Homogenization of Hamilton-Jacobi equations in perforated sets
The problem by the author is one of homogenization of first-order Hamilton-Jacobi equations in a bounded set with small periodic holes. More exactly, the author studies the asymptotic behaviour of the solutions \(u_\varepsilon\) to \[ \partial_t u_\varepsilon+ H\Biggl({x\over\varepsilon}, x,Du_\varepsilon\Biggr)= 0\quad\text{in }\varepsilon\overline{Y^*}\cap \Omega,\quad u_\varepsilon= 0\quad\text{on }\varepsilon\overline{Y^*}\cap \partial\Omega,\tag{\(\text{HJ}_\varepsilon\)} \] when the smooth open set \(Y^*\) and the Hamiltonian \(H(y,x,p)\) are \(\mathbb{Z}^n\)-periodic in \(y\). Periodic homogenization concerns the asymptotic study of a family of partial differential equations that have an oscillatory behaviour with small period of size \(\varepsilon> 0\). For relevance to optimal control, a boundary condition on state constraints on the part of the boundary \(\varepsilon\partial Y^*\cap \Omega\) is imposed. Since the solutions are not smooth in general, the author uses the theory of viscosity solutions that gives sense to the equation for discontinuous functions and is well adapted to asymptotic analysis. In the first section the author proves the convergence of \(u_\varepsilon\) in some weak sense to the solution of a limit Hamilton-Jacobi equation \[ \partial_tu+\overline H(x, Du)= 0\quad\text{in }\Omega\quad\text{and }u= 0\quad\text{on }\partial\Omega.\tag{\(\overline{\text{HJ}}\)} \] The effective Hamiltonian \(\overline H\) that drives \((\overline{\text{HJ}})\) is determined by solving a periodic problem in the \(y\)-variable, called the cell problem. In the second parts applies the preceding results to deterministic optimal control problems, and this approach yields to new representation formula for the effective Hamiltonian. This is the main result of the paper.
scientific article; zbMATH DE number 1417621
scientific article