Relaxation of optimal control problems driven by nonlinear evolution equations
The paper concerns the analysis of an optimal control problem driven by a nonlinear evolutionary ODE. Specifically, the problem is to minimize an integral functional of two variables \(x,u\) depending on time \(t\). The function \(x\) solves a nonlinear differential equation with coefficients depending on the function \(u\). In its turn, \(u(t)\) belongs to a multifunction \(U(t,x(t))\). Since the problem is not convex, optimal pairs need not exist. A classical strategy is then to relax the problem, i.e., to enlarge the system in such a way that any minimizing sequence has a limit. Without resorting to Young measures, the authors relax the problem by introducing the convexifications of the cost and of the multifunction \(U\). The solutions of the convexified problems converge in suitable topologies; moreover also the corresponding cost functions converge.
scientific article; zbMATH DE number 7293860
scientific article