We analyze discrete optimal control problems and their connection with back propagation and deep learning. We consider in particular the symmetric representation of the discrete rigid body equations developed via optimal control analysis and optimal flows on adjoint orbits

This paper is an overview of the work of the authors about generalized polynomial curves and splines on Riemannian manifolds. The emphasis is put on the variational approach that gives rise to such curves, and on the Hamiltonian formulation for the cubic case.

In this paper we investigate the existence and uniqueness of Riemannian cubics under boundary conditions on position and velocity. We restrict the study to cubics close to geodesics at the boundaries. In other words, we consider the boundary data in a neighborhood of geodesic boundary data. We define a map that generalizes the Riemannian exponential, the biexponential. This map is used to establish the correspondence between initial and boundary data. We also emphasize the relation between biconjugate points and bi-Jacobi fields along cubics by means of the biexponential map.

We study local existence and uniqueness for Riemannian cubics satisfying boundary conditions. We define the biexponential map and use it to relate initial and boundary data. We also describe biconjugate points along cubics by means of the biexponential map.

In this paper we define "embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on n-spheres.

In this paper we analyze a class of discrete optimal control problems. These systems are discretizations of a class of optimal control problems defined on invariant submanifolds which we denote embedded optimal control problems. We analyze a particular subset of these called discrete Clebsch optimal control problems where the invariant manifolds are group orbits. The generating Hamiltonian equations for such systems are analyzed. The analysis provides a large class of geometric integrators for mechanical systems. We apply the theory to two example systems: mechanical systems on matrix Lie groups and mechanical systems on the n-sphere.

The classical definition of a rolling map, describing the rolling motion, without slip or twist, of one Euclidean submanifold over another of the same dimension, as given in Sharpe [8], is generalized for the situation when the embedded space is equipped with a pseudo-Riemannian metric and applied to derive the kinematic equations for the constrained rolling motion of a connected pseudo-Riemannian orthogonal group over its affine tangent spaces at a point. The kinematic equations are solved explicitly when the curve along which the first manifold rolls is a geodesic. We also show that rolling motions along a curve with non-holonomic constraints of not-wist and no-slip encode parallel transport, and derive formulas for the tangent and normal parallel transport of a vector along geodesics. Finally, we make a brief reference on how rolling motions can be used to generate smooth interpolating curves on pseudo-orthogonal groups.

This paper is devoted to a detailed analysis of the geodesic problem on matrix Lie groups, with left invariant metric, by examining representations of embeddings of geodesic flows in suitable vector spaces. We show how these representations generate extremals for optimal control problems. In particular we discuss in detail the symmetric representation of the so-called $n$-dimensional rigid body equation and its relation to the more classical Euler description. We detail invariant manifolds of these flows on which we are able to define a strict notion of equivalence between representations, and identify naturally induced symplectic structures.

In this paper we define a class of optimal control problems which we denote “embedded optimal control problems”. These are not true optimal control problems since the control system is not locally controllable on the manifold on which it is defined. Despite this, they allow for a well defined associated optimal control problem which does not admit abnormal extremals. We apply Pontryagin's maximum principle to the embedded optimal control problem to derive the generating differential equations for the normal and abnormal extremals. We show that the normal extremal generating equations in a sense contain the extremal generating equations for the associated optimal control problem. We show that this is not the case for the abnormal extremal generating equations. This has applications to the study of the optimal control of systems constrained to a given submanifold of a configuration space, for example the sphere or hypersphere. We apply the theory to three examples in order to illustrate its applicability and to show how it relates to well known results.

The purpose of this paper is to extend the symmetric representation of the rigid body equations from the group SO ( n ) to other groups. These groups are matrix subgroups of the general linear group that are defined by a quadratic matrix identity. Their corresponding Lie algebras include several classical semisimple matrix Lie algebras. The approach is to start with an optimal control problem on these groups that generates geodesics for a left-invariant metric. Earlier work by Bloch, Crouch, Marsden, and Ratiu defines the symmetric representation of the rigid body equations, which is obtained by solving the same optimal control problem in the particular case of the Lie group SO ( n ). This paper generalizes this symmetric representation to a wider class of matrix groups satisfying a certain quadratic matrix identity. We consider the relationship between this symmetric representation of the generalized rigid body equations and the generalized rigid body equations themselves. A discretization of this symmetric representation is constructed making use of the symmetry, which in turn give rise to numerical algorithms to integrate the generalized rigid body equations for the given class of matrix Lie groups.

In their paper on discrete analogues of some classical systems such as the rigid body and the geodesic flow on an ellipsoid, Moser and Veselov introduced their analysis in the general context of flows on Stiefel manifolds. We consider here a general class of continuous time, quadratic cost, optimal control problems on Stiefel manifolds, which in the extreme dimensions again yield these classical physical geodesic flows. We have already shown that this optimal control setting gives a new symmetric representation of the rigid body flow and in this paper we extend this representation to the geodesic flow on the ellipsoid and the more general Stiefel manifold case. The metric we choose on the Stiefel manifolds is the same as that used in the symmetric representation of the rigid body flow and that used by Moser and Veselov. In the extreme cases of the ellipsoid and the rigid body, the geodesic flows are known to be integrable. We obtain the extremal flows using both variational and optimal control approaches and elucidate the structure of the flows on general Stiefel manifolds.

Given a general nonlinear affine control system with outputs and a torsion-free affine connection defined on its state space, we investigate the gradient realization problem: we give necessary and sufficient conditions under which the control system can be written as a gradient control system corresponding to some pseudo-Riemannian metric whose Levi-Civita connection is equal to the given affine connection. The results rely on a suitable notion of compatibility of the system with respect to the given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role throughout the discussion.

This paper analyses continuous and discrete versions of the generalized rigid body equations and the role of these equations in numerical analysis, optimal control and integrable Hamiltonian systems. In particular, we present a symmetric representation of the rigid body equations on the Cartesian product SO(n)×SO(n) and study its associated symplectic structure. We describe the relationship of these ideas with the Moser-Veselov theory of discrete integrable systems and with the theory of variational symplectic integrators. Preliminary work on the ideas discussed in this paper may be found in Bloch et al (Bloch A M, Crouch P, Marsden J E and Ratiu T S 1998Proc. IEEE Conf. on Decision and Control 37 2249-54).

We continue the work of Crouch and Silva Leite on the geometry of cubic polynomials on Riemannian manifolds. In particular, we generalize the theory of Jacobi fields and conjugate points and present necessary and sufficient optimality conditions.

We present a Hamiltonian formulation of a second order variational problem on a Riemannian manifold (Q, < .,. >), which gives rise to generalized cubic polynomials on Q, and explore the possibility of writing down the extremal solutions of that problem as a flow in the space ∪qϵQTqQ⊕Tq*Q⊕Tq*Q. For that we utilize the connection ∇ on Q, corresponding to the metric < .,. >. We exhibit the extremal equations in Hamiltonian form and identify the correct symplectic form. In general the results depend upon a choice of frame for TQ, but for the special situation when Q is a Lie group G with Lie algebra G, our results are global and the flow reduces to a flow on G × g ×g* ×g*.

In this paper we consider the Hamiltonian formulation of the equations of incompressible ideal fluid flow from the point of view of optimal control theory. The equations are compared to the finite symmetric rigid body equations analyzed earlier by the authors. We discuss various aspects of the Hamiltonian structure of the Euler equations and show in particular that the optimal control approach leads to a standard formulation of the Euler equations - the so-called impulse equations in their Lagrangian form. We discuss various other aspects of the Euler equations from a pedagogical point of view. We show that the Hamiltonian in the maximum principle is given by the pairing of the Eulerian impulse density with the velocity. We provide a comparative discussion of the flow equations in their Eulerian and Lagrangian form and describe how these forms occur naturally in the context of optimal control. We demonstrate that the extremal equations corresponding to the optimal control problem for the flow have a natural canonical symplectic structure.

We consider the nonlinear dynamic interpolation problem on Riemannian manifolds and, in particular, on connected and compact Lie groups. Basically we force the dynamic variables of a control system to pass through specific points in the configuration space, while minimizing a certain energy function, by a suitable choice of the controls. The energy function we consider depends on the velocity and acceleration along trajectories. The solution curves can be seen as generalizations of the classical splines in tension for the Euclidean space. The relations with sub-Riemannian optimal control problems are explained.

We describe a geometric approach to finding the extremal equations for variational problems subject to nonlinear constraints on manifolds. This has applications to optimal control problems and extends earlier work. We formulate the problem on an appropriate vector bundle and show how to describe the extremal equations in terms of symplectic geometry on an appropriate reduced space