Hamilton-Jacobi equations for control problems of parabolic equations

Sophie Gombao; Jean-Pierre Raymond

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 2, page 311-349
  • ISSN: 1292-8119

Abstract

top
We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power ( - A ) β – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power ( - A ) β appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space Y ↪ X . To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable.

How to cite

top

Gombao, Sophie, and Raymond, Jean-Pierre. "Hamilton-Jacobi equations for control problems of parabolic equations." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 311-349. <http://eudml.org/doc/249677>.

@article{Gombao2006,
abstract = { We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power $(-A)^\beta$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power $(-A)^\beta$ appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space $Y\hookrightarrow X$. To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable. },
author = {Gombao, Sophie, Raymond, Jean-Pierre},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamilton-Jacobi-Bellman equation; boundary control; semilinear parabolic equations.; semilinear parabolic equations},
language = {eng},
month = {3},
number = {2},
pages = {311-349},
publisher = {EDP Sciences},
title = {Hamilton-Jacobi equations for control problems of parabolic equations},
url = {http://eudml.org/doc/249677},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Gombao, Sophie
AU - Raymond, Jean-Pierre
TI - Hamilton-Jacobi equations for control problems of parabolic equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 311
EP - 349
AB - We study Hamilton-Jacobi equations related to the boundary (or internal) control of semilinear parabolic equations, including the case of a control acting in a nonlinear boundary condition, or the case of a nonlinearity of Burgers' type in 2D. To deal with a control acting in a boundary condition a fractional power $(-A)^\beta$ – where (A,D(A)) is an unbounded operator in a Hilbert space X – is contained in the Hamiltonian functional appearing in the Hamilton-Jacobi equation. This situation has already been studied in the literature. But, due to the nonlinear term in the state equation, the same fractional power $(-A)^\beta$ appears in another nonlinear term whose behavior is different from the one of the Hamiltonian functional. We also consider cost functionals which are not bounded in bounded subsets in X, but only in bounded subsets in a space $Y\hookrightarrow X$. To treat these new difficulties, we show that the value function of control problems we consider is equal in bounded sets in Y to the unique viscosity solution of some Hamilton-Jacobi-Bellman equation. We look for viscosity solutions in classes of functions which are Hölder continuous with respect to the time variable.
LA - eng
KW - Hamilton-Jacobi-Bellman equation; boundary control; semilinear parabolic equations.; semilinear parabolic equations
UR - http://eudml.org/doc/249677
ER -

References

top
  1. H. Amann, Linear and quasilinear parabolic problems. Vol. I, Abstract linear theory. Birkhäuser Boston Inc., Boston, MA. Monographs Math.89 (1995).  
  2. V. Barbu and G. Da Prato, Hamilton-Jacobi equations in Hilbert spaces, Pitman (Advanced Publishing Program), Boston, MA Res. Notes Math. 86 (1983).  
  3. A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite-dimensional systems. Vol. 1. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1992).  
  4. A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and control of infinite-dimensional systems. Vol. 2. Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1993).  
  5. P. Cannarsa and H. Frankowska, Value function and optimality condition for semilinear control problems. II. Parabolic case. Appl. Math. Optim.33 (1996) 1–33.  
  6. P. Cannarsa and M.E. Tessitore, Cauchy problem for the dynamic programming equation of boundary control. Boundary control and variation (1994) 13–26.  
  7. P. Cannarsa and M.E. Tessitore, Cauchy problem for Hamilton-Jacobi equations and Dirichlet boundary control problems of parabolic type, in Control of partial differential equations and applications (Laredo, 1994), Dekker, New York. Lect. Notes Pure Appl. Math.174 (1996) 31–42.  
  8. P. Cannarsa and M.E. Tessitore, Dynamic programming equation for a class of nonlinear boundary control problems of parabolic type. Cont. Cybernetics25 (1996) 483–495. Distributed parameter systems: modelling and control (1995).  
  9. P. Cannarsa and M.E. Tessitore, Infinite-dimensional Hamilton-Jacobi equations and Dirichlet boundary control problems of parabolic type. SIAM J. Control Optim.34 (1996) 1831–1847.  
  10. M.G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. I. Uniqueness of viscosity solutions. J. Funct. Anal.62 (1985) 379–396.  
  11. M.G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. II. Existence of viscosity solutions. J. Funct. Anal.65 (1986) 368–405.  
  12. M.G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. III. J. Funct. Anal.68 (1986) 214–247.  
  13. M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms. J. Funct. Anal.90 (1990) 237–283.  
  14. M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. V. Unbounded linear terms and B-continuous solutions. J. Funct. Anal.97 (1991) 417–465.  
  15. M.G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. VI. Nonlinear A and Tataru's method refined, in Evolution equations, control theory, and biomathematics (Han sur Lesse 1991), Dekker, New York. Lect. Notes Pure Appl. Math.155 (1994) 51–89.  
  16. M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. VII. The HJB equation is not always satisfied. J. Funct. Anal.125 (1994) 111–148.  
  17. S Gombao, Équations de Hamilton-Jacobi-Bellman pour des problèmes de contrôle d'équations paraboliques semi-linéaires. Approche théorique et numérique. Université Paul Sabatier, Toulouse (2004).  
  18. F. Gozzi, S.S. Sritharan and A. Święch, Viscosity solutions of dynamic-programming equations for the optimal control of the two-dimensional Navier-Stokes equations. Arch. Ration. Mech. Anal.163 (2002) 295–327.  
  19. D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin. Lect. Notes Math. 840 (1981).  
  20. H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces. J. Funct. Anal.105 (1992) 301–341.  
  21. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris (1968).  
  22. S.M. Rankin, III. Semilinear evolution equations in Banach spaces with application to parabolic partial differential equations. Trans. Amer. Math. Soc.336 (1993) 523–535.  
  23. J.-P. Raymond, Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete Contin. Dynam. Syst.3 (1997) 341–370.  
  24. J.-P. Raymond and H. Zidani, Hamiltonian Pontryagin's principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim.39 (1999) 143–177.  
  25. T. Runst and W. Sickel, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, Walter de Gruyter & Co., Berlin, de Gruyter Series in Nonlinear Analysis and Applications3 (1996).  
  26. K. Shimano, A class of Hamilton-Jacobi equations with unbounded coefficients in Hilbert spaces. Appl. Math. Optim.45 (2002) 75–98.  
  27. H.M. Soner, On the Hamilton-Jacobi-Bellman equations in Banach spaces. J. Optim. Theory Appl.57 (1988) 429–437.  
  28. D. Tataru, Viscosity solutions for the dynamic programming equations. Appl. Math. Optim.25 (1992) 109–126.  
  29. D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Differ. Equ.111 (1994) 123–146.  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.