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doi: 10.3934/eect.2021042
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## Well-posed control problems related to second-order differential inclusions

 LMPA Laboratory, Department of Mathematics, Jijel University, 18000, Algeria

* Corresponding author: Mustapha Fateh Yarou

Received  September 2020 Revised  July 2021 Early access August 2021

The paper deals with quadratic optimal control problems, we study the equivalence between well-posed problems and affinity on the control for a second-order differential inclusions with two-points conditions, governed by a maximal monotone operator in a finite dimensional space.

Citation: Doria Affane, Mustapha Fateh Yarou. Well-posed control problems related to second-order differential inclusions. Evolution Equations & Control Theory, doi: 10.3934/eect.2021042
##### References:
 [1] S. Adly, A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics, Springer Briefs in Mathematics, 2017. doi: 10.1007/978-3-319-68658-5.  Google Scholar [2] S. Adly, B. Brogliato and B. K. Le, Well-posedness, robustness and stability analysis of a set-valued controller for Lagrangian systems, SIAM J. Control Optim., 51 (2013), 1592-1614.  doi: 10.1137/120872450.  Google Scholar [3] D. Affane, Quelques Problèmes de Contrôle Optimal pour des Inclusions Différentielles, Ph.D. thesis, MSBY University of Jijel, Algeria, 2012. Google Scholar [4] D. Affane and D. Azzam-Laouir, A control problem governed by a second-order differential inclusion, Applic. Anal., 88 (2009), 1677-1690.  doi: 10.1080/00036810903330520.  Google Scholar [5] E. Asplund, $\breve{C}$eby$\breve{s}$ev sets in Hilbert space, Trans. Amer. Math. Soc., 114 (1969), 235-240.  doi: 10.2307/1995279.  Google Scholar [6] H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.  Google Scholar [7] H. Attouch, R. I. Bot and E. R. Csetnek, Fast optization via inertial dynamics with closed-loop damping, arXiv: 2008.02261v3, [math.OC] (2021), 1–63. Google Scholar [8] H. Attouch and J. B. Wets, Quantitative stability of variational systems: III. $\epsilon$-appoximate solutions, (Title: Lipschitzian stabilty of $\epsilon$-appoximate solutions in convex optimization), IIASA. Laxenburg, 25 (1987), 87. Google Scholar [9] H. Attouch and J. B. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc., 328 (1991), 695-729.  doi: 10.2307/2001800.  Google Scholar [10] H. Attouch and J. B. Wets, Quantitative stability of variational systems: II. A framework for nonlinear conditioning, SIAM J. Optimization, 3 (1993), 359-381.  doi: 10.1137/0803016.  Google Scholar [11] D. Azzam-Laouir and S. Lounis, Nonconvex perturbations of second order maximal monotone differential inclusions, Topol. Meth. Nonlin. Anal., 35 (2010), 305-317.   Google Scholar [12] J. Bastien, Systémes Dynamiques Discrets Avec Frottement et Identification en Biomécanique, Mémoire d'Habilitation à Diriger des Recherches, Université Lyon 1, 2013. Google Scholar [13] G. Beer, Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3.  Google Scholar [14] B. Brogliato and A. Tanwani, Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability, SIAM Review, Society for Industrial and Applied Mathematics, 62 (2020), 3-129.  doi: 10.1137/18M1234795.  Google Scholar [15] A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Springer-Verlag, Berlin Heidelberg, 1993. doi: 10.1007/BFb0084195.  Google Scholar [16] I. Ekeland and R. Temam, Analyse Convexe and Problémes Variationelles, Dunod and Gauthier-Villars, Paris, 1974. Google Scholar [17] A. D. Ioffe and A. J. Zaslavski, Variational principles and well-posedness in optimization and calculus of variations, SIAM J. Control Optim., 38 (2000), 566-581.  doi: 10.1137/S0363012998335632.  Google Scholar [18] R. Lucchetti and T. Zolezzi, On well-posedness and stability analysis in optimization, in: Mathematical programming with data perturbations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 195 (1997), 223-251.  Google Scholar [19] E. Muselli, Affinity and well-posedness for optimal control problems in hilbert spaces, J. Convex Anal., 14 (2007), 767-784.   Google Scholar [20] L. Paoli and A. Petrov, Global solutions to phase change models with heat transfer for a class of smart materials, Nonlinear Anal. Real World Appl., 17 (2014), 47-63.  doi: 10.1016/j.nonrwa.2013.10.005.  Google Scholar [21] T. Zolezzi, A characterization of well-posed optimal control systems, SIAM J. Control Optim., 19 (1981), 604-616.  doi: 10.1137/0319038.  Google Scholar [22] T. Zolezzi, Well-posedness criteria in optimization with application to the calculus of variations, Nonlinear Anal., 25 (1995), 437-453.  doi: 10.1016/0362-546X(94)00142-5.  Google Scholar [23] T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266.  doi: 10.1007/BF02192292.  Google Scholar [24] T. Zolezzi, Well-posedness and conditioning of optimization problems, Pliska Stud. Math. Bulgar., 12 (1998), 267-280.   Google Scholar

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##### References:
 [1] S. Adly, A Variational Approach to Nonsmooth Dynamics: Applications in Unilateral Mechanics and Electronics, Springer Briefs in Mathematics, 2017. doi: 10.1007/978-3-319-68658-5.  Google Scholar [2] S. Adly, B. Brogliato and B. K. Le, Well-posedness, robustness and stability analysis of a set-valued controller for Lagrangian systems, SIAM J. Control Optim., 51 (2013), 1592-1614.  doi: 10.1137/120872450.  Google Scholar [3] D. Affane, Quelques Problèmes de Contrôle Optimal pour des Inclusions Différentielles, Ph.D. thesis, MSBY University of Jijel, Algeria, 2012. Google Scholar [4] D. Affane and D. Azzam-Laouir, A control problem governed by a second-order differential inclusion, Applic. Anal., 88 (2009), 1677-1690.  doi: 10.1080/00036810903330520.  Google Scholar [5] E. Asplund, $\breve{C}$eby$\breve{s}$ev sets in Hilbert space, Trans. Amer. Math. Soc., 114 (1969), 235-240.  doi: 10.2307/1995279.  Google Scholar [6] H. Attouch, Variational Convergence for Functions and Operators, Pitman, Boston, 1984.  Google Scholar [7] H. Attouch, R. I. Bot and E. R. Csetnek, Fast optization via inertial dynamics with closed-loop damping, arXiv: 2008.02261v3, [math.OC] (2021), 1–63. Google Scholar [8] H. Attouch and J. B. Wets, Quantitative stability of variational systems: III. $\epsilon$-appoximate solutions, (Title: Lipschitzian stabilty of $\epsilon$-appoximate solutions in convex optimization), IIASA. Laxenburg, 25 (1987), 87. Google Scholar [9] H. Attouch and J. B. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc., 328 (1991), 695-729.  doi: 10.2307/2001800.  Google Scholar [10] H. Attouch and J. B. Wets, Quantitative stability of variational systems: II. A framework for nonlinear conditioning, SIAM J. Optimization, 3 (1993), 359-381.  doi: 10.1137/0803016.  Google Scholar [11] D. Azzam-Laouir and S. Lounis, Nonconvex perturbations of second order maximal monotone differential inclusions, Topol. Meth. Nonlin. Anal., 35 (2010), 305-317.   Google Scholar [12] J. Bastien, Systémes Dynamiques Discrets Avec Frottement et Identification en Biomécanique, Mémoire d'Habilitation à Diriger des Recherches, Université Lyon 1, 2013. Google Scholar [13] G. Beer, Topologies on Closed and Closed Convex Sets, Mathematics and Its Applications, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8149-3.  Google Scholar [14] B. Brogliato and A. Tanwani, Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability, SIAM Review, Society for Industrial and Applied Mathematics, 62 (2020), 3-129.  doi: 10.1137/18M1234795.  Google Scholar [15] A. L. Dontchev and T. Zolezzi, Well-Posed Optimization Problems, Springer-Verlag, Berlin Heidelberg, 1993. doi: 10.1007/BFb0084195.  Google Scholar [16] I. Ekeland and R. Temam, Analyse Convexe and Problémes Variationelles, Dunod and Gauthier-Villars, Paris, 1974. Google Scholar [17] A. D. Ioffe and A. J. Zaslavski, Variational principles and well-posedness in optimization and calculus of variations, SIAM J. Control Optim., 38 (2000), 566-581.  doi: 10.1137/S0363012998335632.  Google Scholar [18] R. Lucchetti and T. Zolezzi, On well-posedness and stability analysis in optimization, in: Mathematical programming with data perturbations, Lecture Notes in Pure and Appl. Math., Dekker, New York, 195 (1997), 223-251.  Google Scholar [19] E. Muselli, Affinity and well-posedness for optimal control problems in hilbert spaces, J. Convex Anal., 14 (2007), 767-784.   Google Scholar [20] L. Paoli and A. Petrov, Global solutions to phase change models with heat transfer for a class of smart materials, Nonlinear Anal. Real World Appl., 17 (2014), 47-63.  doi: 10.1016/j.nonrwa.2013.10.005.  Google Scholar [21] T. Zolezzi, A characterization of well-posed optimal control systems, SIAM J. Control Optim., 19 (1981), 604-616.  doi: 10.1137/0319038.  Google Scholar [22] T. Zolezzi, Well-posedness criteria in optimization with application to the calculus of variations, Nonlinear Anal., 25 (1995), 437-453.  doi: 10.1016/0362-546X(94)00142-5.  Google Scholar [23] T. Zolezzi, Extended well-posedness of optimization problems, J. Optim. Theory Appl., 91 (1996), 257-266.  doi: 10.1007/BF02192292.  Google Scholar [24] T. Zolezzi, Well-posedness and conditioning of optimization problems, Pliska Stud. Math. Bulgar., 12 (1998), 267-280.   Google Scholar
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