Multivariate Optimal Control with Payoffs Defined by Submanifold Integrals

Abstract

This paper adapts the multivariate optimal control theory to a Riemannian setting. In this sense, a coherent correspondence between the key elements of a standard optimal control problem and several basic geometric ingredients is created, with the purpose of generating a geometric version of Pontryagin’s maximum principle. More precisely, the local coordinates on a Riemannian manifold play the role of evolution variables (“multitime”), the Riemannian structure, and the corresponding Levi–Civita linear connection become state variables, while the control variables are represented by some objects with the properties of the Riemann curvature tensor field. Moreover, the constraints are provided by the second order partial differential equations describing the dynamics of the Riemannian structure. The shift from formal analysis to optimal Riemannian control takes deeply into account the symmetries (or anti-symmetries) these geometric elements or equations rely on. In addition, various submanifold integral cost functionals are considered as controlled payoffs.

Least Squares Approximation of Flatness on Riemannian Manifolds

“…Having in mind the so-called variational approach [1,20,21], in this Subsection we add typical functionals that appear in the theory of geometric and physical fields [2].…”

“…Least squares Lagrangians on Riemannian manifolds and the problem of best approximation of flatness have gained much attention lately [1], especially when they are involved in optimization problems whose objectives are integral functionals. Combining this theory with decomposable multivariate dynamics [2], we get new results in differential geometry and global analysis.…”

Solutions of Optimization Problems on Hadamard Manifolds with Lipschitz Functions

“…The space of diffusion tensors required in these cases is a curved manifold named as a Riemannian symmetric space. In Bejenaru and Udriste [13], the authors extended multivariate optimal control techniques to Riemannian optimization problems in order to derive a Hamiltonian approach.…”

Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport

“…Exemplary early contributions are the studies on generalized Dubins problem [6,7,8] and on the rolling problem in arbitrary dimensions [22]. Recent contribution to this field are the paper [3] that adapts the multivariate optimal control theory to a Riemannian setting, and the paper [32] that proposes a modified integral controller to treat actuation biases in nonholonomic systems.…”

“…Let us recall that, on a Riemannian manifold M endowed with a connection ∇, we may define a Riemannian curvature operator R : (T M) 3…”