no. 2
Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities
ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 271-294.

We study a finite horizon problem for a system whose evolution is governed by a controlled ordinary differential equation, which takes also account of a hysteretic component: namely, the output of a Preisach operator of hysteresis. We derive a discontinuous infinite dimensional Hamilton-Jacobi equation and prove that, under fairly general hypotheses, the value function is the unique bounded and uniformly continuous viscosity solution of the corresponding Cauchy problem.

DOI : 10.1051/cocv:2004007
Classification : 47J40, 49J15, 49L20, 49L25
Mots clés : hysteresis, optimal control, dynamic programming, viscosity solutions 
@article{COCV_2004__10_2_271_0,
     author = {Bagagiolo, Fabio},
     title = {Viscosity solutions for an optimal control problem with {Preisach} hysteresis nonlinearities},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {271--294},
     publisher = {EDP-Sciences},
     volume = {10},
     number = {2},
     year = {2004},
     doi = {10.1051/cocv:2004007},
     mrnumber = {2083488},
     zbl = {1068.49024},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2004007/}
}
TY  - JOUR
AU  - Bagagiolo, Fabio
TI  - Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2004
SP  - 271
EP  - 294
VL  - 10
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2004007/
DO  - 10.1051/cocv:2004007
LA  - en
ID  - COCV_2004__10_2_271_0
ER  - 
%0 Journal Article
%A Bagagiolo, Fabio
%T Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2004
%P 271-294
%V 10
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2004007/
%R 10.1051/cocv:2004007
%G en
%F COCV_2004__10_2_271_0
Bagagiolo, Fabio. Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities. ESAIM: Control, Optimisation and Calculus of Variations, Tome 10 (2004) no. 2, pp. 271-294. doi : 10.1051/cocv:2004007. http://www.numdam.org/articles/10.1051/cocv:2004007/

[1] F. Bagagiolo, An infinite horizon optimal control problem for some switching systems. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 443-462. | MR | Zbl

[2] F. Bagagiolo, Dynamic programming for some optimal control problems with hysteresis. NoDEA Nonlinear Differ. Equ. Appl. 9 (2002) 149-174. | MR | Zbl

[3] F. Bagagiolo, Optimal control of finite horizon type for a multidimensional delayed switching system. Department of Mathematics, University of Trento, Preprint No. 647 (2003). | MR | Zbl

[4] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | Zbl

[5] G. Barles and P.L. Lions, Fully nonlinear Neumann type boundary conditions for first-order Hamilton-Jacobi equations. Nonlinear Anal. 16 (1991) 143-153. | Zbl

[6] S.A. Belbas and I.D. Mayergoyz, Optimal control of dynamic systems with hysteresis. Int. J. Control 73 (2000) 22-28. | MR | Zbl

[7] S.A. Belbas and I.D. Mayergoyz, Dynamic programming for systems with hysteresis. Physica B Condensed Matter 306 (2001) 200-205.

[8] M. Brokate, ODE control problems including the Preisach hysteresis operator: Necessary optimality conditions, in Dynamic Economic Models and Optimal Control, G. Feichtinger Ed., North-Holland, Amsterdam (1992) 51-68. | MR

[9] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions. Springer, Berlin (1997). | MR | Zbl

[10] M.G. Crandall and P.L. Lions, Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of solutions. J. Funct. Anal. 62 (1985) 379-396. | MR | Zbl

[11] E. Della Torre, Magnetic Hysteresis. IEEE Press, New York (1999).

[12] M.A. Krasnoselskii and A.V. Pokrovskii, Systems with Hysteresis. Springer, Berlin (1989). Russian Ed. Nauka, Moscow (1983). | MR

[13] P. Krejci, Convexity, Hysteresis and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo (1996).

[14] I. Ishii, A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1989) 105-135. | Numdam | MR | Zbl

[15] S.M. Lenhart, T. Seidman and J. Yong, Optimal control of a bioreactor with modal switching. Math. Models Methods Appl. Sci. 11 (2001) 933-949. | MR | Zbl

[16] P.L. Lions, Neumann type boundary condition for Hamilton-Jacobi equations. Duke Math. J. 52 (1985) 793-820. | MR | Zbl

[17] P.L. Lions, Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: the case of bounded stochastic evolutions. Acta Math. 161 (1988) 243-278. | MR | Zbl

[18] I.D. Mayergoyz, Mathematical Models of Hysteresis. Springer, New York (1991). | MR | Zbl

[19] X. Tan and J.S. Baras, Optimal control of hysteresis in smart actuators: a viscosity solutions approach. Center for Dynamics and Control of Smart Actuators, preprint (2002). | Zbl

[20] G. Tao and P.V. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley & Sons, New York (1996). | MR | Zbl

[21] A. Visintin, Differential Models of Hysteresis. Springer, Heidelberg (1994). | MR | Zbl

Cité par Sources :