational principles for boundary value problems for given ordinary and partial functional differential equations are obtained. Examples are given illustrating the results.

1.

Introduction

By the problem of construction of integral variational principles for a system of equations of some given model we mean the construction of functionals for which the set of critical (extremal or stationary) points coincides with the set of the solutions of the given system. The search of a functional F , that admits the given equations as its Euler-Lagrange equations is known as the classical inverse problem of the calculus of variations. Since the end of the XIXth century there has been a

90

Functional differential equations

great deal of activity in this field (see Helmholtz [4], Volterra [13], Santilli [9], Tonti [12], Filippov, Savchin and Shorokhov [2] and refs. therein). In the course of a long time advantages of the variational principles have been mainly used for ordinary and partial differential equations with the so-called potential operators [2]. There is a practical need to develop different approaches to the construction of integral variational principles for equations with deviating arguments. The first known Euler-Lagrange equations with deviating arguments corresponding to some variational problems have been considered by El’sgol’tz [1], Hughes [5], Sabbagh [8] and Kamenskii [6]. The main aim of this paper is to find out necessary and sufficient conditions under which the given ordinary and partial functional differential equations with appropriate boundary conditions admit variational formulation. The reader should keep in mind that most of the results can be formulated under weaker smoothness conditions.

2.

Certain auxiliary notations and definitions

Let U, V be normed linear spaces over the field of real numbers R, and OU , OV be their zero elements. Take any operator N : D(N ) → R(N ), where D(N ) ⊆ U , R(N ) ⊆ V . A limit 1 lim [N (u + h) − N (u)] = δN (u, h), u ∈ D(N ), (u + h) ∈ D(N ),

→0

if it exists, is called the Gâteaux differential of N at the point u . If it is linear relative to h, then the operator δN (u, ·) : U → V is called the Gˆ ateaux derivative of N at u and will be denoted by Nu . Its domain of definition D(Nu ) consists of the elements h ∈ U such that (u + h) ∈ D(N ) for all sufficiently small. Let us consider the equation (1)

N (u) = OV ,

u ∈ D(N )

with the Gâteaux differentiable operator N , and a convex set D(N ). In order to consider the existence of its variational formulation we need a non-degenerate bilinear form ·, · : V × U → R.

I. A. Kolesnikova, A. M. Popov and V. M. Savchin

91

Definition. The operator N : D(N ) → V is said to be potential on the set D(N ) relative to a given bilinear form ·, · : V × U → R, if there exists a functional FN : D(FN ) = D(N ) → R such that δFN [u, h] = N (u), h

∀h ∈ D(Nu ).

∀u ∈ D(N ),

The functional FN is called the potential of the operator N , and in turn the operator N is called the gradient of the functional FN . As it is known Volterra [13] the condition for potentiality of the operator N takes the form (2)

Nu h, g = Nu g, h

∀u ∈ D(N ),

∀h, g ∈ D(Nu ).

Under this condition the potential FN is given by 1 (3) N (u0 + λ(u − u0 )), u − u0 dλ + const, FN [u] = 0

where u0 is a fixed element of D(N).

3.

The potentiality conditions for ordinary functional differential equations

Let us consider the equation (4)

N (u) ≡ F (t, u(ωi (γk (t))), u (ωi (γk (t))), u (ωi (γk (t)))) = 0 i, k = 0, n;

t ∈ (t1 , t2 ),

where ω0 (t) ≡ t, γ0 (t) ≡ t; ωi (t) and γi (t) are mutually inverse functions; ωi (t), γi (t) ∈ C 3 [t1 , t2 ], ω(t) ≤ t, ω (t) > 0 ∀t ∈ [t1 , t2 ]; F ∈ C 3 (Rm ), m = 3(n2 + n) + 4. Let us denote αj = min ωi (γk (tj )), βj = max ωi (γk (tj )), j = 1, 2. We i,k

i,k

assume that β1 < α2 . The functions ϕi (i = 1, 2), belonging to the class of functions C 2 , are given on the sets Ej = [αj , βj ], j = 1, 2. Let us denote by U the set of all absolutely continuous functions on [α1 , β2 ] for which the first and the second derivatives exist and are almost everywhere bounded on [α1 , β2 ]. A function u(t) ∈ U , satisfying the equation (4) almost everywhere on [α1 , β2 ], is called its solution. The domain of definition of the given operator N is defined by the equality (5)

D(N ) = {u ∈ U : u(t) = ϕj (t), t ∈ Ej , j = 1, 2},

92


where ϕj (t) are certain given functions, ϕj (t) ∈ C 2 . We define the bilinear form by the relation t2 v, g = (6) v(t)g(t)dt. t1

Theorem 1. In order that the operator N (4) be potential on D(N ) (5) relative to the bilinear form (6), it is necessary and sufficient that ∀t ∈ (t1 , t2 ) ∂F dω (γ (t)) d ∂F dωk (γi (t)) k i (7) − ∂uik ki dt dt ∂uik ki dt 2 ∂F dωk (γi (t)) d ∂F , + 2 = dt ∂uik ki dt ∂uki ∂F dω (γ (t)) 2 d ∂F dωk (γi (t)) dωk (γi (t)) k i (8) + − ∂uik ki dt dt ∂uik ki dt dt 2 d dωk (γi (t)) ∂F ∂F + , = dt ∂uik ki dt ∂uki ∂F dω (γ (t)) 3 ∂F k i (9) = , ∂uik ki dt ∂uki where uik = u(ωi (γk (t)), [Ψ]ik ≡ Ψ(s, u(ωp (γq (s))) . . .), s = ωi (γk (t)), p, q = 0, n, i, k = 0, n. Proof. We shall verify that under the conditions (7)-(9) the Volterra’s criterion of potentiality (2), where (10)

D(Nu ) = {h ∈ U : h = 0, t ∈ E1 ∪ E2 },

will be fulfilled. Using the Gâteaux derivative of the operator (4), we get (11) Nu h, g =

t2

t1

n i,k=0

∂F h(ωi (γk (t))) ∂u(ωi (γk (t)))

∂F h (ωi (γk (t))) i (γk (t)))

∂F h (ωi (γk (t))) g(t)dt. + ∂u (ωi (γk (t)))

+

∂u (ω


93

Let us perform the following transformations in the right-hand side of the equality (11): (a) in the first group of terms we use the change of variable of the form s = ωi (γk (t)); (b) in the second group of terms we use the change of the variable of integration, then the formula of integration by parts; (c) in the third group of terms we use the change of the variable of integration, then twice the formula of integration by parts. Bearing in mind the equality (10) for h and g , and going back to the initial notation of the variable of integration, we get t2 ∂F h(ωi (γk (t)))g(t)dt (12) ∂u(ω i (γk (t))) t1 t2 dω (γ (t)) ∂F k i = g(ωk (γi (t)))h(t)dt, ∂u(ωi (γk (t))) ki dt t1 t2 ∂F h (ωi (γk (t)))g(t)dt (ω (γ (t))) ∂u i k t1 t2 dω (γ (t)) d ∂F k i =− g(ωk (γi (t))) (ω (γ (t))) ki dt ∂u dt i k t1 dω (γ (t)) 2 ∂F k i + g (ω (γ (t))) h(t)dt, k i ∂u (ωi (γk (t))) ki dt t2 ∂F h (ωi (γk (t)))g(t)dt (ω (γ (t))) ∂u i k t1 t2 2 dω (γ (t)) d ∂F k i = g(ωk (γi (t))) dt2 ∂u (ωi (γk (t))) ki dt t1 dω (γ (t)) dω (γ (t)) ∂F d k i k i g (ωk (γi (t))) + dt ∂u (ωi (γk (t))) ki dt dt dω (γ (t)) 2 d ∂F k i + g (ωk (γi (t))) dt ∂u (ωi (γk (t))) ki dt dω (γ (t)) 3 ∂F k i + g (ω (γ (t))) h(t)dt. k i ∂u (ωi (γk (t))) ki dt Now we get (13) Nu g, h t2 n = t1

∂F ∂F g(ωi (γk (t))) + g (ωi (γk (t))) ∂u(ωi (γk (t))) ∂u (ωi (γk (t))) i,k=0

∂F + g (ωi (γk (t))) h(t)dt. ∂u (ωi (γk (t)))

94


Comparing the expressions (11) and (13), and taking into account the above transformations, we conclude that for the fulfillment of the potentiality criterion it is necessary and sufficient that the relations (7)(9) hold. The proof of the theorem is completed. Example 1. Let us consider the equation

t t a2 a3 (14)N (u) ≡ a1 u(t) + a2 u(c1 t) + a3 u(c2 t) + u + u c1 c1 c2 c2 2 t c c c a 4 1 2 t − a5 22 u t + a4 u (c1 t) − 2 u + a5 u c1 c1 c2 c1 c1 t c1 c2 c3 a6 + a6 u (c1 t) + 3 u t + a7 23 u t = 0, + a7 u c1 c1 c2 c1 c1 0 < t 1 < t < t2 , where ai (i = 1, 7), c1 and c2 are constants, such that 0 < c1 < c2 < 1. In accordance with the accepted notations (4) here we have ω1 (t) = c1 t, ω2 (t) = c2 t, γ1 (t) = ct1 , γ2 = ct2 , ω1 (γ2 (t)) = cc12 t, ω2 (γ1 (t)) = c2 c1 t. Let us note that if a6 ≡ 0 then the equation (14) is of neutral type, and if a6 = 0 , then it is of advanced type (see Kamenskii [6]). It is easy to check that the given operator (14) satisfies the conditions of potentiality (7)-(9). Applying the formula (3) for the potential FN of the operator N we get 1 t2 FN [u] = [a1 u2 (t) + a2 u(c1 t)u(t) + a3 u(c2 t)u(t) 2 t1 t t a2 a3 + u u(t) + u u(t) + a4 u (c1 t)u(t) c1 c1 c2 c2 t c1 c2 a4 c2 − 2 u t u(t) − a5 22 u t u(t) u(t) + a5 u c1 c1 c2 c1 c1 t c1 a6 + a6 u (c1 t)u(t) + 3 u t u(t) u(t) + a7 u c1 c1 c2 c2 c3 + a7 23 u t u(t)]dt. c1 c1 Integrating by parts, we diminish the second order of derivatives in the integrand, and finally we come to the following potential t2 a1 2 u (t) + a2 u(t)u(c1 t) + a3 u(t)u(c2 t) + a4 u(t)u (c1 t) FN [u] = 2 t1

t c22 a6 + a5 c2 u (c1 t)u(c2 t) − u (t)u − a7 u (c1 t)u (c2 t) dt c1 c1 c1 + const.


4.

95

On variational formulation for partial differential difference equations

Let us consider the following equation (15) N (u) ≡ f (x, u(k) α (x, t + λτ )) = 0, t ∈ (t0 , t1 ),

x = (x1 , . . . , xm ) ∈ Ω,

λ = −1, 0, 1; τ > 0; m |α| = αi , |α| = 0, s,

k = 0, l;

α = (α1 , . . . , αm ),

t1 − t0 > 2τ,

i=1

where u is an unknown function, f is a given smooth function, u(k) α =

∂ k ∂α u ; ∂tk

∂α =

∂ |α| . (∂x1 )α1 . . . (∂xm )αm

The domain of definition D(N ) is given by the equality s,l (16) D(N ) = u ∈ U = Cx,t (Ω × [t0 − τ, t1 + τ ]) : ∂ku = ϕ1k (x, t), (x, t) ∈ E1 = Ω × [t0 − τ, t0 ]; k = 0, l0 ∂tk ∂ku = ϕ2k (x, t), (x, t) ∈ E2 = Ω × [t1 , t1 + τ ]; k = 0, l0 , ∂tk ∂ ν u = ψ (x, t), ν = 0, s ν 0 . ∂nνx ∂Γτ Here ϕ10 , ϕ20 , ψν are given sufficiently smooth functions, ϕjk

∂ k ϕj0 ∂tk (j

=

= 1, 2; k = 1, lo ), Γτ = ∂Ω × [t0 − τ, t1 + τ ]. The numbers l0 and s0 depend on l and s, respectively. If l, s are even, then l0 = 2l − 1, s0 = s l+1 s0 = s+1 2 − 1. For odd l, s we set l0 = 2 − 1, 2 − 1. We shall use the notations αm α1 α β1 . . . βm , if αi ≥ βi = β 0, if αi < βi , and Ckν = for binomial coefficients.

k! k = ν ν!(k − ν)!

96


Assume that there is given the bilinear form t1 v, g = (17) v(x, t)g(x, t)dtdx. Ω

t0

Theorem 2. For the potentiality of the operator (15) on the given set (16) with respect to the bilinear form (17), it is necessary and sufficient that the following conditions hold (18)

s l

|α|+k

(−1)

Ckν

k=0 |α|=0

=

α ∂f k−ν Dt Dα−β (k) β ∂uα (x, t − λτ ) t→t+λτ

∂f (ν) ∂uβ (x, t

− λτ )

∀u ∈ D(N ), ∀x ∈ Ω, ∀t ∈ (t0 , t1 ), λ = −1, 0, 1, ν = 0, l, |β| = 0, s, where the notation (. . .)|t→t+λτ means that in the expression within the parentheses one has to change t by t + λτ. Proof. Taking into account formulas (15),(17), we get (19)

Nu h, g t1 l s 1 = t0

∂f

(k) Ω λ=−1 k=0 |α|=0 ∂uα (x, t

− λτ )

h(k) α (x, t − λτ )g(x, t)dxdt.

By integrating by parts and taking into account that the functions g, h ∈ D(Nu ) are subject to the conditions ∂kg k ∂t ν ∂ g ∂nνx Γτ

∂kh = 0, (x, t) ∈ E1 ∪ E2 , k = 0, l0 , ∂tk ∂ ν h = 0, ν = 0, s0 , ∂nνx Γτ

= =

we obtain from (19)

Nu h, g =

t1

t0

×

l s 1

(−1)|α|+k Dtk Dα

Ω λ=−1 k=0 |α|=0

∂f (k)

∂uα (x, t − λτ )

g(x, t) h(x, t − λτ )dxdt,


97

where Dtk is the total derivative of order k with respect to the variable t; Dα is the total derivative corresponding to the multiindex α. From here by using the Leibnitz formula there follows that (20)

Nu h, g =

t1

t0

l 1

s

|α|+k

(−1)

Ω λ=−1 k,ν=0 |α|,|β|=0

× Dtk−ν Dα−β

Ckν

α β

∂f (k)

∂uα (x, t − λτ )

(ν)

× gβ (x, t)h(x, t − λτ )dxdt. By using the change t − λτ = t , from (20) we get (21)

1

Nu h, g =

λ=−1

t1 −λτ

t0 −λτ

(−1)|α|+k Ckν

Ω k,ν=0 |α|,|β|=0

Dα−β × Dtk−ν (ν) × gβ (x, t

s

l

∂f (k)

∂uα (x, t − λτ )

α β

t→t +λτ

+ λτ )h(x, t )dxdt .

Denoting t by t, we obtain from (21) (22)

1

Nu h, g =

λ=−1

t1 −λτ

t0 −λτ

(−1)

∂f (k)

∂uα (x, t − λτ )

t1 +τ

h(x, t)dxdt

t1

Ω

|α|+k

Ckν

α β

t→t+λτ

+ λτ )h(x, t)dxdt.

Taking into account that t0 h(x, t)dxdt = t0 −τ

s

Ω k,ν=0 |α|,|β|=0

Dα−β ×Dtk−ν (ν) × gβ (x, t

l

Ω

∀ h ∈ D(Nu ).

the equality (22) can be written as (23) Nu h, g=

t1

t0

l 1

s

(−1)|α|+k Ckν

Ω λ=−1 k,ν=0 |α|,|β|=0


∂f (k)

∂uα (x, t + λτ )

α β

g(x, t + λτ ) h(x, t)dxdt.

98


Bearing in mind the equality (19) we obtain

Nu h, g =

(24)

t1

t0

1 s l

Ω λ=−1 ν=0 |β|=0

∂f (ν)

∂uβ (x, t − λτ )

(ν) × gβ (x, t − λτ ) h(x, t)dxdt. By using (23), (24) we get

Nu h, g −

Nu g, h

t1

= t0

s

l0 1

|α|+k+ν

(−1)

Ω λ=−1 k,ν=0 |α|,|β|=0


Ckν

α β

∂f (k)

∂uα (x, t − λτ ) t→t+λτ

∂f (ν) − (ν) gβ (x, t + λτ )h(x, t)dxdt ∂uβ (x, t − λτ ) ∀u ∈ D(N ), ∀g, h ∈ D(Nu ).

Since g and h are arbitrary functions from D(Nu ) then Nu h, g − Nu g, h = 0 ∀u ∈ D(N ), ∀h, g ∈ D(Nu ) if and only if conditions (18) hold. Example 2. Let us consider the equation (25) N1 (u) ≡

1

(aλ utt (x, t + λτ ) − bij λ uxi xj (x, t + λτ )) = 0,

λ=−1

(x, t) ∈ Q = Ω × (t0 , t1 ), t1 − t0 > 2τ. Here u = u(x, t) is an unknown function, utt = 0,2 Cx,t (Q),

bij λ

2,0 Cx,t (Q),

∂2u ∂t2 ,

uxi ,xj = m

∂2u ∂xi ∂xj ;

aλ ∈

∈ Ω is a bounded domain in R with a piecewise smooth boundary ∂Ω; Ω is the closure of Ω in Rm , and repeated indices of factors situated at different levels denote summation, i, j = 1, m. Let us define the domain of definition of N1 by setting (26) D(N1 )

=

u ∈ U = C 2 (Qτ ) : ∂ k u(x, t) = ϕ1k (x, t), (x, t) ∈ E1 = Ω × [t0 − τ, t0 ], k = 0, 1, ∂tk


99

∂ k u(x, t) = ϕ2k (x, t), (x, t) ∈ E2 = Ω × [t1 , t1 + τ ], k = 0, 1, ∂t k ∂u = ψ(x, t) , ∂nx Γτ

where Γτ = ∂Ω×[t0 −τ, t1 +τ ], Qτ = Ω×(t0 −τ, t1 +τ ), ϕi0 ∈ C 1 (Ei ), ϕi1 = ∂ϕi0 ∂t (i = 1, 2), ψ ∈ C(Γτ ). First, let us study the existence of the classical variational formulation of problem (25), (26). For that aim, we introduce the notation V = C(Qτ ) and use the bilinear form (17). The conditions of potentiality (18) for the given case take the form a−λ (x) bij −λ (t) t→t+λτ

= aλ (x),

x ∈ Ω,

λ = −1, 0, 1,

= bij λ (t),

t ∈ [t0 , t1 ],

λ = −1, 0, 1,

i, j = 1, m.

If this relations are hold the given problem (25), (26) allows the classical variational formulation and the corresponding functional has the form 1 t2 FN [u] = − [a1 (x)ut (x, t − τ )ut (x, t) + a0 (x)u2t (x, t) 2 t1 Ω + a1 (x)ut (x, t + τ )ut (x, t) + bij 1 (t − τ )uxi (x, t − τ )uxj (x, t) ij + bij 0 (t)uxi (x, t)uxj (x, t) + b1 (t + τ )uxi (x, t + τ )uxj (x, t)]dxdt

+ const. Example 3. Let us consider the equation (27) N2 (u) ≡ b(ux )2 (x, t + τ ) − a(ut )2 (x, t) − 2autt (x, t) − 2beu(x,t−τ )−u(x,t){ux(x, t − τ )ux (x, t) + uxx (x, t)} = 0, (x, t) ∈ Q = (c, d) × (t1 , t2 ), t2 − t1 > 2τ, τ > 0, where a, b are constants,u is an unknown function, ux = ∂u ∂x , ut = ∂2u ∂t2 . Let us definite the domain of definition of N2 by setting

∂u ∂t ,

utt =

(28)

2,2 D(N2 ) = u ∈ U = Cx,t (Qτ ) : ∂ k u(x, t) = ϕ1k (x, t), (x, t) ∈ E1 = [c, d] × [t1 − τ, t1 ]; k = 0, 1, ∂tk

100

Functional differential equations ∂ku = ϕ2k (x, t), (x, t) ∈ E2 = [c, d] × [t2 , t2 + τ ]; k = 0, 1, ∂tk u(c, t) = ψ1 (t), u(d, t) = ψ2 (t), t ∈ [t1 − τ, t2 + τ ] ,

i0 where Qτ = (c, d) × (t1 − τ, t2 + τ ), ϕi0 ∈ C 1 (Ei ), ϕi1 = ∂ϕ ∂t , ψi ∈ C([t1 − τ, t2 + τ ])(i = 1, 2). It is easy to check that operator (27) is not potential on the domain (28) with respect to the classical bilinear form (17). In that connection we search for the function M = M (x, t, u, ut ) = 0 and the functional FN [u]such that

t2

δF [u, h] = t1

Ω

M (x, t, u, ut )N (u)δhdxdt

∀u ∈ D(N ),

∀h ∈ D(Nu ).

Using the theorem 2 it is easy to find that M = eu(x,t) . The operator N (u) = eu(x,t) N2 (u) is potential. The corresponding functional is given by t2 FN [u] = eu(x,t−τ ) [a(ut )2 (x, t − τ ) + b(ux )2 (x, t)]dxdt. t1

Ω

Remark 1. Differential equations with several variable deviating arguments occur for example in the technical cybernetics (see [11]). Remark 2. The theorem 1 generalizes appropriate results by Popov [7]. Remark 3. Different approaches to the definition of solutions of different types of functional differential equations are discussed by Hale [3]. Remark 4. The study of the inverse problems of the calculus of variations for equations with deviating arguments has been initiated by Savchin [10]. Acknowledgements. This research was partially supported by the Russian Fund of Basic Research under grant no.05.01.00422A.

References [1] L. E. El’sgol’tz, Qualitative methods in mathematical analysis, Trans. Math. Mono., Amer. Math. Soc., 12 (1964). [2] V. M. Filippov, V. M. Savchin and S. G. Shorokhov, Variational principles for nonpotential operators, J. Math. Sci, 68 (3) (1994), 275– 398.


101

[3] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [4] H. Helmholtz, Ueber die physikalische bedeutung des prinzips der kleinsten wirkung, J.Reine und Angew. Math. 100 (1887), 137–166. [5] D. K. Hughes, Variational and optimal control problem with delayed arguments, J. Optimization Theory Appl., 2 (1968), 1–14. [6] G. A. Kamenskii, Variational and boundary-value problems with deviating argument, Differents. Uravn, 69 (8) (1970), 1349–1358. [7] A. M. Popov, Conditions for potentiality of differential difference equations, Differents. Uravn., 34 (3) (1998), 422–424. [8] L. D. Sabbagh, Variational problems with lags, J. Optimization Theory Appl., 3 (1969), 34–51. [9] R. M. Santilli, Foundations of Theoretical Mechanics 1, The Inverse Problem in Newtonian Mechanics, Springer-Verlag, 1978. [10] V. M. Savchin, Helmholtz’s conditions of potentiality for PDE with deviating arguments, in Theses of the 35th scientific conference of the departement of physico-mathematical and natural science, 1994, May 16-24, Part 2, Mathematical sections. - M.:RUDN, 1994, p. 25. [11] A. V. Solodov and E. A. Solodova, Systems with variable retardation, [in Russian] Nauka, Moscow, 1980. [12] E. Tonti, A general solution of the inverse problem of the calculus of variations, Hadronic J., 5 (4) (1982), 1404–1450. [13] V. Volterra, Lecons sur les Fonctions de Lignes, Paris, Gautier-Villars, 1913. Peoples’ Friendship University of Russia Miklukho-Maklaya str., 6 117198, Moscow Russia (E-mail: [email protected] Email: [email protected] E-mail: [email protected]) (Received : August 2005 )

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