-
Previous Article
Local exact controllability to positive trajectory for parabolic system of chemotaxis
- MCRF Home
- This Issue
-
Next Article
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.
doi: 10.1016/0022-0396(79)90088-3. |
[2] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems,, Academic Press, (1993).
|
[3] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.1,, Birkhäuser, (1992).
|
[4] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.2,, Birkhäuser, (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.
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.
doi: 10.1016/j.jde.2011.09.007. |
[7] |
L. Boltzmann, Zur theorie der elastischen nachwirkung,, Wien. Ber., 70 (1874), 275. Google Scholar |
[8] |
L. Boltzmann, Zur theorie der elastischen nachwirkung,, Wied. Ann., 5 (1878), 430. Google Scholar |
[9] |
P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation,, Springer-Verlag, (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.
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. Google Scholar |
[12] |
F. H. Clarke, Optimization and Nonsmooth Analysis,, SIAM, (1990).
doi: 10.1137/1.9781611971309. |
[13] |
C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Ration. Mech. Anal., 37 (1970), 297.
|
[14] |
H. O. Fattorini, Infinite-dimensional Optimization and Control Theory,, Cambridge University Press, (1999).
doi: 10.1017/CBO9780511574795. |
[15] |
H. Frankowska, A priori estimates for operational differential inclusions,, J. Differential Equations, 84 (1990), 100.
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.
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.
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.
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.
doi: 10.1023/A:1004668504089. |
[20] |
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis,, Dover, (1975).
|
[21] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories,, Cambridge University Press, (2000).
|
[22] |
X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems,, Birkhäuser, (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.
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.
|
[25] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (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.
doi: 10.1016/0022-1236(90)90147-D. |
[27] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1983).
|
[28] |
H. M. Soner, Optimal control with state-space constraints,, SIAM J. Control Optim., 24 (1986), 552.
doi: 10.1137/0324032. |
[29] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (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.
doi: 10.1007/BF02418820. |
[31] |
V. Volterra, Leçons sur les Fonctions De Lignes,, Gauthier-Villars, (1913). Google Scholar |
[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.
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.
doi: 10.1016/0022-0396(79)90088-3. |
[2] |
V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems,, Academic Press, (1993).
|
[3] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.1,, Birkhäuser, (1992).
|
[4] |
A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.2,, Birkhäuser, (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.
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.
doi: 10.1016/j.jde.2011.09.007. |
[7] |
L. Boltzmann, Zur theorie der elastischen nachwirkung,, Wien. Ber., 70 (1874), 275. Google Scholar |
[8] |
L. Boltzmann, Zur theorie der elastischen nachwirkung,, Wied. Ann., 5 (1878), 430. Google Scholar |
[9] |
P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation,, Springer-Verlag, (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.
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. Google Scholar |
[12] |
F. H. Clarke, Optimization and Nonsmooth Analysis,, SIAM, (1990).
doi: 10.1137/1.9781611971309. |
[13] |
C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Ration. Mech. Anal., 37 (1970), 297.
|
[14] |
H. O. Fattorini, Infinite-dimensional Optimization and Control Theory,, Cambridge University Press, (1999).
doi: 10.1017/CBO9780511574795. |
[15] |
H. Frankowska, A priori estimates for operational differential inclusions,, J. Differential Equations, 84 (1990), 100.
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.
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.
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.
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.
doi: 10.1023/A:1004668504089. |
[20] |
A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis,, Dover, (1975).
|
[21] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories,, Cambridge University Press, (2000).
|
[22] |
X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems,, Birkhäuser, (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.
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.
|
[25] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (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.
doi: 10.1016/0022-1236(90)90147-D. |
[27] |
J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1983).
|
[28] |
H. M. Soner, Optimal control with state-space constraints,, SIAM J. Control Optim., 24 (1986), 552.
doi: 10.1137/0324032. |
[29] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (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.
doi: 10.1007/BF02418820. |
[31] |
V. Volterra, Leçons sur les Fonctions De Lignes,, Gauthier-Villars, (1913). Google Scholar |
[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.
doi: 10.1137/0315003. |
[1] |
Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020110 |
[2] |
Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392 |
[3] |
Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020440 |
[4] |
Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129 |
[5] |
Xuemei Chen, Julia Dobrosotskaya. Inpainting via sparse recovery with directional constraints. Mathematical Foundations of Computing, 2020, 3 (4) : 229-247. doi: 10.3934/mfc.2020025 |
[6] |
Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2020 doi: 10.3934/jgm.2021001 |
[7] |
Junkee Jeon. Finite horizon portfolio selection problems with stochastic borrowing constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 733-763. doi: 10.3934/jimo.2019132 |
[8] |
Nan Zhang, Linyi Qian, Zhuo Jin, Wei Wang. Optimal stop-loss reinsurance with joint utility constraints. Journal of Industrial & Management Optimization, 2021, 17 (2) : 841-868. doi: 10.3934/jimo.2020001 |
[9] |
Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104 |
[10] |
Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021027 |
[11] |
Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020297 |
[12] |
Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002 |
[13] |
Dominique Chapelle, Philippe Moireau, Patrick Le Tallec. Robust filtering for joint state-parameter estimation in distributed mechanical systems. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 65-84. doi: 10.3934/dcds.2009.23.65 |
[14] |
Shuxing Chen, Jianzhong Min, Yongqian Zhang. Weak shock solution in supersonic flow past a wedge. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 115-132. doi: 10.3934/dcds.2009.23.115 |
[15] |
Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020 doi: 10.3934/dcdsb.2020340 |
[16] |
Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040 |
[17] |
Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 |
[18] |
Darko Dimitrov, Hosam Abdo. Tight independent set neighborhood union condition for fractional critical deleted graphs and ID deleted graphs. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 711-721. doi: 10.3934/dcdss.2019045 |
[19] |
Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230 |
[20] |
Tomáš Bodnár, Philippe Fraunié, Petr Knobloch, Hynek Řezníček. Numerical evaluation of artificial boundary condition for wall-bounded stably stratified flows. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 785-801. doi: 10.3934/dcdss.2020333 |
2019 Impact Factor: 0.857
Tools
Metrics
Other articles
by authors
[Back to Top]