Hamilton-Jacobi equations for control problems of parabolic equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 2, pp. 311-349.

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 YX. 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.

DOI: 10.1051/cocv:2006004
Classification: 49K20, 49L25
Keywords: Hamilton-Jacobi-Bellman equation, boundary control, semilinear parabolic equations
@article{COCV_2006__12_2_311_0,
     author = {Gombao, Sophie and Raymond, Jean-Pierre},
     title = {Hamilton-Jacobi equations for control problems of parabolic equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {311--349},
     publisher = {EDP-Sciences},
     volume = {12},
     number = {2},
     year = {2006},
     doi = {10.1051/cocv:2006004},
     mrnumber = {2209356},
     zbl = {1113.49031},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2006004/}
}
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
PY  - 2006
SP  - 311
EP  - 349
VL  - 12
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2006004/
DO  - 10.1051/cocv:2006004
LA  - en
ID  - COCV_2006__12_2_311_0
ER  - 
%0 Journal Article
%A Gombao, Sophie
%A Raymond, Jean-Pierre
%T Hamilton-Jacobi equations for control problems of parabolic equations
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2006
%P 311-349
%V 12
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2006004/
%R 10.1051/cocv:2006004
%G en
%F COCV_2006__12_2_311_0
Gombao, Sophie; Raymond, Jean-Pierre. Hamilton-Jacobi equations for control problems of parabolic equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 12 (2006) no. 2, pp. 311-349. doi : 10.1051/cocv:2006004. http://archive.numdam.org/articles/10.1051/cocv:2006004/

[1] H. Amann, Linear and quasilinear parabolic problems. Vol. I, Abstract linear theory. Birkhäuser Boston Inc., Boston, MA. Monographs Math. 89 (1995). | MR | Zbl

[2] V. Barbu and G. Da Prato, Hamilton-Jacobi equations in Hilbert spaces, Pitman (Advanced Publishing Program), Boston, MA Res. Notes Math. 86 (1983). | MR | Zbl

[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). | MR | Zbl

[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). | MR | Zbl

[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. | Zbl

[6] P. Cannarsa and M.E. Tessitore, Cauchy problem for the dynamic programming equation of boundary control. Boundary control and variation (1994) 13-26. | Zbl

[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. | Zbl

[8] P. Cannarsa and M.E. Tessitore, Dynamic programming equation for a class of nonlinear boundary control problems of parabolic type. Cont. Cybernetics 25 (1996) 483-495. Distributed parameter systems: modelling and control (1995). | Zbl

[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. | Zbl

[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. | Zbl

[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. | Zbl

[12] M.G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions. III. J. Funct. Anal. 68 (1986) 214-247. | Zbl

[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. | Zbl

[14] M.G. Crandall and P.-L. Lions, Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions 97 (1991) 417-465. | Zbl

[15] M.G. Crandall and P.-L. Lions, Hamilton-Jacobi equations in infinite dimensions 155 (1994) 51-89. | Zbl

[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. | Zbl

[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. Świȩch, 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. | Zbl

[19] D. Henry, Geometric theory of semilinear parabolic equations, Springer-Verlag, Berlin. Lect. Notes Math. 840 (1981). | MR | Zbl

[20] H. Ishii, Viscosity solutions for a class of Hamilton-Jacobi equations in Hilbert spaces. J. Funct. Anal. 105 (1992) 301-341. | Zbl

[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). | MR | Zbl

[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. | Zbl

[23] J.-P. Raymond, Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete Contin. Dynam. Syst. 3 (1997) 341-370. | Zbl

[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. | Zbl

[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 Applications 3 (1996). | MR | Zbl

[26] K. Shimano, A class of Hamilton-Jacobi equations with unbounded coefficients in Hilbert spaces. Appl. Math. Optim. 45 (2002) 75-98. | Zbl

[27] H.M. Soner, On the Hamilton-Jacobi-Bellman equations in Banach spaces. J. Optim. Theory Appl. 57 (1988) 429-437. | Zbl

[28] D. Tataru, Viscosity solutions for the dynamic programming equations. Appl. Math. Optim. 25 (1992) 109-126. | Zbl

[29] D. Tataru, Viscosity solutions for Hamilton-Jacobi equations with unbounded nonlinear term: a simplified approach. J. Differ. Equ. 111 (1994) 123-146. | Zbl

Cited by Sources: