A characterization of the existence of solutions for Hamilton-Jacobi equations in ergodic control problems with applications
In this paper is considered the first-order Hamilton-Jacobi equation \(H(x,u,Du)= 0\) in \(\Omega\), where \(\Omega\) is an open subset of a real Hilbert space \({\mathcal H}\), \(u\) is the unknown which is a real-valued function on \(\overline\Omega\), \(Du(x)\) denotes the Fréchet derivative at \(x\) which is regarded as an element of \({\mathcal H}\) via Riesz theorem, and \(H\) is a real-valued function on \(\Omega\times\mathbb{R}\times{\mathcal H}\) which is called a Hamiltonian. For this equation, the so-called state-constraint problem is considered and the existence of a viscosity solution is studied. This problem is in the category of ergodic control problems and hence, the Hamiltonian \(H(x,u,p)\) is not assumed to be strictly increasing with respect to \(u\). One of the main results characterizes the existence of a bounded viscosity solution in terms of a sort of value functions of the state-constraint problem associated with the Hamilton-Jacobi equation. One of the applications of the main results is the reexamination of an example and the other is concerned with calculation of effective Hamiltonians in periodic homogenization of Hamilton-Jacobi equations.
scientific article; zbMATH DE number 1485112