Dynamic programming for some optimal control problems with hysteresis

Abstract

We study an infinite horizon optimal control problem for a system with two state variables. One of them has the evolution governed by a controlled ordinary differential equation and the other one is related to the latter by a hysteresis relation, represented here by either a play operator or a Prandtl-Ishlinskii operator. By dynamic programming, we derive the corresponding (discontinuous) first order Hamilton-Jacobi equation, which in the first case is of finite dimension and in the second case is of infinite dimension. In both cases we prove that the value function is the only bounded uniformly continuous viscosity solution of the equation.2000 Mathematics Subject Classification: 47J40, 49J15, 49L20, 49L25.

About an Optimal Visiting Problem

“…through which we conclude (see also Bagagiolo [1] for explicit calculations in the case of the Play operator).…”

“…See also Day [8] for the case of re ected dynamics, which is linked to our hysteresis problem (also see Bagagiolo [1]). …”

“…For what concerns mathematical hysteresis models, and in particular the so-called hysteresis operators, we refer to the book by Visintin [14]. An optimal control problem with hysteresis evolution, similar to the present one, was studied in Bagagiolo [1]. In that article, however, (besides an in nite dimensional problem) the state variable is only one-dimensional, only one hysteresis operator is applied (instead of m operators as in the present case), and the optimal control is of in nite horizon type, i.e.…”

Viscosity solutions for an optimal control problem with Preisach hysteresis nonlinearities

“…Finally we observe that the present author, in [1] and [2], studied two optimal control problems for ODE with hysteresis. In those papers, a Hamilton-Jacobi equation satisfied by the value function is derived and studied in the framework of viscosity solutions.…”

“…However, those settings are different from the present one. In particular, in [2] the case of the Play/Prandtl-Ishlinskii model and in [1] the case of a finite sum of delayed relays are, respectively, studied (they respectively lead to a finite/infinite dimensional discontinuous Hamilton-Jacobi equation and to a suitably coupled system of finite dimensional continuous Hamilton-Jacobi equations). See also [3] for an extension of [1] to a vectorial case.…”