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Optimal control for a phase field system with a possibly singular potential
A relaxation result for state constrained inclusions in infinite dimension
1. | CNRS, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Universités, UPMC Univ Paris 06, Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris, France |
2. | Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy |
3. | UPMC Univ Paris 06, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Sorbonne Universités, CNRS, Univ Paris Diderot, Sorbonne Paris Cité, Case 247, 4 Place Jussieu, 75252 Paris, France |
References:
[1] |
N. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225.
doi: 10.1016/0022-0396(79)90088-3. |
[2] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, New York, 1993. |
[3] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.1, Birkhäuser, Boston, 1992. |
[4] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.2, Birkhäuser, Boston, 1993.
doi: 10.1007/978-0-8176-4581-6. |
[5] |
P. Bettiol, A. Bressan and R. B. Vinter, On trajectories satisfying a state constraint: $W^{1,1}$ estimates and counter-examples, SIAM J. Control Optim., 48 (2010), 4664-4679.
doi: 10.1137/090769788. |
[6] |
P. Bettiol, H. Frankowska and R. B. Vinter, $L^\infty$ estimates on trajectories confined to a closed subset, J. Differential Equations, 252 (2012), 1912-1933.
doi: 10.1016/j.jde.2011.09.007. |
[7] |
L. Boltzmann, Zur theorie der elastischen nachwirkung, Wien. Ber., 70 (1874), 275-306. |
[8] |
L. Boltzmann, Zur theorie der elastischen nachwirkung, Wied. Ann., 5 (1878), 430-432. |
[9] |
P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation, Springer-Verlag, New York, 1967.
doi: 10.1007/978-3-642-64981-3. |
[10] |
P. Cannarsa, H. Frankowska and E. M. Marchini, On Bolza optimal control problems with constraints, Discrete Contin. Dyn. Syst., 11 (2009), 629-653.
doi: 10.3934/dcdsb.2009.11.629. |
[11] |
P. Cannarsa, H. Frankowska and E. M. Marchini, Optimal control for evolution equations with memory, J. Evol. Equ., 13 (2013), 197-227. |
[12] |
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
doi: 10.1137/1.9781611971309. |
[13] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. |
[14] |
H. O. Fattorini, Infinite-dimensional Optimization and Control Theory, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511574795. |
[15] |
H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations, 84 (1990), 100-128.
doi: 10.1016/0022-0396(90)90129-D. |
[16] |
H. Frankowska and M. Mazzola, Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints, Calc. Var. Partial Differential Equations, 46 (2013), 725-747.
doi: 10.1007/s00526-012-0501-8. |
[17] |
H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, Nonlinear Differ. Equ. Appl., 20 (2013), 361-383.
doi: 10.1007/s00030-012-0183-0. |
[18] |
H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Equations, 161 (2000), 449-478.
doi: 10.1006/jdeq.2000.3711. |
[19] |
H. Frankowska and R. B. Vinter, Existence of neighbouring feasible trajectories: Applications to dynamic programming for state constrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 21-40.
doi: 10.1023/A:1004668504089. |
[20] |
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover, New York, 1975. |
[21] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Cambridge University Press, Cambridge, 2000. |
[22] |
X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Birkhäuser, Boston, 1995.
doi: 10.1007/978-1-4612-4260-4. |
[23] |
R. H. Martin, Jr., Invariant sets for perturbed semigroups of linear operators, Ann. Mat. Pura Appl., 105 (1975), 221-239.
doi: 10.1007/BF02414931. |
[24] |
J. V. Outrata and Z. Schindler, An augmented Lagrangian method for a class of convex continuous optimal control problems, Problems Control Inform. Theory, 10 (1981), 67-81. |
[25] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[26] |
D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Functional Anal., 91 (1990), 312-345.
doi: 10.1016/0022-1236(90)90147-D. |
[27] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. |
[28] |
H. M. Soner, Optimal control with state-space constraints, SIAM J. Control Optim., 24 (1986), 552-561.
doi: 10.1137/0324032. |
[29] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4612-0645-3. |
[30] |
V. Volterra, Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.
doi: 10.1007/BF02418820. |
[31] |
V. Volterra, Leçons sur les Fonctions De Lignes, Gauthier-Villars, Paris, 1913. |
[32] |
A. P. Wierzbicki and S. Kurcyusz, Projection on a cone, penalty functionals and duality theory for problems with inequality constraints in Hilbert space, SIAM J. Control Optim., 15 (1977), 25-56.
doi: 10.1137/0315003. |
show all references
References:
[1] |
N. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225.
doi: 10.1016/0022-0396(79)90088-3. |
[2] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems, Academic Press, Boston, New York, 1993. |
[3] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.1, Birkhäuser, Boston, 1992. |
[4] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.2, Birkhäuser, Boston, 1993.
doi: 10.1007/978-0-8176-4581-6. |
[5] |
P. Bettiol, A. Bressan and R. B. Vinter, On trajectories satisfying a state constraint: $W^{1,1}$ estimates and counter-examples, SIAM J. Control Optim., 48 (2010), 4664-4679.
doi: 10.1137/090769788. |
[6] |
P. Bettiol, H. Frankowska and R. B. Vinter, $L^\infty$ estimates on trajectories confined to a closed subset, J. Differential Equations, 252 (2012), 1912-1933.
doi: 10.1016/j.jde.2011.09.007. |
[7] |
L. Boltzmann, Zur theorie der elastischen nachwirkung, Wien. Ber., 70 (1874), 275-306. |
[8] |
L. Boltzmann, Zur theorie der elastischen nachwirkung, Wied. Ann., 5 (1878), 430-432. |
[9] |
P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation, Springer-Verlag, New York, 1967.
doi: 10.1007/978-3-642-64981-3. |
[10] |
P. Cannarsa, H. Frankowska and E. M. Marchini, On Bolza optimal control problems with constraints, Discrete Contin. Dyn. Syst., 11 (2009), 629-653.
doi: 10.3934/dcdsb.2009.11.629. |
[11] |
P. Cannarsa, H. Frankowska and E. M. Marchini, Optimal control for evolution equations with memory, J. Evol. Equ., 13 (2013), 197-227. |
[12] |
F. H. Clarke, Optimization and Nonsmooth Analysis, SIAM, Philadelphia, 1990.
doi: 10.1137/1.9781611971309. |
[13] |
C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37 (1970), 297-308. |
[14] |
H. O. Fattorini, Infinite-dimensional Optimization and Control Theory, Cambridge University Press, Cambridge, 1999.
doi: 10.1017/CBO9780511574795. |
[15] |
H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations, 84 (1990), 100-128.
doi: 10.1016/0022-0396(90)90129-D. |
[16] |
H. Frankowska and M. Mazzola, Discontinuous solutions of Hamilton-Jacobi-Bellman equation under state constraints, Calc. Var. Partial Differential Equations, 46 (2013), 725-747.
doi: 10.1007/s00526-012-0501-8. |
[17] |
H. Frankowska and M. Mazzola, On relations of the adjoint state to the value function for optimal control problems with state constraints, Nonlinear Differ. Equ. Appl., 20 (2013), 361-383.
doi: 10.1007/s00030-012-0183-0. |
[18] |
H. Frankowska and F. Rampazzo, Filippov's and Filippov-Wazewski's theorems on closed domains, J. Differential Equations, 161 (2000), 449-478.
doi: 10.1006/jdeq.2000.3711. |
[19] |
H. Frankowska and R. B. Vinter, Existence of neighbouring feasible trajectories: Applications to dynamic programming for state constrained optimal control problems, J. Optim. Theory Appl., 104 (2000), 21-40.
doi: 10.1023/A:1004668504089. |
[20] |
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis, Dover, New York, 1975. |
[21] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Cambridge University Press, Cambridge, 2000. |
[22] |
X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems, Birkhäuser, Boston, 1995.
doi: 10.1007/978-1-4612-4260-4. |
[23] |
R. H. Martin, Jr., Invariant sets for perturbed semigroups of linear operators, Ann. Mat. Pura Appl., 105 (1975), 221-239.
doi: 10.1007/BF02414931. |
[24] |
J. V. Outrata and Z. Schindler, An augmented Lagrangian method for a class of convex continuous optimal control problems, Problems Control Inform. Theory, 10 (1981), 67-81. |
[25] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[26] |
D. Preiss, Differentiability of Lipschitz functions on Banach spaces, J. Functional Anal., 91 (1990), 312-345.
doi: 10.1016/0022-1236(90)90147-D. |
[27] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983. |
[28] |
H. M. Soner, Optimal control with state-space constraints, SIAM J. Control Optim., 24 (1986), 552-561.
doi: 10.1137/0324032. |
[29] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Springer, New York, 1988.
doi: 10.1007/978-1-4612-0645-3. |
[30] |
V. Volterra, Sur les équations intégro-différentielles et leurs applications, Acta Math., 35 (1912), 295-356.
doi: 10.1007/BF02418820. |
[31] |
V. Volterra, Leçons sur les Fonctions De Lignes, Gauthier-Villars, Paris, 1913. |
[32] |
A. P. Wierzbicki and S. Kurcyusz, Projection on a cone, penalty functionals and duality theory for problems with inequality constraints in Hilbert space, SIAM J. Control Optim., 15 (1977), 25-56.
doi: 10.1137/0315003. |
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