A relaxation result for state constrained inclusions in infinite dimension

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

March 2016, 6(1): 113-141. doi: 10.3934/mcrf.2016.6.113

A relaxation result for state constrained inclusions in infinite dimension

Helene Frankowska ^1, , Elsa M. Marchini ^2, and Marco Mazzola ^3,

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

Received December 2014 Revised July 2015 Published January 2016

Abstract
Related Papers

In this paper we consider a state constrained differential inclusion $\dot x\in \mathbb A x+ F(t,x)$, with $\mathbb A$ generator of a strongly continuous semigroup in an infinite dimensional separable Banach space. Under an ``inward pointing condition'' we prove a relaxation result stating that the set of trajectories lying in the interior of the constraint is dense in the set of constrained trajectories of the convexified inclusion $\dot x\in \mathbb A x+ \overline{\textrm{co}}F(t,x)$. Some applications to control problems involving PDEs are given.

Keywords: relaxation, state constraints, mild solution, Differential inclusion, inward pointing condition..

Mathematics Subject Classification: 34G25, 34K09, 49J4.

Citation: Helene Frankowska, Elsa M. Marchini, Marco Mazzola. A relaxation result for state constrained inclusions in infinite dimension. Mathematical Control and Related Fields, 2016, 6 (1) : 113-141. doi: 10.3934/mcrf.2016.6.113

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.

[1]	Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[2]	Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018
[3]	Nidhal Gammoudi, Hasnaa Zidani. A differential game control problem with state constraints. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022008
[4]	Piernicola Bettiol, Hélène Frankowska. Lipschitz regularity of solution map of control systems with multiple state constraints. Discrete and Continuous Dynamical Systems, 2012, 32 (1) : 1-26. doi: 10.3934/dcds.2012.32.1
[5]	Wojciech Kryszewski, Dorota Gabor, Jakub Siemianowski. The Krasnosel'skii formula for parabolic differential inclusions with state constraints. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 295-329. doi: 10.3934/dcdsb.2018021
[6]	Hans-Otto Walther. On solution manifolds of differential systems with discrete state-dependent delays. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022108
[7]	Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations and Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024
[8]	Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287
[9]	Jean-François Babadjian, Clément Mifsud, Nicolas Seguin. Relaxation approximation of Friedrichs' systems under convex constraints. Networks and Heterogeneous Media, 2016, 11 (2) : 223-237. doi: 10.3934/nhm.2016.11.223
[10]	Francesca Faraci, Antonio Iannizzotto. Three nonzero periodic solutions for a differential inclusion. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 779-788. doi: 10.3934/dcdss.2012.5.779
[11]	Yongjian Liu, Zhenhai Liu, Dumitru Motreanu. Differential inclusion problems with convolution and discontinuous nonlinearities. Evolution Equations and Control Theory, 2020, 9 (4) : 1057-1071. doi: 10.3934/eect.2020056
[12]	Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051
[13]	Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825
[14]	Yejuan Wang, Tongtong Liang. Mild solutions to the time fractional Navier-Stokes delay differential inclusions. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3713-3740. doi: 10.3934/dcdsb.2018312
[15]	Yumi Yahagi. Construction of unique mild solution and continuity of solution for the small initial data to 1-D Keller-Segel system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (3) : 1497-1510. doi: 10.3934/dcdsb.2021099
[16]	Nguyen Thi Hoai. Asymptotic approximation to a solution of a singularly perturbed linear-quadratic optimal control problem with second-order linear ordinary differential equation of state variable. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 495-512. doi: 10.3934/naco.2020040
[17]	T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[18]	Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure and Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020
[19]	Mikhail Gusev. On reachability analysis for nonlinear control systems with state constraints. Conference Publications, 2015, 2015 (special) : 579-587. doi: 10.3934/proc.2015.0579
[20]	M. Arisawa, P.-L. Lions. Continuity of admissible trajectories for state constraints control problems. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 297-305. doi: 10.3934/dcds.1996.2.297

2020 Impact Factor: 1.284

Tools

Metrics

Other articles
by authors

on AIMS
Helene Frankowska Elsa M. Marchini Marco Mazzola
on Google Scholar
Helene Frankowska Elsa M. Marchini Marco Mazzola

[Back to Top]