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 : https://doi.org/10.1051/cocv:2004007
Classification : 47J40,  49J15,  49L20,  49L25
Mots clés : hysteresis, optimal control, dynamic programming, viscosity solutions
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     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},
     zbl = {1068.49024},
     mrnumber = {2083488},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2004007/}
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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 1876884 | Zbl 1036.49005

[2] F. Bagagiolo, Dynamic programming for some optimal control problems with hysteresis. NoDEA Nonlinear Differ. Equ. Appl. 9 (2002) 149-174. | MR 1905823 | Zbl 1009.47071

[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 2129376 | Zbl 1120.49021

[4] M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Boston (1997). | Zbl 0890.49011

[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 0736.35023

[6] S.A. Belbas and I.D. Mayergoyz, Optimal control of dynamic systems with hysteresis. Int. J. Control 73 (2000) 22-28. | MR 1736053 | Zbl 0995.49015

[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 1213692

[9] M. Brokate and J. Sprekels, Hysteresis and Phase Transitions. Springer, Berlin (1997). | MR 1411908 | Zbl 0951.74002

[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 794776 | Zbl 0627.49013

[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 987431

[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 1056130 | Zbl 0701.35052

[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 1850557 | Zbl 1013.92049

[16] P.L. Lions, Neumann type boundary condition for Hamilton-Jacobi equations. Duke Math. J. 52 (1985) 793-820. | MR 816386 | Zbl 0599.35025

[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 971797 | Zbl 0757.93082

[18] I.D. Mayergoyz, Mathematical Models of Hysteresis. Springer, New York (1991). | MR 1083150 | Zbl 0723.73003

[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 1044.82021

[20] G. Tao and P.V. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley & Sons, New York (1996). | MR 1482524 | Zbl 0953.93002

[21] A. Visintin, Differential Models of Hysteresis. Springer, Heidelberg (1994). | MR 1329094 | Zbl 0820.35004

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