Multitime optimal control and equilibrium deformations

Abstract: This paper uses optimal control theory in order to describe the deforming processes which, at some future moment optimize a certain energy. Section 1 analyses an optimal control problem with interior cost and terminal cost differential forms. The proof of the corresponding multitime maximum principle relies on a certain temporal order and a corresponding type of needle-shaped control variations. The main result in Section 2 refers to a variational problem with mixed Lagrangian. In Section 3, the equilibrium equations of an elastic body are derived from the mixed Euler-Lagrange PDEs of a variational problem with mixed Lagrangian. Section 4 applies again the results of Section 2 in order to obtain the multitime maximum principle for a control problem with terminal running cost differential form. Again, we use needle-shaped control variations and an adapted temporal order. In the last section, we analyze the evolution of a curve which, at some given future moment, minimizes the energy.