联合系统的几何方法 (Geometric Methods for Adjoint Systems)

Adjoint systems are widely used to inform control, optimization, and design in systems described by ordinary differential equations or differential-algebraic equations. In this paper, we explore the geometric properties and develop methods for such adjoint systems. In particular, we utilize symplectic and presymplectic geometry to investigate the properties of adjoint systems associated with ordinary differential equations and differential-algebraic equations, respectively. We show that the adjoint variational quadratic conservation laws, which are key to adjoint sensitivity analysis, arise from (pre)symplecticity of such adjoint systems. We discuss various additional geometric properties of adjoint systems, such as symmetries and variational characterizations. For adjoint systems associated with a differential-algebraic equation, we relate the index of the differential-algebraic equation to the presymplectic constraint algorithm of Gotay et al. [18]. As an application of this geometric framework, we discuss how the adjoint variational quadratic conservation laws can be used to compute sensitivities of terminal or running cost functions. Furthermore, we develop structure-preserving numerical methods for such systems using Galerkin Hamiltonian variational integrators (Leok and Zhang [23]) which admit discrete analogues of these quadratic conservation laws. We additionally show that such methods are natural, in the sense that reduction, forming the adjoint system, and discretization all commute, for suitable choices of these processes. We utilize this naturality to derive a variational error analysis result for the presymplectic variational integrator that we use to discretize the adjoint DAE system. Finally, we discuss the application of adjoint systems in the context of optimal control problems, where we prove a similar naturality result.

翻译：在普通差分方程式或差异-藻类方程式所描述的系统中,共合系统被广泛用于为控制、优化和设计提供信息。在本文件中,我们探讨了几何特性,并为此类连合系统开发了方法。特别是,我们使用共振和预正偏偏偏偏偏对等测量,分别用于调查与普通差异方程式和差异-藻类方程式相关的连合系统特性。我们表明,对调离感应分析至关重要的共通互异方程式保护法源于这种联合系统的(预)近似性。我们讨论了对联合系统的各种额外几何特性,如对称和变异特性等。对于与差异-正对等等等方程式相关的连合系统特性,我们把差异-正对等方方程式的指数与Gotay etal-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-al-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-x-l-x-x-l-l-l-x-x-x-l-l-l-l-x-x-x-x-x-l-l-l-l-l-l-x---x-x-x-x-x-l-l-l-l-l-x-x-l-l-l--------------l-l-l-l-------l-l---l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l-l--l-l-l-