We present an indirect method to solve state-constrained optimal control problems. The presented method is based on the maximum principle in Gamkrelidze's form. We focus on a class of problems having a certain regularity condition, which will ensure the continuity of the measure Lagrange multiplier associated with the state constraint. This property of the multiplier will play a key role in solving the two-point boundary value problem resulting from the maximum principle. Several illustrative applications to time optimal control problems are considered.
joint work with Francisco Silva-Alvarez, Dante Kalise
We address the numerical approximation of Mean Field Games with local couplings. For power-like Hamiltonians, we consider both unconstrained and constrained stationary systems with density constraints in order to model hard congestion effects. For finite difference discretizations of the Mean Field Game system, we follow a variational approach. We prove that the aforementioned schemes can be obtained as the optimality system of suitably defined optimization problems. Next, we study and compare several efficiently convergent first-order methods.
We study an inverse problem of crack identification formulated as an optimal control problem with a PDE constraint describing the anti-plane equilibrium of an elastic body with a stress-free crack under the action of a traction force. The optimal control problem is formulated by minimizing the $L^2$-distance between the displacement and the observation and then is solved by shape optimization techniques via a Lagrangian approach. An algorithm is proposed based on a gradient step procedure and several numerical experiments are carried out to show its performance in diverse situations.