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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

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 & 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.  doi: 10.1016/0022-0396(79)90088-3.  Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems,, Academic Press, (1993).   Google Scholar

[3]

A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.1,, Birkhäuser, (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Ration. Mech. Anal., 37 (1970), 297.   Google Scholar

[14]

H. O. Fattorini, Infinite-dimensional Optimization and Control Theory,, Cambridge University Press, (1999).  doi: 10.1017/CBO9780511574795.  Google Scholar

[15]

H. Frankowska, A priori estimates for operational differential inclusions,, J. Differential Equations, 84 (1990), 100.  doi: 10.1016/0022-0396(90)90129-D.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis,, Dover, (1975).   Google Scholar

[21]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories,, Cambridge University Press, (2000).   Google Scholar

[22]

X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems,, Birkhäuser, (1995).  doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[23]

R. H. Martin, Jr., Invariant sets for perturbed semigroups of linear operators,, Ann. Mat. Pura Appl., 105 (1975), 221.  doi: 10.1007/BF02414931.  Google Scholar

[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.   Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[26]

D. Preiss, Differentiability of Lipschitz functions on Banach spaces,, J. Functional Anal., 91 (1990), 312.  doi: 10.1016/0022-1236(90)90147-D.  Google Scholar

[27]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1983).   Google Scholar

[28]

H. M. Soner, Optimal control with state-space constraints,, SIAM J. Control Optim., 24 (1986), 552.  doi: 10.1137/0324032.  Google Scholar

[29]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[30]

V. Volterra, Sur les équations intégro-différentielles et leurs applications,, Acta Math., 35 (1912), 295.  doi: 10.1007/BF02418820.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. Barbu, Analysis and Control of Nonlinear Infinite Dimensional Systems,, Academic Press, (1993).   Google Scholar

[3]

A. Bensoussan, G. Da Prato, M. Delfour and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, vol.1,, Birkhäuser, (1992).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

C. M. Dafermos, Asymptotic stability in viscoelasticity,, Arch. Ration. Mech. Anal., 37 (1970), 297.   Google Scholar

[14]

H. O. Fattorini, Infinite-dimensional Optimization and Control Theory,, Cambridge University Press, (1999).  doi: 10.1017/CBO9780511574795.  Google Scholar

[15]

H. Frankowska, A priori estimates for operational differential inclusions,, J. Differential Equations, 84 (1990), 100.  doi: 10.1016/0022-0396(90)90129-D.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

A. N. Kolmogorov and S. V. Fomin, Introductory Real Analysis,, Dover, (1975).   Google Scholar

[21]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories,, Cambridge University Press, (2000).   Google Scholar

[22]

X. Li and J. Yong, Optimal Control Theory for Infinite-Dimensional Systems,, Birkhäuser, (1995).  doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[23]

R. H. Martin, Jr., Invariant sets for perturbed semigroups of linear operators,, Ann. Mat. Pura Appl., 105 (1975), 221.  doi: 10.1007/BF02414931.  Google Scholar

[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.   Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer-Verlag, (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[26]

D. Preiss, Differentiability of Lipschitz functions on Banach spaces,, J. Functional Anal., 91 (1990), 312.  doi: 10.1016/0022-1236(90)90147-D.  Google Scholar

[27]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer-Verlag, (1983).   Google Scholar

[28]

H. M. Soner, Optimal control with state-space constraints,, SIAM J. Control Optim., 24 (1986), 552.  doi: 10.1137/0324032.  Google Scholar

[29]

R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer, (1988).  doi: 10.1007/978-1-4612-0645-3.  Google Scholar

[30]

V. Volterra, Sur les équations intégro-différentielles et leurs applications,, Acta Math., 35 (1912), 295.  doi: 10.1007/BF02418820.  Google Scholar

[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.  Google Scholar

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