On a class of inverse optimal control problems (Q5894659)

The goal is to find a disturbance \(y^*\) such that a given control \(u^*\) for the ODE: \(\dot x=f(y,x,u,t)\) minimizes a cost \(J(y^*,u)\). The proposed solution consists in minimizing over \(y\) a cost \(\Phi(y) =J(y,u^*) -J(y, \overline u(y))\) where \(\overline u(y)\) minimizes \(J(y,u)\). An intermediate step consists in finding a zero of \(\Phi_y(z) =J(z,u^*)- J(z, \overline u(y))\); the set of such zeros is shown to contain the required solution. An iterative algorithm based on necessary conditions of optimality with respect to a perturbed exogeneous input \(y\) is proposed. The formulation is refined under stronger regularity assumptions. Some simple examples show that the initial problem may not have a solution.