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Sequential optimization for semilinear divergent hyperbolic equation with a boundary control and state inequality constraint

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Języki publikacji
EN
Abstrakty
EN
An optimal control problem with a state constraint of inequality type and with dynamics described by a semilinear hyperbolic equation in divergence form with the non-homogeneous boundary condition of the third kind is considered. The state constraint contains a functional parameter that belongs to the class of continuous functions and occurs as an additive term. We study the properties of solutions of linear hyperbolic equations in divergence form with measures in the original data and compute the first variations of functionals on the basis of a so-called two-parameter needle variation of controls. We consider the necessary conditions for minimizing sequences in an optimal control problem with a pointwise in time state constraint of inequality type and with dynamics described by a semilinear hyperbolic equation in divergence form with the non-homogeneous boundary condition of the third kind. For the parametric optimization problem, we also consider regularity and normality conditions stipulated by the differential properties of its value function.
Rocznik
Strony
183--226
Opis fizyczny
Bibliogr. 47 poz.
Twórcy
  • Mathematical Department, Nizhnii Novgorod State University, Gagarin street 23, 603950 Nizhnii Novgorod, Russia
autor
  • Mathematical Department, Nizhnii Novgorod State University, Gagarin street 23, 603950 Nizhnii Novgorod, Russia
Bibliografia
  • 1. BERGOUNIOUX, M. (1992) A Penalization Method for Optimal Control of Elliptic Problems with State Constraints. SIAM J. Control Optim., 30, 2, 305-323.
  • 2. BONNANS, J.F. and CASAS, E. (1995) An Extension of Pontryagin’s Principle for State-Constrained Optimal Control of Semilinear Elliptic Equations and Variational Inequalities. SIAM J. Control Optim., 33, 1, 274-298.
  • 3. CASAS, E. (1993) Boundary Control of Semilinear Elliptic Equations with Point wise State Constraints. SIAM J. Control Optim., 31, 4, 993– 1006.
  • 4. CASAS, E. (1997) Pontryagin’s Principle for State-Constrained Boundary Control Problems of Semilinear Parabolic Equations. SIAM J. Control Optim., 35, 4, 1297-1327.
  • 5. CASAS, E., RAYMOND, J.-P., and ZIDANI, H. (2000) Pontryagin’s Principle for Local Solutions of Control Problems with Mixed Control-State Constraints.
  • 6. SIAM J. Control Optim., 39, 4, 1182-1208.
  • 7. CLARKE, F.H. (1983) Optimization and Nonsmooth Analysis. Wiley, New York.
  • 8. EKELAND, I. (1974) On the Variational Principle. J. Math. Anal. Appl., 47, 324–353.
  • 9. GAVRILOV, V.S. (2012) Divergent hyperbolic differential equation with various boundary value conditions on various parts of boundary. Vestnik Nizhegorodskogo universiteta im. N.I. Lobachevskogo, 4, 183-192 [in Russian].
  • 10. GAVRILOV, V.S. and SUMIN, M.I. (2004) Parametric Optimization of Nonlinear Goursat-Darboux Systems with State Constraints. Computational Mathematics and Mathematical Physics, 44, 6, 949-968.
  • 11. GAVRILOV, V.S. and SUMIN, M.I. (2005) A Parametric Problem of the Suboptimal Control of the Goursat Darboux System with a Pointwise Phase Constraint. Russian Math. (Izv. vuzov), 49, 6, 37-48.
  • 12. GAVRILOV, V.S. and SUMIN, M.I. (2011a) Parametric Optimization for a Hyperbolic Equation in Divergence Form with a Pointwise State Constraint: I. Differential Equations, 47, 4, 547-559.
  • 13. GAVRILOV, V.S. and SUMIN, M.I. (2011b) Parametric Optimization for a Hyperbolic Equation in Divergence Form with a Pointwise State Constraint: II. Differential Equations, 47, 5, 726-737.
  • 14. GAVRILOV, V.S. and SUMIN, M.I. (2011c) Perturbation Method in the Theory of Pontryagin Maximum Principle for Optimal Control of Divergent Semilinear Hyperbolic Equations with Pointwise State Constraints. In: Control Theory and its Applications, Nova Science Publishers Inc., New York: Chapter 4, 83-144.
  • 15. LADYZHENSKAYA, O.A. (1973) Boundary–Value Problems of Mathematical Physics. Nauka, Moscow [in Russian].
  • 16. LI, X. and YONG, J (1995) Optimal Control Theory for Infinite Dimensional Systems. Birkh¨auser Verlag, Basel.
  • 17. LIONS, J.-L., MAGENES, E. (1968) Problemes aux limites non homog`enes et applications, volume 1. Dunod, Paris.
  • 18. MACKENROTH, U. (1982) Convex Parabolic Boundary Control Problems with Pointwise State Constraints. J. Math. Anal. Appl., 87, 256-277.
  • 19. MACKENROTH, U. (1986) On Some Elliptic Optimal Control Problems with State Constraints. Optimization, 17, 595-607.
  • 20. MORDUKHOVICH, B.S. (1976) Maximum principle in problems of time optimal control with state constraints. J. Applied Math. Mech., 40, 960-969.
  • 21. MORDUKHOVICH, B.S. (1980) Metric approximation and necessary optimality conditions for general classes of extremal problems. Soviet. Math. Dokl., 22, 526-530.
  • 22. MORDUKHOVICH, B.S. (1988) Approximation Methods in Problems of Optimization and Control. Nauka, Moscow [in Russian].
  • 23. MORDUKHOVICH, B.S. (2006a) Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin.
  • 24. MORDUKHOVICH, B.S. (2006b) Variational Analysis and Generalized Differentiation, II: Applications. Springer, Berlin.
  • 25. MORDUKHOVICH, B.S. and RAYMOND, J.-P. (2004) Dirichlet Boundary Control of Hyperbolic Equations in the Presence of State Constraints. Appl. Math. Optim., 49, 145-157.
  • 26. MORDUKHOVICH, B.S. and RAYMOND, J.-P. (2005) Neumann Boundary Control of Hyperbolic Equations with Pointwise State Constraints. SIAM J. Control Optim., 43, 4, 1354-1372.
  • 27. MORDUKHOVICH, B.S. and SHAO, Y. (1996) Nonsmooth sequential analysis in Asplund spaces. Trans. Amer. Math. Soc., 346, 4, 1235-1280.
  • 28. NOVOZHENOV, M.M. and PLOTNIKOV, V.I. (1982) Generalized Lagrange Multipliers Rule for Distributed Systems with State Constraints. Diff. Uravn., 18, 4, 584-692. [in Russian]
  • 29. OSIPOV, Yu.S., VASIL’EV, F.P., and POTAPOV, M.M. (1999) Fundamentals of the Dynamic Regularization Method. Mosk. Gos. Univ., Moscow [in Russian].
  • 30. RAYMOND, J.-P. and ZIDANI, H. (1998) Pontryagin’s Principle for State- Constrained Control Problems Governed by Parabolic Equations with Unbounded Controls. SIAM J. Control Optim., 36,6, 1853-1879.
  • 31. STANE, E.M. (1970) Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton.
  • 32. SUMIN, M.I. (1983) Optimal Control Problems for Lumped and Distributed Systems with Differentiable and Nondifferentiable Functionals and Functions Defining Systems. Candidate’s Dissertation in Mathematics and Physics. Gor’kov. Gos. Univ., Gor’kii [in Russian].
  • 33. SUMIN, M.I. (1986) On Minimizing Sequences in Optimal Control Problems with Constrained State Coordinates. Diff. Uravn., 22, 10, 1719-1731 [In Russian].
  • 34. SUMIN, M.I. (1989) Optimal Control of Objects Described by Quasilinear Elliptic Equations. Differ. Uravn., 25, 8, 1406-1416 [in Russian].
  • 35. SUMIN, M.I. (1991) On First Variation in Optimal Control Theory for Distributed Parameters Systems. Diff. Uravn., 27, 12. 2179-2181 [in Russian].
  • 36. SUMIN, M.I. (1997) Suboptimal Control of Distributed Parameters Systems: Minimizing Sequences and the Value Function. Computational Mathematics and Mathematical Physics, 37, 1, 21-39.
  • 37. SUMIN, M.I. (2000a) Optimal Control of Semilinear Elliptic Equation with State Constraint: Maximum Principle for Minimizing Sequence, Regularity, Normality, Sensitivity. Control and Cybernetics, 29, 2, 449-472.
  • 38. SUMIN, M.I. (2000b) Mathematical Theory of Suboptimal Control for Distributed Systems. Doctoral Dissertation in Mathematics and Physics, Nizhegorodskii Gos. Univ., Nizhnii Novgorod [in Russian].
  • 39. SUMIN, M.I. (2000c) Suboptimal Control of Semilinear Elliptic Equations with Phase Constraints, I: The Maximum Principle for Minimizing Sequences and Normality. Russian Math. (Izv. vuzov), 44, 6, 31-42.
  • 40. SUMIN, M.I. (2000d) Suboptimal Control of Semilinear Elliptic Equations with Phase Constraints, II: Sensitivity, Genericity of the Regular Maximum Principle. Russian Math. (Izv. vuzov), 44, 8, 50-60.
  • 41. SUMIN, M.I. (2001) Suboptimal Control of a Semilinear Elliptic Equation with a Phase Constraint and a Boundary Control. Differential Equations, 37, 2, 281-300.
  • 42. SUMIN, M.I. (2009) The First Variation and Pontryagin’s Maximum Principle in Optimal Control for Partial Differential Equations. Computational Mathematics and Mathematical Physics, 49, 6, 958-978.
  • 43. SUMIN, M.I. (2011) Regularized parametric Kuhn–Tucker theorem in a Hilbert space. Computational Mathematics and Mathematical Physics, 51, 9, 1489-1509.
  • 44. SUMIN, M.I. (2012) On the Stable Sequential Kuhn–Tucker Theorem and its Applications. Applied Mathematics, 3, 10A (Special issue “Optimization”), 1334-1350.
  • 45. SUMIN, M.I. and TRUSHINA, E.V. (2008) Minimizing sequences in optimal control with approximately given input data and the regularizing properties of the Pontryagin maximum principle. Computational Mathematics and Mathematical Physics, 48, 2, 209-224.
  • 46. WARD, A.L. (1935) Differentiability of VectorMonotone Functions. Proc. Lon- don Math. Soc., 32, 2, 339-362.
  • 47. WARGA, J. (1972) Optimal Control of Differential and Functional Equations. Academic Press, New York.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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