A geometric integration approach to optimal control problems

COLOMBO, LEONARDO; FERRARO, SEBASTIÁN JOSÉ; MARTÍN DE DIEGO, DAVID

Rio de Janeiro

Congreso; International Congress of Mathematicians ICM 2018; 2018

International Mathematical Union, IMPA, SBM

We will discuss a geometric integrator for optimal trajectories of mechanical systems which we proposed in Colombo, Ferraro and Martin de Diego (J Nonlinear Sci 26(6), 2016).Given a mechanical system with a regular Lagrangian function $L: TQ \to \mathbb{R}$, its controlled Euler--Lagrange equations are, in local coordinates,\begin{equation*}\frac{d}{dt}\frac{\partial L}{\partial \dot{q}^{i}}-\frac{\partial L}{\partial q^{i}}=u_{i},\end{equation*} where $u_{i}$ are the control parameters. Given a cost function $C: TQ\times U \to \mathbb{R}$, we seek to minimize $\int_{t_0}^{t_f} C(q^{i},\dot{q}^{i},u_{i})\,dt$. We can see this as a new Lagrangian system where the second-order Lagrangian $\widetilde{L}: T^{(2)}Q\to \mathbb{R}$ is given by\[\widetilde{L}(q^{i},\dot{q}^{i},\ddot{q}^{i})=C\left(q^{i},\dot{q}^{i},u_i=\frac{d}{dt}\frac{\partial L}{\partial\dot{q}^{i}}-\frac{\partial L}{\partial q^{i}}\right).\]In previous approaches, the theory of discrete variational mechanicsfor second-order systems was derived using a discrete Lagrangian$L_d:  Q\times Q\times Q\to \mathbb{R}$, where three points are used to approximate the positions, velocities and accelerations.Instead of this, we use a different kind of discrete Lagrangian $L_d: TQ\times TQ \to{\mathbb R}$, effectively treating the second-order problem as a discrete Lagrangian system on $TQ$ with a Moser-Veselov discretization, and discuss some of the advantages, such as the lack of need of discretization of the boundary conditions and the proof of existence of an exact discrete Lagrangian.We will show some concrete examples and simulations corresponding to the optimal control of fully actuated systems, supplemented by computer animations.