Variational integrators for a new Lagrangian approach to control affine systems with a quadratic Lagrange term

Abstract

In this work, we analyse the discretisation of a recently proposed new Lagrangian approach to optimal control problems of affine-controlled second-order differential equations with cost functions quadratic in the controls. We propose exact discrete and semi-discrete versions of the problem, providing new tools to develop numerical methods. Discrete necessary conditions for optimality are derived and their equivalence with the continuous version is proven. A family of low-order integration schemes is devised to find approximate optimality conditions, and used to solve a low-thrust orbital transfer problem. Non-trivial equivalent standard direct methods are constructed. Noether's theorem for the new Lagrangian approach is investigated in the exact and approximate cases.

Figures (3)

• Figure 1: Example solutions of a low-thrust orbital transfer from $q(0) = (4,0)$ to $q(T) = (-5,0)$ for $T = 28$ with $d=1.5$, $M=10, g = 1, h=0.1$, $K_q = K_v = \mathrm{Id}_{2 \times 2}$ with the parameters for the a) row being $\alpha=1=\beta=\gamma$, bottom row b) $\alpha=0.5=\beta=\gamma$. Shown for each choice of the integration scheme are the control-dependent case $\tilde{\mathcal{L}}_d^\mathcal{E}$ (blue lines) and the independent case $\tilde{\mathcal{L}}_d$ (orange lines). The rows contain from left to right: ($x,y$)-trajectory with example state-velocity vectors $v$, control $u$ evolution, ($\lambda_x,\lambda_y$)-trajectory with example costate-velocity vectors $v_\lambda$, error evolution of the conserved quantity $I-I(0)$.
• Figure 2: Comparison of solving the optimal control problem via the new approach \ref{['apUopts']} and a standard solution algorithm. The parameters are the same as in Figure \ref{['fig1']}, for step size $h=0.1$ and midpoint evaluations $\alpha=\beta=\gamma=1/2$. The new approach yields the same optimal solution.
• Figure 3: Convergence of the method as a function of the number of steps of the discretisation, $N$. Here the first-order methods with a) $\alpha=\beta=\gamma=1$ (left two columns) and b) $\alpha=1,~ \beta=\gamma=0$ (right two columns) are considered with otherwise the same parameters as in Figure \ref{['fig1']}. The figures for a) and b) are the same, being in the first row the $(x,y)$-trajectory and $(v_x,v_y)$-trajectory, in the second row the $(\lambda_x,\lambda_y)$-trajectory and $(v_{\lambda_x},v_{\lambda_y})$-trajectory. The third row contains the control $u$ evolution and the evolution of the control Hamiltonian $\tilde{\mathcal{H}}$. The number of steps correspond to step-sizes of $h=[0.1,0.4,0.8]$. The reference trajectories correspond to the solution of the problem for $h=0.01$ with the second-order method $\alpha=\beta=\gamma = 0.5$. Both methods approach the reference solution for decreasing $h$ in all aspects.

Theorems & Definitions (46)

• Remark 2.1
• Remark 2.2
• Definition 2.3
• Remark 2.4
• Remark 2.5
• Definition 2.6
• Remark 2.7
• Theorem 2.8
• proof
• Remark 2.9