Parametric synthesis of nonlinear controlled systems (Q1342584)

The author studies a nonlinear -- but analytic -- control system: \(x'= f(t, x, \alpha, u)\) which depends on a parameter \(\alpha\). The control \(u\) is a discontinuous law of \((t, x, \alpha)\). The value of the problem is the infimum over \(\alpha\) of the supremum over initial conditions of some cost (a sum of an integral cost and an end point cost). It is well- known that this volume function is in general not differentiable. So it does not satisfy the Hamilton-Jacobi equation in the usual sense. Instead of studying nonsmooth solutions to this PDE (in a contingent or viscosity sense for example) the author studies suboptimal solutions to the problem and he gives an algorithm solving the suboptimal problem.