A Gradient Flow Equation for Optimal Control Problems With End-point Cost

Abstract In this paper, we consider a control system of the form $\dot x = F(x)u$ x ̇ = F ( ) u , linear in variable u . Given fixed starting point, study finite-horizon optimal problem, where want to minimize weighted sum an end-point cost and squared 2-norm control. This functional induces gradient flow on Hilbert space admissible controls, prove convergence result by means Lojasiewicz-Simon inequality. Finally, show that, if let weight tend infinity, resulting family functionals is Γ -convergent, it turns out that limiting problem consists joining point minimizer with horizontal length-minimizer path.