Sequences of L ∞ -optimal control problems: Γ-convergence and Hamilton–Jacobi equations

We consider the sequence of optimal control problems having as state equation y^′(t)=a_n(t,y)+b_n(t,u) (t∈(0,T], y(0)=x) and cost functional $\[$J_{n}(y,u)=\mathop{\mathrm{ess}}\mathop{\mathrm{sup}}\nolimits_{t\in[0,T]}f_{n}(t,y(t),u(t))$$ . We prove a Γ-convergence result and we study the entailed properties on the stability for the related Hamilton–Jacobi equations.