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Motivated by recent developments in Hamiltonian variational principles, Hamiltonian variational integrators, and their applications such as to optimization and control, we present a new Type II variational approach for Hamiltonian systems, based on a virtual work principle that enforces the Type II boundary conditions through a combination of essential and natural boundary conditions; particularly, this approach allows us to define this variational principle intrinsically on manifolds. We first develop this variational principle on vector spaces and subsequently extend it to parallelizable manifolds, general manifolds, as well as to the infinite-dimensional setting. Furthermore, we provide a review of variational principles for Hamiltonian systems in various settings as well as their applications. 

Scanning electron microscopy yields high-resolution insight into the perovskite to non-perovskite phase transformations of materials such as cesium lead iodide and formamidinium lead iodide. 

Abstract 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. (J Math Phys 19(11):2388–2399, 1978). 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 in IMA J. Numer. Anal. 31(4):1497–1532, 2011) 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.