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Viscosity solutions for an optimal control problemwith Preisach hysteresis nonlinearities

Published online by Cambridge University Press:  15 March 2004

Fabio Bagagiolo
Fabio Bagagiolo*
Affiliation:
Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050 Povo-Trento, Italy; [email protected].
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Abstract

We study a finite horizon problem for a system whose evolution isgoverned by a controlled ordinary differential equation, which takesalso account of a hysteretic component: namely, the outputof a Preisach operator of hysteresis. We derive a discontinuousinfinitedimensional Hamilton–Jacobi equation and prove that, under fairlygeneral hypotheses, the value function is the unique bounded anduniformly continuous viscosity solution of the corresponding Cauchyproblem.

Keywords

Hysteresisoptimal controldynamic programmingviscositysolutions.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 10 , Issue 2 , April 2004 , pp. 271 - 294
Copyright
© EDP Sciences, SMAI, 2004

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References

F. Bagagiolo, An infinite horizon optimal control problem for some switching systems. Discrete Contin. Dyn. Syst. Ser. B 1 (2001) 443-462.
Bagagiolo, F., Dynamic programming for some optimal control problems with hysteresis. NoDEA Nonlinear Differ. Equ. Appl. 9 (2002) 149-174. CrossRef
F. Bagagiolo, Optimal control of finite horizon type for a multidimensional delayed switching system. Department of Mathematics, University of Trento, Preprint No. 647 (2003).
M. Bardi and I. Capuzzo Dolcetta, Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Birkhäuser, Boston (1997).
Barles, G. and Lions, P.L., Fully nonlinear Neumann type boundary conditions for first-order Hamilton–Jacobi equations. Nonlinear Anal. 16 (1991) 143-153. CrossRef
Belbas, S.A. and Mayergoyz, I.D., Optimal control of dynamic systems with hysteresis. Int. J. Control 73 (2000) 22-28. CrossRef
Belbas, S.A. and Mayergoyz, I.D., Dynamic programming for systems with hysteresis. Physica B Condensed Matter 306 (2001) 200-205. CrossRef
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).
Crandall, M.G. and Lions, P.L., Hamilton-Jacobi equations in infinite dimensions. Part I: Uniqueness of solutions. J. Funct. Anal. 62 (1985) 379-396. CrossRef
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).
Ishii, I., A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 16 (1989) 105-135.
Lenhart, S.M., Seidman, T. and Yong, J., Optimal control of a bioreactor with modal switching. Math. Models Methods Appl. Sci. 11 (2001) 933-949. CrossRef
Lions, P.L., Neumann type boundary condition for Hamilton-Jacobi equations. Duke Math. J. 52 (1985) 793-820. CrossRef
Lions, P.L., 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. CrossRef
I.D. Mayergoyz, Mathematical Models of Hysteresis. Springer, New York (1991).
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).
G. Tao and P.V. Kokotovic, Adaptive Control of Systems with Actuator and Sensor Nonlinearities. John Wiley & Sons, New York (1996).
A. Visintin, Differential Models of Hysteresis. Springer, Heidelberg (1994).