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：伴随系统广泛应用于由常微分方程或微分代数方程描述的系统的控制、优化和设计中。本文探讨了此类伴随系统的几何性质并发展了相应方法。具体而言，我们利用辛几何与预辛几何分别研究常微分方程和微分代数方程所对应的伴随系统性质。研究表明，伴随变分二次守恒律——这一伴随灵敏度分析的关键要素——源于此类伴随系统的（预）辛性。我们讨论了伴随系统的其他几何性质，如对称性与变分表征。对于与微分代数方程相关的伴随系统，我们将该微分代数方程的指标与Gotay等人[18]提出的预辛约束算法相关联。作为该几何框架的应用，我们阐述了如何利用伴随变分二次守恒律计算终端或运行成本函数的灵敏度。此外，我们采用Galerkin哈密顿变分积分器（Leok和Zhang [23]）为这类系统开发了保结构数值方法，该方法能保持这些二次守恒律的离散模拟形式。我们进一步证明此类方法具有自然性：在适当选择简化、伴随系统构建和离散化过程时，三者可交换。利用这一自然性，我们推导了用于离散化伴随微分代数方程系统的预辛变分积分器的变分误差分析结果。最后，我们讨论了伴随系统在最优控制问题中的应用，并证明了类似自然性结论。