Control theory
Some Regularity Properties on Bolza problems in the Calculus of Variations
Comptes Rendus. Mathématique, Volume 360 (2022) no. G3, pp. 205-218.

The paper summarizes the main core of the last results that we obtained in [8, 4, 17] on the regularity of the value function for a Bolza problem of a one-dimensional, vectorial problem of the calculus of variations. We are concerned with a nonautonomous Lagrangian that is possibly highly discontinuous in the state and velocity variables, nonconvex in the velocity variable and non coercive. The main results are achieved under the assumption that the Lagrangian is convex on the one-dimensional lines of the velocity variable and satisfies a local Lipschitz continuity condition w.r.t. the time variable, known in the literature as Property (S), and strictly related to the validity of the Erdmann–Du-Bois Reymond equation.

Under our assumptions, there exists a minimizing sequence of Lipschitz functions. A first consequence is that we can exclude the presence of the Lavrentiev phenomenon. Moreover, under a further mild growth assumption satisfied by the minimal length functional, fully described in the paper, the above sequence may be taken with the same Lipschitz rank, even when the initial datum and initial value vary on a compact set. The Lipschitz regularity of the value function follows.

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DOI: 10.5802/crmath.220
Classification: 49J05, 49J15, 49J30, 49N60
Bernis, Julien 1; Bettiol, Piernicola 1; Mariconda, Carlo 2

1 Université de Brest, UMR CNRS 6205, Laboratoire de Mathématiques de Bretagne Atlantique, 6 Avenue Victor Le Gorgeu, 29200 Brest, France
2 University of Padua, Dipartimento di Matematica “Tullio Levi-Civita”, Padova 35121, Italy
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Bernis, Julien; Bettiol, Piernicola; Mariconda, Carlo. Some Regularity Properties on Bolza problems in the Calculus of Variations. Comptes Rendus. Mathématique, Volume 360 (2022) no. G3, pp. 205-218. doi : 10.5802/crmath.220. http://www.numdam.org/articles/10.5802/crmath.220/

[1] Alberti, Giovanni; Serra Cassano, Francesco Non-occurrence of gap for one-dimensional autonomous functionals, Calculus of variations, homogenization and continuum mechanics (Marseille, 1993) (Series on Advances in Mathematics for Applied Sciences), Volume 18, World Scientific, 1994, pp. 1-17 | MR | Zbl

[2] Ambrosio, Luigi; Ascenzi, Oscar; Buttazzo, Giuseppe Lipschitz regularity for minimizers of integral functionals with highly discontinuous integrands, J. Math. Anal. Appl., Volume 142 (1989), pp. 301-316 | DOI | MR | Zbl

[3] Ball, John M.; Mizel, Victor J. One-dimensional variational problems whose minimizers do not satisfy the Euler–Lagrange equation, Arch. Ration. Mech. Anal., Volume 90 (1985), pp. 325-388 | DOI | MR | Zbl

[4] Bernis, Julien; Bettiol, Piernicola; Mariconda, Carlo Lipschitz regularity of the value function and solutions to the Hamilton–Jacobi equation for nonautonomous problems in the calculus of variations (2020) (in preparation)

[5] Bettiol, Piernicola; Mariconda, Carlo A new variational inequality in the calculus of variations and Lipschitz regularity of minimizers, J. Differ. Equations, Volume 268 (2020) no. 5, pp. 2332-2367 | DOI | MR | Zbl

[6] Bettiol, Piernicola; Mariconda, Carlo Regularity and necessary conditions for a Bolza optimal control problem, J. Math. Anal. Appl., Volume 489 (2020) no. 1, p. 124123, 17 | DOI | MR | Zbl

[7] Bettiol, Piernicola; Mariconda, Carlo A Du Bois-Reymond convex inclusion for non-autonomous problems of the Calculus of Variations and regularity of minimizers, Appl. Math. Optim., Volume 83 (2021), pp. 2083-2107 | DOI | Zbl

[8] Bettiol, Piernicola; Mariconda, Carlo Uniform boundedness for the optimal controls of a discontinuous, non-convex Bolza problem (2021) (submitted)

[9] Buttazzo, Giuseppe; Giaquinta, Mariano; Hildebrandt, Stefan One-dimensional variational problems. An introduction, Oxford Lecture Series in Mathematics and its Applications, 15, Clarendon Press, 1998, viii+262 pages

[10] Cellina, Arrigo The classical problem of the calculus of variations in the autonomous case: relaxation and Lipschitzianity of solutions, Trans. Am. Math. Soc., Volume 356 (2004) no. 1, pp. 415-426 | DOI | MR | Zbl

[11] Cellina, Arrigo; Ferriero, Alessandro Existence of Lipschitzian solutions to the classical problem of the calculus of variations in the autonomous case, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 20 (2003) no. 6, pp. 911-919 | DOI | Numdam | MR | Zbl

[12] Cellina, Arrigo; Treu, Giulia; Zagatti, Sandro On the minimum problem for a class of non-coercive functionals, J. Differ. Equations, Volume 127 (1996) no. 1, pp. 225-262 | DOI | MR | Zbl

[13] Clarke, Francis H. An indirect method in the calculus of variations, Trans. Am. Math. Soc., Volume 336 (1993) no. 2, pp. 655-673 | DOI | MR | Zbl

[14] Clarke, Francis H.; Vinter, Richard B. Regularity properties of solutions to the basic problem in the calculus of variations, Trans. Am. Math. Soc., Volume 289 (1985), pp. 73-98 | DOI | MR | Zbl

[15] Dal Maso, Gianni; Frankowska, Hélène Autonomous integral functionals with discontinuous nonconvex integrands: Lipschitz regularity of minimizers, DuBois-Reymond necessary conditions, and Hamilton–Jacobi equations, Appl. Math. Optim., Volume 48 (2003) no. 1, pp. 39-66 | MR | Zbl

[16] Manià, Basilio Sopra un esempio di Lavrentieff, Boll. Unione Mat. Ital., Volume 13 (1934), pp. 146-153 | Zbl

[17] Mariconda, Carlo Equi-Lipschitz minimizing trajectories for non coercive, discontinuous, non convex Bolza controlled-linear optimal control problems (2020) (submitted)

[18] Mariconda, Carlo; Treu, Giulia Lipschitz regularity of the minimizers of autonomous integral functionals with discontinuous non-convex integrands of slow growth, Calc. Var. Partial Differ. Equ., Volume 29 (2007) no. 1, pp. 99-117 | DOI | MR | Zbl

[19] Natanson, Isidor P. Theory of functions of real variable, Frederick Ungar Publishing Co., 1955, 277 pages (translated by Leo F. Boron with the collaboration of Edwin Hewitt)

[20] Serrin, James; Varberg, Dale E. A general chain rule for derivatives and the change of variables formula for the Lebesgue integral, Am. Math. Mon., Volume 76 (1969), pp. 514-520 | DOI | MR | Zbl

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