Abstract
The present work was primarily motivated by our findings in the literature of some flaws within the proof of the second-order Legendre necessary optimality condition for fractional calculus of variations problems. Therefore, we were eager to elaborate a correct proof and it turns out that this goal is highly nontrivial, especially when considering final constraints. This paper is the result of our reflections on this subject. Precisely, we consider here a constrained minimization problem of a general Bolza functional that depends on a Caputo fractional derivative of order \(0 < \alpha \le 1\) and on a Riemann–Liouville fractional integral of order \(\beta > 0\), the constraint set describing general mixed initial/final constraints. The main contribution of our work is to derive corresponding first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the transversality conditions and, of course, the Legendre condition. A detailed discussion is provided on the obstructions encountered with the classical strategy, while the new proof that we propose here is based on the Ekeland variational principle. Furthermore, we underline that some subsidiary contributions are provided all along the paper. In particular, we prove an independent and intrinsic result of fractional calculus stating that it does not exist a nontrivial function which is, together with its Caputo fractional derivative of order \(0< \alpha <1\), compactly supported. Moreover, we also discuss some evidences claiming that Riemann–Liouville fractional integrals should be considered in the formulation of fractional calculus of variations problems in order to preserve the existence of solutions.
Similar content being viewed by others
Notes
As a first step toward the open challenge of proving a fractional version of the second-order Legendre necessary optimality condition, the cost functional considered in [47] is the most basic one, containing (only) a classical Lagrange cost depending on a fractional derivative of order \(0 < \alpha \le 1\), under fixed initial/final conditions.
References
Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, vol. 87. Springer, Berlin (2004)
Agrawal, O.P.: Formulation of Euler-Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 272(1), 368–379 (2002)
Agrawal, O.P.: A general formulation and solution scheme for fractional optimal control problems. Nonlinear Dyn. 38(1–4), 323–337 (2004)
Agrawal, O.P., Defterli, O., Baleanu, D.: Fractional optimal control problems with several state and control variables. J. Vib. Control 16(13), 1967–1976 (2010)
Almeida, R.: Variational problems involving a Caputo-type fractional derivative. J. Optim. Theory Appl. 174(1), 276–294 (2017)
Almeida, R.: Optimality conditions for fractional variational problems with free terminal time. Discrete Contin. Dyn. Syst. Ser. S 11(1), 1–19 (2018)
Almeida, R., Malinowska, A.B., Torres, D.F.M.: A fractional calculus of variations for multiple integrals with application to vibrating string. J. Math. Phys. 51(3), 033503 (2010)
Almeida, R., Morgado, M.L.: The Euler-Lagrange and Legendre equations for functionals involving distributed-order fractional derivatives. Appl. Math. Comput. 331, 394–403 (2018)
Almeida, R., Torres, D.F.M.: Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun. Nonlinear Sci. Numer. Simul. 16(3), 1490–1500 (2011)
Almeida, R., Torres, D.F.M.: A discrete method to solve fractional optimal control problems. Nonlinear Dyn. 80(4), 1811–1816 (2015)
Atanacković, T.M., et al.: Variational problems with fractional derivatives: invariance conditions and Noether’s theorem. Nonlinear Anal. 71(5–6), 1504–1517 (2009)
Bagley, R.L., Torvik, P.J.: A theoretical basis for the application of fractional calculus in viscoelasticity. J. Rheol. 27, 201–210 (1983)
Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelasticity behavior. J. Rheol. 30, 133–155 (1986)
Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29(2), 417–437 (2011)
Bergounioux, M., Bourdin, L.: Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints. ESAIM Control Optim. Calc. Var. 26, 38 (2020)
Bourdin, L.: Existence of a weak solution for fractional Euler-Lagrange equations. J. Math. Anal. Appl. 399(1), 239–251 (2013)
Bourdin, L., Cresson, J., Greff, I.: A continuous/discrete fractional Noether’s theorem. Commun. Nonlinear Sci. Numer. Simul. 18(4), 878–887 (2013)
Bourdin, L., Idczak, D.: A fractional fundamental lemma and a fractional integration by parts formula–Applications to critical points of Bolza functionals and to linear boundary value problems. Adv. Differ. Equ. 20(3–4), 213–232 (2015)
Bourdin, L., Odzijewicz, T., Torres, D.F.M.: Existence of minimizers for fractional variational problems containing Caputo derivatives. Adv. Dyn. Syst. Appl. 8(1), 3–12 (2013)
Bourdin, L., Odzijewicz, T., Torres, D.F.M.: Existence of minimizers for generalized Lagrangian functionals and a necessary optimality condition–application to fractional variational problems. Differ. Integral Equ. 27(7–8), 743–766 (2014)
Bressan, A., Piccoli, B.: Introduction to the mathematical theory of control. AIMS Series on Applied Mathematics, 2, American Institute of Mathematical Sciences. AIMS), Springfield, MO (2007)
Brezis, H.: Functional analysis. Universitext, Springer, New York, Sobolev spaces and partial differential equations (2011)
Bullo, F., Lewis, A.D.: Geometric Control of Mechanical Systems, Texts in Applied Mathematics, vol. 49. Springer, New York (2005)
Comte, F.: Opérateurs fractionnaires en économétrie et en finance. Prépublication MAP5 (2001)
Cong, N.D., Tuan, H.T.: Generation of nonlocal fractional dynamical systems by fractional differential equations. J. Integral Equ. Appl. 29(4), 585–608 (2017)
Cresson, J.: Fractional embedding of differential operators and Lagrangian systems. J. Math. Phys. 48(3), 033504 (2007)
Cresson, J., Inizan, P.: Variational formulations of differential equations and asymmetric fractional embedding. J. Math. Anal. Appl. 385(2), 975–997 (2012)
Diethelm, K., Ford, N.J.: Volterra integral equations and fractional calculus: do neighboring solutions intersect? J. Integral Equ. Appl. 24(1), 25–37 (2012)
Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)
Ferreira, R.A.C.: Fractional calculus of variations: a novel way to look at it. Fract. Calc. Appl. Anal. 22(4), 1133–1144 (2019)
Ferreira, R.A.C., Malinowska, A.B.: A counterexample to a Frederico–Torres fractional Noether-type theorem. J. Math. Anal. Appl. 429(2), 1370–1373 (2015)
Ferreira, R.A.C., Torres, D.F.M.: Fractional \(h\)-difference equations arising from the calculus of variations. Appl. Anal. Discrete Math. 5(1), 110–121 (2011)
Frederico, G.S.F., Torres, D.F.M.: Fractional Noether’s theorem in the Riesz-Caputo sense. Appl. Math. Comput. 217(3), 1023–1033 (2010)
Frederico, G.S.F., Torres, D.F.M.: Fractional isoperimetric Noether’s theorem in the Riemann-Liouville sense. Rep. Math. Phys. 71(3), 291–304 (2013)
Gelfand, I.M., Fomin, S.V.: Calculus of variations, revised English edition translated and edited by Richard A. Silverman, Prentice-Hall Inc, Englewood Cliffs, NJ (1963)
Gerolymatou, E., Vardoulakis, I., Hilfer, R.: Modelling infiltration by means of a nonlinear fractional diffusion model. J. Phys. D Appl. Phys. 39, 4104–4110 (2006)
Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J . 68, 46–53 (1995)
Guo, T.L.: The necessary conditions of fractional optimal control in the sense of Caputo. J. Optim. Theory Appl. 156(1), 115–126 (2013)
Hélie, T., Matignon, D.: Diffusive representations for the analysis and simulation of flared acoustic pipes with visco-thermal losses. Math. Models Methods Appl. Sci. 16(4), 503–536 (2006)
Herzallah, M.A.E., Baleanu, D.: Fractional Euler–Lagrange equations revisited. Nonlinear Dyn. 69(3), 977–982 (2012)
Hestenes, M.R.: Calculus of variations and optimal control theory. Robert E. Krieger Publishing Co., Inc., Huntington, N.Y. (1980). Corrected reprint of the 1966 original
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, River Edge (2000)
Hilfer, R.: Fractional calculus and regular variation in thermodynamics. In: Applications of fractional calculus in physics, pp. 429–463. World Sci. Publ., River Edge, NJ (2000)
Jelicic, Z.D., Petrovacki, N.: Optimality conditions and a solution scheme for fractional optimal control problems. Struct. Multidiscip. Optim. 38(6), 571–581 (2009)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V, Amsterdam (2006)
Klimek, M.: Solutions of Euler–Lagrange equations in fractional mechanics. In: XXVI Workshop on Geometrical Methods in Physics, 73–78, AIP Conf. Proc., 956, Amer. Inst. Phys., Melville, NY
Lazo, M.J., Torres, D.F.M.: The Legendre condition of the fractional calculus of variations. Optimization 63(8), 1157–1165 (2014)
Lévy, P.: L’addition des variables aléatoires définies sur un circonférence. Bull. Soc. Math. France 67, 1–41 (1939)
Liberzon, D.: Calculus of Variations and Optimal Control Theory. Princeton University Press, Princeton, NJ (2012)
Magin, R.L.: Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59(5), 1586–1593 (2010)
Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, no. 1, 77 pp (2000)
Oldham, K.B., Spanier, J.: The replacement of Fick’s laws by a formulation involving semidifferentiation. J. Electroanal. Chem. 26, 331–341 (1970)
Pfitzenreiter, T.: A physical basis for fractional derivatives in constitutive equations. Z. Angew. Math. Mech. 84(4), 284–287 (2004)
Pontryagin, L. S. et al.: The mathematical theory of optimal processes, translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt. Interscience Publishers John Wiley & Sons, Inc. New York (1962)
Pooseh, S., Almeida, R., Torres, D.F.M.: Fractional order optimal control problems with free terminal time. J. Ind. Manag. Optim. 10(2), 363–381 (2014)
Riewe, F.: Nonconservative Lagrangian and Hamiltonian mechanics. Phys. Rev. E 53(2), 1890–1899 (1996)
Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in fractional calculus. Springer, Dordrecht (2007)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional integrals and derivatives, translated from the 1987 Russian original. Gordon and Breach Science Publishers, Yverdon (1993)
Schättler, H., Ledzewicz, U.: Geometric Optimal Control, Interdisciplinary Applied Mathematics, vol. 38. Springer, New York (2012)
Song, C.-J., Zhang, Y.: Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications. Fract. Calc. Appl. Anal. 21(2), 509–526 (2018)
van Brunt, B.: The Calculus of Variations. Universitext. Springer, New York (2004)
Zaslavsky, G.M.: Chaos, fractional kinetics, and anomalous transport. Phys. Rep. 371(6), 461–580 (2002)
Zaslavsky, G.M.: Hamiltonian chaos and fractional dynamics, reprint of the 2005 original. Oxford University Press, Oxford (2008)
Zelikin,. M. I.: Control theory and optimization. I, a translation of Homogeneous spaces and the Riccati equation in the calculus of variations (Russian), “Faktorial”, Moscow, 1998, translation by S. A. Vakhrameev, Encyclopaedia of Mathematical Sciences, 86, Springer-Verlag, Berlin (2000)
Zoia, A., Néell, M.-C., Cortis, A.: Continuous-time random-walk model of transport in variably saturated heterogeneous porous media. Phys. Rev. E, 81(3) (2010)
Zoia, A., Néell, M.-C., Joelson, M.: Mass transport subject to time-dependent flow with nonuniform sorption in porous media. Phys. Rev. E 80,(2009)
Acknowledgements
Rui A. C. Ferreira was supported by the “Fundação para a Ciência e a Tecnologia (FCT)” through the program “Stimulus of Scientific Employment, Individual Support-2017 Call” with reference CEECIND/00640/2017.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Nikolai Osmolovskii.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
A Proof of Proposition 3.3
In this section we provide a detailed proof of Proposition 3.3. In order to accomplish it, recall that for all \(\rho \in {\mathbb {R}}\), there exists a real sequence \((\rho _k)_{k \in {\mathbb {N}}}\) such that
for all \(\xi \in (-1,1)\). We also recall that the series convergence is uniform on all compact subsets included in \((-1,1)\). Moreover, if \(\rho \in {\mathbb {R}}\backslash {\mathbb {N}}\), the terms \(\rho _k\) are all different from zero.
Lemma A.1
Let \(c > a\) be a real number and \(u \in \mathrm {L}^1([a,c],{\mathbb {R}})\). If
for all \(k \in {\mathbb {N}}\), then \(u=0\).
Proof
This result easily follows from the density of polynomial functions in \(\mathrm {C}([a,c],{\mathbb {R}})\) and from [22, Corollary 4.24 p.110]. \(\square \)
Lemma A.2
Let \(c > a\) be a real number and \(u \in \mathrm {L}^1([a,c],{\mathbb {R}})\). Let us consider the function
where \(\mu \in {\mathbb {R}}\backslash {\mathbb {N}}\). If the function \(\varPsi \) is polynomial over a subinterval \(I \subset (c,+\infty )\) with a nonempty interior, then \(u = 0\).
Proof
Without loss of generality, we can assume that \(I = [c_1,c_2]\) is compact with \(c< c_1 < c_2\). From the LDC theorem, one can easily see that \(\varPsi \) is of class \(\mathrm {C}^\infty \) with
for all \(t > c\) and all \(r \ge 1\). In the sequel we fix some \(r \ge 1\) larger than the degree of \(\varPsi \) (polynomial over I) plus one. Since \(\mu \in {\mathbb {R}}\backslash {\mathbb {N}}\), we get that
and thus
for all \(t \in I\). From Equality (9) (with \(\rho := \mu -r \in {\mathbb {R}}\backslash {\mathbb {N}}\)) and the uniform convergence of the power series (since \(0 \le \frac{s-a}{t-a} \le \frac{c-a}{c_1 - c } <1\) for all \((t,s) \in I \times [a,c]\)), we get that
for all \(t \in I\). Finally, using the change of variable \(T=\frac{1}{t-a}\), we can write that \( \sum _{k \in {\mathbb {N}}} \lambda _k T^k = 0\) for all \(T \in [\frac{1}{c_2 - a},\frac{1}{c_1-a}]\), where \( \lambda _k := \rho _k \int _a^c (s-a)^k u(s) \, ds \) for all \(k \in {\mathbb {N}}\). Since the zeros of a nonzero power series are isolated, we deduce that \(\lambda _k = 0\) for all \(k \in {\mathbb {N}}\). Since all \(\rho _k\) are different from zero, Lemma A.1 concludes the proof. \(\square \)
We are now in a position to prove Proposition 3.3. Actually we can even prove the more general following statement.
Proposition A.1
Let \(0< \alpha <1\) and \(x \in {}_{\mathrm {c}} \mathrm {AC}^\alpha _{a+}\). If there exist two real numbers \(a \le c < d \le b\) such that:
-
(i)
x is polynomial over [c, d];
-
(ii)
\({}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x](t) = 0_{{\mathbb {R}}^n}\) for almost every \(t \in [c,d]\);
then x is constant over [a, d].
Proof
Without loss of generality, we assume in this proof that \(n=1\). From Proposition 2.5, it holds that \( x(t) = x(a) + \mathrm {I}^\alpha _{a+}[ {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x] ](t)\) for all \(t \in [a,b]\), and thus
for all \(t \in [c,d]\). Let us denote by \(u \in \mathrm {L}^1([a,c],{\mathbb {R}})\) the restriction of \({}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]\) over [a, c]. From the hypothesis, one can easily deduce that the function
is polynomial over (c, d]. From Lemma A.2, we deduce that \(u=0\) and thus \({}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x](t) = 0\) for almost every \(t \in [a,d]\). We deduce that \(x(t)=x(a)\) for all \(t \in [a,d]\) which completes the proof. \(\square \)
B Proof of Theorem 3.2
Our strategy of proof is based on the Ekeland variational principle [29]. Let us enunciate hereafter a simplified version (but sufficient for our purposes).
Proposition B.1
(Ekeland variational principle) Let \((\mathrm {E},\mathrm {d}_\mathrm {E})\) be a complete metric space and \(\mathrm {J}: \mathrm {E}\rightarrow {\mathbb {R}}^+\) be a continuous nonnegative map. Let \(\varepsilon >0\) and \(\lambda \in \mathrm {E}\) such that \(\mathrm {J}(\lambda ) = \varepsilon \). Then there exists \(\lambda _\varepsilon \in \mathrm {E}\) such that \(\mathrm {d}_\mathrm {E}( \lambda _\varepsilon , \lambda ) \le \sqrt{\varepsilon }\) and \(-\sqrt{\varepsilon } \; \mathrm {d}_\mathrm {E}( \lambda ' , \lambda _\varepsilon ) \le \mathrm {J}(\lambda ')-\mathrm {J}(\lambda _\varepsilon )\) for all \(\lambda ' \in \mathrm {E}\).
In Section B.1 we give some recalls about convex analysis. In Section B.2 we investigate the sensitivity analysis of the Bolza functional \({\mathcal {L}}\). Finally the proof of Theorem 3.2 is detailed in Section B.3 by applying the Ekeland variational principle on a penalized functional.
1.1 B.1 Basics of Convex Analysis
Let \(\mathrm {d}_\mathrm {S}: {\mathbb {R}}^j \rightarrow {\mathbb {R}}_{+}\) denote the standard distance function to the nonempty closed convex subset \(\mathrm {S}\subset {\mathbb {R}}^j\) defined by \(\mathrm {d}_\mathrm {S}(z):= \inf _{z' \in \mathrm {S}} \Vert z-z' \Vert _{{\mathbb {R}}^j}\) for all \(z \in {\mathbb {R}}^j\). We recall that for all \(z \in {\mathbb {R}}^j\), there exists a unique element \(\mathrm {P}_\mathrm {S}(z) \in \mathrm {S}\) (called the projection of z onto \(\mathrm {S}\)) such that \(\mathrm {d}_\mathrm {S}(z)=\Vert z-\mathrm {P}_\mathrm {S}(z) \Vert _{{\mathbb {R}}^j}\). It can easily be shown that the map \(\mathrm {P}_\mathrm {S}:{\mathbb {R}}^j \rightarrow \mathrm {S}\) is 1-Lipschitz continuous. Moreover it holds that \(( z- \mathrm {P}_\mathrm {S}(z)) \cdot (z'-\mathrm {P}_\mathrm {S}(z)) \le 0\) for all \(z'\in \mathrm {S}\), that is, \(z-\mathrm {P}_\mathrm {S}(z)\in \mathrm {N}_\mathrm {S}[\mathrm {P}_\mathrm {S}(z)]\) for all \(z \in {\mathbb {R}}^j\). Let us recall the two following required lemmas, whose proofs are detailed for the reader’s convenience.
Lemma B.1
Let \((z_k)_{k\in {\mathbb {N}}}\) be a sequence in \({\mathbb {R}}^j\) converging to some point \(z\in \mathrm {S}\) and let \((\zeta _k)_{k\in {\mathbb {N}}}\) be a positive real sequence. If \(\zeta _k(z_k-\mathrm {P}_\mathrm {S}(z_k))\) converges to some \({\overline{z}} \in {\mathbb {R}}^j\), then \({\overline{z}} \in \mathrm {N}_\mathrm {S}[z]\).
Proof
Since \(z_k - \mathrm {P}_\mathrm {S}(z_k) \in \mathrm {N}_\mathrm {S}[\mathrm {P}_\mathrm {S}(z_k)]\) and \(\zeta _k > 0\) for all \(k \in {\mathbb {N}}\), we obtain that \( \zeta _k (z_k - \mathrm {P}_\mathrm {S}(z_k)) \cdot (z' - \mathrm {P}_\mathrm {S}(z_k)) \le 0\) for all \(z' \in \mathrm {S}\) and all \(k \in {\mathbb {N}}\). Passing to the limit \(k \rightarrow \infty \), and since \(z \in \mathrm {S}\), we obtain that \( {\overline{z}} \cdot (z' - z) \le 0\) for all \(z' \in \mathrm {S}\) which exactly means that \({\overline{z}} \in \mathrm {N}_\mathrm {S}[z]\). \(\square \)
Lemma B.2
The map
is Fréchet-differentiable on \({\mathbb {R}}^j\), and its differential \({\mathcal {D}}\mathrm {d}^2_\mathrm {S}(z)\) at every \(z \in {\mathbb {R}}^j\) can be expressed as
for all \(z' \in {\mathbb {R}}^j\).
Proof
Let \(z \in {\mathbb {R}}^j\) and let us prove that \(\mathrm {d}^2_\mathrm {S}\) is Fréchet-differentiable at z with \( {\mathcal {D}}\mathrm {d}^2_\mathrm {S}(z)(z') = 2 ( z-\mathrm {P}_\mathrm {S}(z) ) \cdot z'\). One has
and, from Cauchy–Schwarz inequality and 1-Lipschitz continuity of \(\mathrm {P}_\mathrm {S}\), one gets
for all \(z' \in {\mathbb {R}}^j\). Using both inequalities, the proof is complete. \(\square \)
1.2 B.2 Sensitivity Analysis of the Bolza Functional
In the proof of Theorem 3.2 (see Section B.3), we denote:
-
by \(r_\alpha \) some real number satisfying \(r_\alpha > \frac{1}{\alpha }\) and by \(r'_\alpha := \frac{r_\alpha }{r_\alpha - 1}\) the classical conjugate of \(r_\alpha \) satisfying \(\frac{1}{r_\alpha } + \frac{1}{r'_\alpha } = 1\);
-
and, for all \((u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n\), by \(x(\cdot ,u,y) \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}\) the function defined by
$$\begin{aligned} x(t,u,y) := y + \mathrm {I}^\alpha _{a+}[u](t), \end{aligned}$$for all \(t \in [a,b]\);
-
and, for all \((u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n\), by \({\mathcal {P}}(u,y)\) the set of Lebesgue points \(\tau \in (a,b)\) of both the functions u and \(L(x(\cdot ,u,y),u,\cdot )\).
Remark B.1
Note that \(r'_\alpha (\alpha -1) +1 > 0\).
Remark B.2
Let \(x \in \mathrm {L}^1\). From Proposition 2.5 and Remark 2.5, note that \(x \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}\) if and only if there exists \((u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n\) such that \(x=x(\cdot ,u,y)\). In that case, the couple (u, y) is unique and is given by \(u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]\) and \(y=x(a)\).
Remark B.3
For all \((u,y) \in \mathrm {L}^\infty \times {\mathbb {R}}^n\), note that the set \({\mathcal {P}}(u,y)\) is of full measure in [a, b].
We introduce the set
for all \(R \ge 0\), where \({\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R )\) denotes the standard closed ball of \({\mathbb {R}}^n\) centered at the origin \(0_{{\mathbb {R}}^n}\) with radius \(R \ge 0\). We endow the set \(\mathrm {L}^\infty _R \times {\mathbb {R}}^n\) with the distance
for all \((u_1,y_1)\), \((u_2,y_2) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\).
Lemma B.3
Let \(R \ge 0\). The following assertions are true:
-
(i)
The metric space \(( \mathrm {L}^\infty _R \times {\mathbb {R}}^n , \mathrm {d}_{ \mathrm {L}^\infty _R \times {\mathbb {R}}^n } )\) is complete;
-
(ii)
The map \((u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n \longmapsto x(\cdot ,u,y) \in \mathrm {C}\) is continuous;
-
(iii)
The map
$$\begin{aligned}{}\begin{array}[t]{lrcl}\varPhi _R :&{}\mathrm {L}^\infty _R \times {\mathbb {R}}^n &{}\longrightarrow &{}{\mathbb {R}}\\ {} &{}(u,y)&{} \longmapsto &{}\varPhi _R (u,y) := {\mathcal {L}}(x(\cdot ,u,y)), \end{array} \end{aligned}$$is continuous.
Proof
The first item can be easily derived from the PCLDC theorem. Secondly, it can be proved from the classical Hölder inequality that
for all \(t \in [a,b]\) and all \((u_1,y_1)\), \((u_2,y_2) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\), which concludes the proof of the second item. The third item can be derived by contradiction and by using the LDC and PCLDC theorems. \(\square \)
The rest of this section is devoted to the sensitivity analysis of the Bolza functional \(\varPhi _R\) under perturbations of the couple (u, y) (see Propositions B.2 and B.3). Before coming to these points, we first introduce the following notion of needle-perturbation of u.
Definition B.1
(Needle-perturbation of u) Let \(R \ge 0\) and let \((u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\). A needle-perturbation of u associated with \((\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\) and \(0 < h \le b-\tau \) is the function \(u^{(\tau ,v)}(\cdot ,h) \in \mathrm {L}^\infty _R\) defined by
for almost every \(t \in [a,b]\).
Lemma B.4
Let \(R \ge 0\) and let \((u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\). It holds that
for all \(t \in [a,b]\) and
for all \(t \in (\tau +h,b]\), all \((\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\) and all \(0 < h \le b-\tau \).
Proof
Let \((\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\) and \(0 < h \le b-\tau \) being fixed for the whole proof. It holds that
for all \(t \in [a,b]\). Since \(u \in \mathrm {L}^\infty _R\) and \(v \in {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\), we obtain that
for all \(t \in [a,b]\). To prove Inequality (10), one has just to see that in all of the three above cases, the right-hand side term is less than \(\frac{2R}{\varGamma (\alpha +1)}h^\alpha \). For the last case, one has just to invoke the basic inequality \(\chi _2^\alpha - \chi _1^\alpha \le (\chi _2 - \chi _1 )^\alpha \) which is satisfied for all \(0 \le \chi _1 \le \chi _2\) (see the proof of Lemma 3.3 for some details). Now let us prove Inequality (11). Using similar arguments, one has
for all \(t \in (\tau +h,b]\). Since \(u \in \mathrm {L}^\infty _R\) and \(v \in {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\), we obtain that
for all \(t \in (\tau +h,b]\). Noting that \((t-s)^{\alpha -1} \le (t-(\tau +h))^{\alpha -1}\) for all \(s \in [\tau ,\tau +h]\) and all \(t \in (\tau +h,b]\), the proof of Inequality (11) is complete. \(\square \)
Proposition B.2
(Sensitivity analysis under needle-perturbation of u) Let \(R \ge 0\) and let \((u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\). It holds that
for all \((\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\), where \(x = x(\cdot ,u,y) \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}\).
Proof
Let \((\tau ,v) \in {\mathcal {P}}(u,y) \times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\). For simplicity of notations in this proof, we denote by
for all \(0 < h \le b-\tau \). From Inequality (11) and since \(\tau \) is a Lebesgue point of u, it is clear that
Moreover the term
can be decomposed as:
for all \(0 < h \le b-\tau \). From the uniform convergence obtained in Inequality (10), the first term tends to zero when \(h \rightarrow 0^+\). Since \(\tau \) is a Lebesgue point of the function \(L(x,u,\cdot )\), it is clear that the second term tends to
when \(h \rightarrow 0^+\). From a Taylor expansion with integral rest, the last term can be decomposed as:
where
for all \(\theta \in [0,1]\) and all \(0 < h \le b-\tau \). The last above term clearly tends to
when \(h \rightarrow 0^+\). From the LDC theorem, the second above term clearly tends to zero when \(h \rightarrow 0^+\). Finally one can prove from Inequality (11) that the norm of the first above term can be bounded by
which tends to zero when \(h \rightarrow 0^+\) (in particular since \(\tau \) is a Lebesgue point of the function u). The proof is complete. \(\square \)
Proposition B.3
(Sensitivity analysis under perturbation of y) Let \(R \ge 0\) and let \((u,y) \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\). It holds that
for all \(y' \in {\mathbb {R}}^n\), where \(x = x(\cdot ,u,y) \in {}_\mathrm {c}\mathrm {AC}^{\alpha ,\infty }_{a+}\).
Proof
We apply Proposition 3.1 with the constant variation \(\eta = y' \in {}_{\mathrm {c}} \mathrm {AC}^{\alpha ,\infty }_{a+}\). \(\square \)
1.3 B.3 Proof of Theorem 3.2 by Applying the Ekeland Variational Principle
Let \(x \in \mathrm {K}\) be a solution to Problem (P). Using the notations introduced in Section B.2, it holds that \(x = x(\cdot ,u,y)\) where \(u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]\) and \(y=x(a)\). In particular note that \(g(y,x(b,u,y))=g(x(a),x(b)) \in \mathrm {S}\). Let \(R := \Vert u \Vert _{\mathrm {L}^\infty } + 1 \ge 0\) and let us consider a positive sequence \((\varepsilon _k)_{k \in {\mathbb {N}}}\) which tends to zero when \(k \rightarrow \infty \). We introduce the penalized functional
for all \(k \in {\mathbb {N}}\). From Lemma B.3 and the continuities of g and \(\mathrm {d}^2_\mathrm {S}\), it is clear that \(J_k\) is a continuous nonnegative map defined on a complete metric space for all \(k \in {\mathbb {N}}\) . Since \(\mathrm {J}_k (u,y) = \varepsilon _k\) for all \(k \in {\mathbb {N}}\), we deduce from the Ekeland variational principle (see Proposition B.1) that there exists a sequence \((u_k,y_k)_{k \in {\mathbb {N}}} \subset \mathrm {L}^\infty _R \times {\mathbb {R}}^n\) such that
and
for all \((u',y') \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\) and all \(k \in {\mathbb {N}}\). In the sequel, we denote by \(x_k := x(\cdot ,u_k,y_k)\) for all \(k \in {\mathbb {N}}\). Note that the sequence \((u_k,y_k)_{k \in {\mathbb {N}}}\) converges to (u, y) in \(\mathrm {L}^\infty _R \times {\mathbb {R}}^n\), and thus the sequence \((x_k)_{k \in {\mathbb {N}}}\) converges to x in \(\mathrm {C}\) (see Lemma B.3).
From the optimality of x, one can easily see that \(\mathrm {J}_k (u',y') > 0\) for all \((u',y') \in \mathrm {L}^\infty _R \times {\mathbb {R}}^n\) and all \(k \in {\mathbb {N}}\). As a consequence, we can correctly define
and
which satisfy \(\vert \psi ^0_k \vert ^2 + \Vert \psi _k \Vert _{{\mathbb {R}}^j}^2 = 1\) for all \(k \in {\mathbb {N}}\). From a standard compactness argument and from the PCLDC theorem, we can extract subsequences (that we do not relabel) such that \((\psi ^0_k)_{k \in {\mathbb {N}}}\) converges to some \(\psi ^0 \le 0\), \((\psi _k)_{k \in {\mathbb {N}}}\) converges to some \(\psi \in {\mathbb {R}}^j \) satisfying \(-\psi \in \mathrm {N}_\mathrm {S}[ g(x(a),x(b)) ]\) (see Lemma B.1) and \((u_k)_{k \in {\mathbb {N}}}\) converges to u pointwisely almost everywhere on [a, b]. Moreover, note that \(\vert \psi ^0 \vert ^2 + \Vert \psi \Vert _{{\mathbb {R}}^j}^2 = 1\) and thus the couple \((\psi ^0,\psi )\) is not trivial.
Perturbation of \(y_k\). Let \(y' \in {\mathbb {R}}^n\) and let us fix some \(k \in {\mathbb {N}}\). From Inequality (12), it holds that
for all \(h > 0\). Letting \(h \rightarrow 0^+\), we exactly get from Proposition B.3 that
Finally, letting \(k \rightarrow \infty \), we obtain (using in particular the LDC theorem) that
Since the above inequality is satisfied for all \(y' \in {\mathbb {R}}^n\) and we can write
we deduce the crucial equality given by
Needle-perturbation of \(u_k\). Let \((\tau ,v) \in {\mathcal {P}}\times {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\) where
Note that \({\mathcal {P}}\) is of full measure in [a, b]. Let us fix some \(k \in {\mathbb {N}}\). From Inequality (12), it holds that
for all \(0 < h \le b - \tau \). Letting \(h \rightarrow 0^+\), we exactly get from Lemma B.4 and Proposition B.2 that
Finally, letting \(k \rightarrow \infty \), we obtain (using in particular the LDC theorem and the fact that \((u_k(\tau ))_{k \in {\mathbb {N}}}\) converges to \(u(\tau )\)) that
Note that the above crucial inequality is satisfied for almost all \(\tau \in [a,b]\) and all \(v \in {\overline{\mathrm {B}}}_{{\mathbb {R}}^n}(0_{{\mathbb {R}}^n},R)\).
Introduction of an adjoint vector. Let us introduce the adjoint vector p defined by
for almost every \(t \in [a,b]\). In particular it holds that
for all \(t \in [a,b]\). We deduce that \(p \in \mathrm {AC}^\alpha _{b-}\) with
for almost every \(t \in [a,b]\). Moreover, in particular from Equality (13), it holds that
From Inequality (14), it is clear that
for almost every \(t \in [a,b]\). One can easily deduce that
for almost every \(t \in [a,b]\).
Normalization. By contradiction, let us assume that \(\psi ^0 = 0\). In that case, one can easily deduce from the above equalities that \( \partial _1 g(x(a),x(b))^\top \times \psi = \partial _2 g(x(a),x(b)) )^\top \times \psi = 0_{{\mathbb {R}}^n}\). Since g is assumed to be regular at (x(a), x(b)), we deduce that \(\psi = 0_{{\mathbb {R}}^j}\) which raises a contradiction with the nontriviality of the couple \((\psi ^0,\psi )\). We deduce that \(\psi ^0 < 0\). Moreover, since the couple \((\psi ^0,\psi )\) is defined up to a positive multiplicative constant, we now normalize the couple \((\psi ^0,\psi )\) such that \(\psi ^0 = -1\).
End of the proof. We deduce from the previous paragraphs that \(p = \frac{(b-\cdot )^{\beta -1}}{\varGamma (\beta )} \partial _2 L (x,u,\cdot ) \in \mathrm {AC}^\alpha _{b-}\) with
for almost every \(t \in [a,b]\), which exactly corresponds to the Euler–Lagrange equation stated in Theorem 3.2 since \(u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]\). We also deduce from the equalities on \(\mathrm {I}^{1-\alpha }_{b-} [p](a)\) and \(\mathrm {I}^{1-\alpha }_{b-} [p](b)\) in a previous paragraph that the transversality conditions given in Theorem 3.2 are satisfied. Finally, from the maximization condition (15), it is clear that the matrix \(\frac{(b-t)^{\beta -1}}{\varGamma (\beta )} \partial ^2_{22} L(x(t),u(t),t)\) is positive semi-definite for almost all \(t \in [a,b]\), which exactly corresponds to the Legendre condition given in Theorem 3.2 since \(u = {}_{\mathrm {c}} \mathrm {D}^\alpha _{a+}[x]\).
Rights and permissions
About this article
Cite this article
Bourdin, L., Ferreira, R.A.C. Legendre’s Necessary Condition for Fractional Bolza Functionals with Mixed Initial/Final Constraints. J Optim Theory Appl 190, 672–708 (2021). https://doi.org/10.1007/s10957-021-01908-w
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10957-021-01908-w
Keywords
- Fractional calculus of variations
- Bolza functional
- Mixed initial/final constraints
- Euler–Lagrange equation
- Transversality conditions
- Legendre condition
- Riemann–Liouville and Caputo fractional operators
- Ekeland variational principle