Optimality conditions for reflecting boundary control problems

详细信息    查看全文

• 作者：Oana Silvia Serea (1)
  
• 关键词：34K35 ; 49J24 ; 49L20 ; 49L25 ; 93B18 ; 93C15 ; Boundary reflection ; Pontryagin鈥檚 principle ; Necessary conditions ; Dynamic programming ; Synthesis for optimal solutions
• 刊名：NoDEA : Nonlinear Differential Equations and Applications
• 出版年：2013
• 出版时间：June 2013
• 年：2013
• 卷：20
• 期：3
• 页码：1225-1242
• 全文大小：311KB
• 参考文献：1. Aubin J.P., Cellina A.: Differential Inclusions, Set-valued Maps and Viability Theory. Springer, Berlin (1984) CrossRef
  2. Aubin J.P., Frankowska H.: Set-valued Analysis. Birkh盲user, Boston (1990)
  3. Bardi, M., Capuzzo-Dolcetta, I.: Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. In: Systems and Control: Foundations and Applications. Birkh盲user, Boston
  4. Barles G.: Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154(1), 191鈥?24 (1999) CrossRef
  5. Barron E.N., Jensen R.: Semicontinuous viscosity solutions for Hamilton-Jacobi equations with convex Hamiltonians. Commun. Partial Differ. Equ. 15(12), 1713鈥?742 (1990) CrossRef
  6. Benoist J.: On ergodic problem for Hamilton-Jacobi-Isaacs equations. Comptes rendus de l鈥橝cad茅mie des sciences. S茅rie 1 Math茅matique 315(8), 941鈥?44 (1992)
  7. Bergounioux M., Zidani H.: Pontryagin maximum principle for optimal control of variational inequalities.. SIAM J. Control Optim. 37, 1273鈥?290 (1999) CrossRef
  8. Bonnans F., Tiba D.: Control problems with mixed constraints and application to an optimal investment problem. Math. Rep. (Rom. Acad Sci.) 11(4), 293鈥?06 (2009)
  9. Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. In: Springer Series in Operations Research. Springer, New York (2000)
  10. Brogliato B., Ten Dam A.A., Paoli L., G茅not F., Abadie M.: Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. ASME Appl. Mech. Rev. 55(2), 107鈥?50 (2002) CrossRef
  11. Cannarsa, P., Sinestrari, C.: Semiconcave functions, Hamilton-Jacobi equations, and optimal control. In: Progress in Nonlinear Differential Equations and their Applications, vol. 58. Birkh盲user, Boston (2004)
  12. Clarke F.: Optimization and Nonsmooth Analysis. Wiley Interscience, New York (1983)
  13. Cornet B.: Existence of slow solutions for a class of differential inclusions. J. Math. Anal. Appl. 96, 130鈥?47 (1983) CrossRef
  14. Crandall M.G., Ishii H., Lions P.L.: User鈥檚 guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. New Ser. 27(1), 1鈥?7 (1992) CrossRef
  15. de Pinho M.R., Rosenblueth J.F.: Necessary conditions for constrained problems under mangasarian-fromowitz conditions. SIAM J. Control Optim. 47(1), 535鈥?52 (2008) CrossRef
  16. Dmitruk, A.V.: Maximum principle for the general optimal control problem with phase and regular mixed constraints. Comput. Math. Model. 4(4), 364鈥?77. Software and models of systems analysis. Optimal control of dynamical systems (1993)
  17. Frankowska H.: A viability approach to the Skorohod problem. Stochastics 14, 227鈥?44 (1985) CrossRef
  18. Frankowska H.: The maximum principle for an optimal solution to a differential inclusion with end points constraints. SIAM J. Control Optim. 25, 145鈥?57 (1987) CrossRef
  19. Frankowska H., Cernea A.: A connection between the maximum principle and dynamic programming for constrained control problems. SIAM J. Control Optim. 44(2), 673鈥?03 (2005) CrossRef
  20. Hiriart-Urruty J.B., Strodiot J.J., Nguyen V.H.: Generalized hessian matrix and second-order optimality conditions for problems with / c ^1,1 data. Appl. Math. Optim. 11, 43鈥?6 (1984) CrossRef
  21. Lions P.L.: Generalized Solutions of Hamilton-Jacobi Equations. Pitman Advanced Publishing Program, Boston (1982)
  22. Lions P.L.: Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52, 793鈥?20 (1985) CrossRef
  23. Lions P.L., Sznitman A.S.: Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37, 511鈥?37 (1984) CrossRef
  24. Milyutin, A.A., Osmolovskii, N.P.: Calculus of variations and optimal control. Translations of Mathematical Monographs. American Mathematical Society, vol 180. Providence (1998). Translated from the Russian manuscript by Dimitrii Chibisov
  25. Mordukhovich, B.S.: Variational analysis and generalized differentiation. I: Basic theory. II: Applications. Springer, Berlin (2005)
  26. Moreau J.J.: Liaisons unilat茅rales sans frottement et chocs in茅lastiques. C. R. Acad. Sci. Paris, Sr. II 296, 1473鈥?476 (1938)
  27. Poliquin R.A., Rockafellar R.T.: Prox-regular functions in variational analysis. Trans. Am. Math. Soc. 348(5), 1805鈥?838 (1996) CrossRef
  28. Poliquin R.A., Rockafellar R.T., Thibault L.: Local differentiability of distance functions. Trans. Am. Math. Soc. 352(11), 5231鈥?249 (2000) CrossRef
  29. Polovinkin E.S., Smirnov G.V.: An approach to differentiation of many-valued mapping and necessary optimality conditions for optimization of solutions of differential inclusions. Differ. Equ. 22, 660鈥?68 (1986)
  30. Rockafellar R.T., Wets R.J.-B.: Variational Analysis. Springer, Berlin (1998) CrossRef
  31. Serea O.S.: On reflecting boundary problem for optimal control. SIAM J. Control Optim. 42(2), 559鈥?75 (2003) CrossRef
  32. Thibault L.: Sweeping process with regular and nonregular sets. J. Differ. Equ. 193(1), 1鈥?6 (2003) CrossRef
  33. Vinter, R.: Optimal control. In: Modern Birkh盲user Classics. Birkh盲user, Boston (2010)
  
• 作者单位：Oana Silvia Serea (1)
  
  1. Laboratoire de Mathematiques et Physique, University of Perpignan, Via Domitia EA 4217, 66860, Perpignan, France
  
• ISSN：1420-9004

文摘

We consider a control problem with reflecting boundary and obtain necessary optimality conditions in the form of the maximum Pontryagin principle. To derive these results we transform the constrained problem in an unconstrained one or we use penalization techniques of Morreau-Yosida type to approach the original problem by a sequence of optimal control problems with Lipschitz dynamics. Then nonsmooth analysis theory is used to study the convergence of the penalization in order to obtain optimality conditions.