Geometric integrators for higher-order variational systems and their application to optimal control (1410.5766v2)

Abstract: Numerical methods that preserve geometric invariants of the system, such as energy, momentum or the symplectic form, are called geometric integrators. In this paper we present a method to construct symplectic-momentum integrators for higher-order Lagrangian systems. Given a regular higher-order Lagrangian $L\colon T^{(k)}Q\to\mathbb{R}$ with $k\geq 1$, the resulting discrete equations define a generally implicit numerical integrator algorithm on $T^{(k-1)}Q\times T^{(k-1)}Q$ that approximates the flow of the higher-order Euler--Lagrange equations for $L$. The algorithm equations are called higher-order discrete Euler--Lagrange equations and constitute a variational integrator for higher-order mechanical systems. The general idea for those variational integrators is to directly discretize Hamilton's principle rather than the equations of motion in a way that preserves the invariants of the original system, notably the symplectic form and, via a discrete version of Noether's theorem, the momentum map. We construct an exact discrete Lagrangian $L_d^e$ using the locally unique solution of the higher-order Euler--Lagrange equations for $L$ with boundary conditions. By taking the discrete Lagrangian as an approximation of $L_d^e$, we obtain variational integrators for higher-order mechanical systems. We apply our techniques to optimal control problems since, given a cost function, the optimal control problem is understood as a second-order variational problem.

• The paper introduces novel geometric integrators that preserve symplectic structure and momentum in higher-order variational systems.
• It extends discrete variational mechanics to address the challenges of second and higher-order systems with proven existence and uniqueness for boundary conditions.
• The methods enable stable and accurate numerical approximations in applications such as robotics, aerospace, and other dynamic control systems.

Geometric Integrators for Higher-Order Variational Systems and Their Application to Optimal Control

The paper by Leonardo Colombo, Sebastián Ferraro, and David Martínez de Diego focuses on the development of numerical methods known as geometric integrators for handling higher-order variational systems, with specific applications to optimal control problems. The importance of preserving geometric invariants in numerical simulations is paramount, particularly for applications in fields such as robotics and aerospace control systems. The authors extend frameworks from discrete variational mechanics, a method that has demonstrated success in maintaining critical system properties such as energy, momentum, and symplectic structure.

Higher-Order Variational Systems

Higher-order variational systems are characterized by Lagrangians defined on tangent bundles of order k, mapped from $T^{(k)}Q$ to $\mathbb{R}$, where $k \geq 1$. These systems result in implicit $2k$-order differential equations. The existing literature, often tackling first-order systems, finds new challenges when extending to higher-order problems, particularly in the field of optimal control where the problems are treated as second-order variational issues.

Construction of Geometric Integrators

Geometric integrators are constructed to maintain symplectic and momentum maps derived from Hamilton’s principle discretization, rather than through direct discretization of equations of motion. The authors address the existence and uniqueness of solutions concerning boundary conditions—an essential aspect for defining these integrators. This is achieved using a direct variational proof along with a regularization procedure to handle the boundary value problem complexity.

Algorithm Development

The discrete Lagrangian, crucial for algorithmic development, is described as an approximation of what the paper names "exact discrete Lagrangian", capturing how closely the action integral approximates the flow of continuous systems. This construction is foundational for ensuring that the numerical methods are consistent with the physical laws governing the system.

Key numerical methods addressed:

• Discrete Euler–Lagrange equations: These equations guide the development of solvers for discrete problems, emphasizing implicit flow approximations on reduced spaces ($T^{(k-1)}Q \times T^{(k-1)}Q$).
• Higher-order systems application: The authors show detailed applications to mechanical systems, providing an exact correspondence between continuous and discrete systems, ensuring regularity within discrete Lagrangians.

Implications and Applications

The broader implications of these methodological advancements suggest enhanced stability and fidelity in simulations necessary for optimal control problems. Since these systems often control complex dynamical processes, the advantages of using geometric integrators for air traffic management, computational anatomy, and other robotic control tasks are profound.

Optimal Control

Specific application to optimal control illustrates how these systems can transition from theoretically modelled continuous systems to practically implemented numerical schemes. The discrete Lagrangian-based approach provides pathways to conceptualize control laws in mechanical systems via accurate, stable simulation of dynamics over discretized time frames.

Conclusion and Future Directions

The paper's contributions lie in extending higher-order Lagrangian mechanics to geometric integrators. Future efforts mentioned include adapting these frameworks for variational integrators on constrained systems, which is pertinent in many practical engineering systems, and exploring higher-order interactions in nonholonomic contexts. This step could further expand the applicability and robustness of these methods in real-world scenarios, particularly in areas necessitating precision control of dynamic systems.