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

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

Received  December 2014 Revised  July 2015 Published  January 2016

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.
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.

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