Dynamic programming for some optimal control problems with hysteresis (Q1601717)

With a hysteresis operator \(\mathcal F\) and a measurable control \(\alpha\) the dynamical system \[ y'(t)=f(y(t), {\mathcal F}[y,\xi_0](t),\alpha(t))\qquad (t\geq 0) \] is studied with initial state \(\xi_0\) and \(y(0)=y_0\). For the problem of minimizing the infinite horizon discount cost \[ J=\int_0^{+\infty} e^{-\lambda t}l(y(t), {\mathcal F}[y,\xi_0](t),\alpha(t)) dt \] (with given \(\lambda>0\) and \(l\)) the corresponding (discontinuous) Hamilton-Jacobi equation is obtained for the particular cases that \(\mathcal F\) is either a Play operator or a Prandtl-Ishlinskii operator. Under natural assumptions, the function \((y_0,\xi_0)\mapsto\inf_\alpha J\) is the only viscosity solution of this equation, and the equation can also be reformulated as a viscosity Neumann type boundary value problem.