# Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 10, Issue: 2, page 271-294
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topBagagiolo, Fabio. "Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities." ESAIM: Control, Optimisation and Calculus of Variations 10.2 (2010): 271-294. <http://eudml.org/doc/90730>.

@article{Bagagiolo2010,

abstract = {
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.
},

author = {Bagagiolo, Fabio},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Hysteresis; optimal control; dynamic programming; viscosity
solutions.; hysteresis; viscosity solutions},

language = {eng},

month = {3},

number = {2},

pages = {271-294},

publisher = {EDP Sciences},

title = {Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities},

url = {http://eudml.org/doc/90730},

volume = {10},

year = {2010},

}

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

DA - 2010/3//

PB - EDP Sciences

VL - 10

IS - 2

SP - 271

EP - 294

AB -
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.

LA - eng

KW - Hysteresis; optimal control; dynamic programming; viscosity
solutions.; hysteresis; viscosity solutions

UR - http://eudml.org/doc/90730

ER -

## References

top- F. Bagagiolo, An infinite horizon optimal control problem for some switching systems. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 443-462. Zbl1036.49005
- F. Bagagiolo, Dynamic programming for some optimal control problems with hysteresis. NoDEA Nonlinear Differ. Equ. Appl.9 (2002) 149-174. Zbl1009.47071
- F. Bagagiolo, Optimal control of finite horizon type for a multidimensional delayed switching system. Department of Mathematics, University of Trento, Preprint No. 647 (2003). Zbl1120.49021
- M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997). Zbl0890.49011
- G. Barles and P.L. Lions, Fully nonlinear Neumann type boundary conditions for first-order Hamilton–Jacobi equations. Nonlinear Anal.16 (1991) 143-153. Zbl0736.35023
- S.A. Belbas and I.D. Mayergoyz, Optimal control of dynamic systems with hysteresis. Int. J. Control73 (2000) 22-28.
- S.A. Belbas and I.D. Mayergoyz, Dynamic programming for systems with hysteresis. Physica B Condensed Matter306 (2001) 200-205.
- 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.
- M. Brokate and J. Sprekels, Hysteresis and Phase Transitions. Springer, Berlin (1997). Zbl0951.74002
- 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. Zbl0627.49013
- E. Della Torre, Magnetic Hysteresis. IEEE Press, New York (1999).
- M.A. Krasnoselskii and A.V. Pokrovskii, Systems with Hysteresis. Springer, Berlin (1989). Russian Ed. Nauka, Moscow (1983).
- P. Krejci, Convexity, Hysteresis and Dissipation in Hyperbolic Equations. Gakkotosho, Tokyo (1996).
- 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. Zbl0701.35052
- 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. Zbl1013.92049
- P.L. Lions, Neumann type boundary condition for Hamilton-Jacobi equations. Duke Math. J.52 (1985) 793-820.
- 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. Zbl0757.93082
- I.D. Mayergoyz, Mathematical Models of Hysteresis. Springer, New York (1991). Zbl0723.73003
- 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). Zbl1044.82021
- G. Tao and P.V. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley & Sons, New York (1996). Zbl0953.93002
- A. Visintin, Differential Models of Hysteresis. Springer, Heidelberg (1994). Zbl0820.35004

## NotesEmbed ?

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