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Composition Functionals in Calculus of Variations. Application to Products and Quotients

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Composition Functionals in Calculus of Variations. Application to Products and Quotients Preprint 11/2005 COMPOSITION FUNCTIONALS IN CALCULUS OF VARIATIONS. APPLICATION TO PRODUCTS AND QUOTIENTS Enrique Castillo, Alberto Luceno,~ Pablo Pedregal E.T.S. Ingenieros Industriales October 2005 Universidad de Castilla - La Mancha http://matematicas.uclm.es/omeva/ 13071 Ciudad Real Composition Functionals in Calculus of Variations. Application to Products and Quotients. Enrique Castillo¤, Alberto Luceno~ ¤ and Pablo Pedregaly ¤ Department of Applied Mathematics and Computational Sciences,University of Cantabria, Spain y Department of Mathematics, University of Castilla-La Mancha, Spain Abstract This paper deals with the problem of the Calculus of Variations for a functional which is the composition of a certain scalar function H with the integral of a vector valued ¯eld f, i.e., of the form x1 H f(x; y(x); y0(x))dx ; x µZ 0 ¶ n 3 n where H : IR ! IR and f : IR ! IR . The integral of f is calculated here componentwise. We examine su±cient conditions for the existence of optimal solutions, and provide rules to ¯nd the Euler-Lagrange and the natural, transversality, Weirstrass-Erdmann and junction conditions for such a functional. Particular attention is paid to the cases of the product and the quotient as we take these as model situations. Finally, the theory is illustrated with a slope stability problem, and an example coming from Economics. Key Words: coercivity, weak lower semicontinuity, Euler-Lagrange equations, product func- tionals, quotient functionals, transversality conditions, natural conditions, Weierstrass -Erdman conditions, slope stability. 1 Introduction Many problems in science and technology can be formulated as problems of the Calculus of Vari- ations where a certain functional is minimized. The Calculus of Variations has been traditionally concerned with functionals of the form x1 0 I(x0; x1; y(x)) = f(x; y(x); y (x))dx; (1) Zx0 where (x0; x1; y(x)) is assumed to belong to adequate spaces for the problem to be tractable. Necessary and su±cient conditions for the existence of a minimum are well-known (see for example Euler (1744), Elsgolc (1962); Gelfand and Fomin (1963); Forray (1968); Bolza (1973)). In particular, ¯rst order necessary conditions such as the following appear in the existing literature: 1. The Euler-Lagrange equation, to be satis¯ed by any extremal that minimizes (1): d f ¡ f 0 = 0: (2) y dx y 2. The natural condition, to be satis¯ed at an end point x = x0 when y(x0) is free: 0 0 fy (x0; y0(x0); y0(x0)) = 0: (3) 3. The transversality condition, to be satis¯ed when an end point x = x0 must be on a given curve y = y¹(x): 0 0 0 0 0 f(x0; y0(x0); y0(x0)) + (y¹0(x0) ¡ y0(x0))fy (x0; y0(x0); y0(x0)) = 0: (4) 1 4. The Weierstrass-Erdmann condition, to be satis¯ed at any angular point x = c: 0 0 0 0 fy (x; y0(x); y0(x)) x=c¡0 ¡ fy (x; y0(x); y0(x)) x=c+0 = 0 (5) 0 0 0 (f(x; y (x); y (¯x)) ¡ y f 0 (x; y (x); y (x)))¯ 0 0 ¯ y 0 0 ¯x=c¡0 0 0 0 ¡ (f(x; y (x); y (x)) ¡ y f 0 (x; y (x); y (x)))¯ = 0 (6) 0 0 y 0 0 ¯x=c+0 ¯ 5. The junction condition for unilateral constraints of the type ¯y ¸ Á(x), to be satis¯ed at the junction point x¹: 0 0 0 0 0 f(x; y0(x); y0(x)) ¡ f(x; y0(x); Á(x)) ¡ (Á (x) ¡ y0(x))fy (x; y0(x); y0(x)) x=x¹ = 0: (7) £ ¤¯ In addition, other second order necessary conditions, such as Legendre's or ¯Jacobi's, and su±cient conditions, such as Legendre's or Hilbert's conditions, are well known for functionals (1). Moreover, su±cient conditions for the existence of optimal solutions involve the so-called \direct method" (Dacorogna (1989, 1992)) which amounts basically to two main ingredients: coercivity and convexity. More precisely, if the integrand f(x; y; ¸) : (x0; x1) £ IR £ IR ! IR is continuous in (y; ¸), measurable in x, convex in ¸ (for every ¯xed pair (x; y)) and satis¯es the coercivity condition f(x; y; ¸) lim = +1 ¸!1 j¸j uniformly in (x; y), then there are global optimal solutions for the minimization problem (1). This is classical and has been extended to the much more complex vector situation (Dacorogna (1989)). There are however interesting problems in which the functional to be minimized is not of the form (1). In this paper we deal with general (non-classical) functionals of the form x1 0 H(x0; x1; y(x)) = H f(x; y(x); y (x))dx ; (8) µZx0 ¶ where f has n components f = (f1; : : : ; fn) and H has n independent variables. In particular, two interesting cases for n = 2 are the product functional: x1 x1 0 0 P (x0; x1; y(x)) = f1(x; y(x); y (x))dx f2(x; y(x); y (x))dx ; (9) µZx0 ¶ µZx0 ¶ and the ratio or quotient functional: x1 0 x0 f1(x; y(x); y (x))dx Q(x0; x1; y(x)) = x1 0 : (10) Rx0 f2(x; y(x); y (x))dx Functionals of the form (8) were dealt witRh by Euler (1744). However, it is really surprising that the scienti¯c community seems to be unaware of it. In fact only very few authors, as Petrov (1968) and Goldstine (1980), cite this important Euler's work, but do it as if it were a secondary contribution. Functionals of the form (10) have appeared in the past in relation to Soil Mechanics problems (see for example Garber (1973), Revilla and Castillo (1977) and Luceno~ (1979)). The product 2 functional is a very particular case of a non-local cost functional, as it can trivially be written as a double integral x1 x1 0 0 P (x0; x1; y(x)) = f1(t; y(t); y (t))f2(s; y(s); y (s)) dt ds: Zx0 Zx0 In general, some non-local integral functionals can be expressed as a double (or multiple) integral of the form x1 x1 F (t; s; y(t); y(s); y0(t); y0(s)) dt ds: Zx0 Zx0 See Pedregal (1997a) for more information on these non-local examples. In this paper, we deal with the problem of minimizing this general type of functionals. In particular, one is interested in answering basic questions such as: 1. What are the corresponding Euler-Lagrange equations for these problems? 2. What are the corresponding natural, transversality, Weierstrass-Erdmann and junction conditions for these problems? 3. Can these problems be reduced to other calculus of variation problems? 4. Sets of su±cient conditions ensuring the existence of optimal solutions. In this paper we answer some of these questions. The paper is organized as follows. Section 2 is concerned with changes needed in the classical direct method so that it can be applied to this class of functionals. We will see that the coercivity issue is very di®erent for the product and the quotient functionals. Indeed, as far as coercivity is concerned we can divide functionals in these two kinds: product-type and quotient-type func- tionals. Section 3 deals with the problem of obtaining a general formula for the Euler-Lagrange equations for the general functional (8), which is then applied to the product (9), the quotient (10) and another more general functional. In Section 4 we give the natural, transversality, Weierstrass-Erdmann and junction conditions for the general functional (8). Sections 5 and 6 illustrate the proposed methods using a slope stability and an economics problem, respectively, Notice that many of our results can be easily generalized and are also valid for much more general situations, such as those functionals involving multiple integrals or multiple unknown functions. Proofs and techniques are very similar, and formally the same, to the ones used here. In addition, the Pontryagin maximum principle can be easily generalized for these more general functionals. 2 Su±cient conditions for existence We ¯rst give a basic, general existence theorem, and then examine with care the two cases of product and quotient of functionals. This result is an adaptation of the direct method to the sort of functionals we are considering. Theorem 1 (First existence theorem.) Let n n f(x; y; ¸) : (x0; x1) £ IR £ IR ! IR ; H(z) : IR ! IR; be given satisfying the following three sets of assumptions: 3 ² regularity and boundedness: f is continuous in (y; ¸) and measurable in x, and H is lower semicontinuous and bounded from below over the subset of IRn n n (x1 ¡ x0)co(im(f)) = z 2 IR : z = sizi; si ¸ 0; si = (x1 ¡ x0); zi 2 im(f) ( ) Xi=0 X by a constant c; ² coercivity: the level sets of H enjoy the following requirement: for each C ¸ c (c has just been determined above) and every sequence z(j) such that z(j) ! 1; c · H(z(j)) · C; ¯ ¯ ¯ ¯ we can ¯nd a subsequence (no¯t rela¯ beled), a constant M and an index i 2 f1; 2; : : : ; ng (all depending possibly on C) such that (j) zi · M and f (x; y; ¸) lim i = +1 j¸j!1 j¸j uniformly in (x; y); ² convexity-monotonicity: each component fi(x; y; ¸) is convex in ¸ for ¯xed (x; y), and H is non-decreasing on each variable zi when the other variables zj are ¯xed values in (x1 ¡ x0)co(im(fj)) for j 6= i. Then there exist global optimal solutions for the variational problem consisting in minimizing the functional x1 0 I(x0; x1; y(x)) = H f(x; y(x); y (x)) dx µZx0 ¶ over the functions y that are absolutely continuous in (x0; x1) complying with suitable boundary conditions and/or other types of restrictions (which should be stable by weak convergence).
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