Abstract:

In this article necessary optimality conditions for Saint-Venant equations coupled to ordinary differential equations (ODE) are derived rigorously. The Saint-Venant equations are first-order hyperbolic partial differential equations (PDE) and model here the fluid in a container that is moved by a truck that is subject to Newton's law of motion. The acceleration of the truck may be controlled. We describe the mathematical model and the corresponding tracking-type optimal control problem. First we prove existence and uniqueness for the coupled ODE-PDE problem locally in time. For sufficiently small times, we derive the first-order necessary optimality conditions in the corresponding function spaces. Furthermore we prove existence of optimal controls. The optimality system is formulated in the setting of a semi-smooth operator equation in Hilbert spaces which we solve numerically by a semi-smooth Newton method. We close with a numerical example for a typical driving maneuver.

Reference:

• Kimmerle, S.-J., Gerdts, M.: Necessary optimality conditions and a semi-smooth Newton approach for an optimal control problem of a coupled system of Saint-Venant equations and ordinary differential equations, Pure Appl. Funct. Anal. 1(2) (2016), 231-256 (Link http://www.ybook.co.jp/online2/oppafa/vol1/p231.html)