Riemannian optimal control

ESAIM: Control, Optimisation and Calculus of Variations, 2006

The motivation for this work is the real-time solution of a standard optimal control problem arising in robotics and aerospace applications. For example, the trajectory planning problem for air vehicles is naturally cast as an optimal control problem on the tangent bundle of the Lie Group SE(3), which is also a parallelizable Riemannian manifold. For an optimal control problem on the tangent bundle of such a manifold, we use frame co-ordinates and obtain first-order necessary conditions employing calculus of variations. The use of frame co-ordinates means that intrinsic quantities like the Levi-Civita connection and Riemannian curvature tensor appear in the equations for the co-states. The resulting equations are singularity-free and considerably simpler (from a numerical perspective) than those obtained using a local co-ordinates representation, and are thus better from a computational point of view. The first order necessary conditions result in a two point boundary value problem which we successfully solve by means of a Modified Simple Shooting Method.

Applied Numerical Analysis & Computational Mathematics, 2004

The aim of this dissertation is to solve numerically the following problem, denoted by P : given a Riemannian manifold and two points a and b belonging to that manifold, find a tangent vector T at a, such that exp a (T ) = b, assuming that T exists. This problem is set under an optimal control formulation, which requires the definition of an objective function and a space of control, the choice of a method for the calculation of the descent direction of that function in the space of control and the use of an optimization algorithm to find its minimum, which corresponds to the solution of the original problem by construction. Several techniques are necessary to be put together, coming from the fields of geometry, numerical analysis and optimization.

Lecture Notes in Mathematics, 2008

These notes are based on the mini-course given in June 2004 in Cetraro, Italy, in the frame of a C.I.M.E. school. Of course, they contain much more material that I could present in the 6 hours course. The goal was to give an idea of the general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian Geometry. This concerns Jacobi fields, Morse's index formula, Levi Civita connection, Riemannian curvature and related topics. I tried to make the presentation as light as possible: gave more details in smooth regular situations and referred to the literature in more complicated cases. There is an evidence that the results described in the notes and treated in technical papers we refer to are just parts of a united beautiful subject to be discovered on the crossroads of Differential Geometry, Dynamical Systems, and Optimal Control Theory. I will be happy if the course and the notes encourage some young ambitious researchers to take part in the discovery and exploration of this subject. Contents I Lagrange multipliers' geometry 3

Journal of Mathematical Analysis and Applications, 2003

Dynamic programming identifies the value function of continuous time optimal control with a solution to the Hamilton-Jacobi equation, appropriately defined. This relationship in turn leads to sufficient conditions of global optimality, which have been widely used to confirm the optimality of putative minimisers. In continuous time optimal control, the dynamic programming methodology has been used for problems with state space a vector space. However there are many problems of interest in which it is necessary to regard the state space as a manifold. This paper extends dynamic programming to cover problems in which the state space is a general finite-dimensional C ∞ manifold. It shows that, also in a manifold setting, we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous, lower bounded, generalised solution of the Hamilton-Jacobi equation. The application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum.

Journal of Dynamical and Control Systems

The goal of this paper is to describe a method to solve a class of time optimal control problems which are equivalent to finding the sub-Riemannian minimizing geodesics on a manifold M. In particular, we assume that the manifold M is acted upon by a group G which is a symmetry group for the dynamics. The action of G on M is proper but not necessarily free. As a consequence, the orbit space M/G is not necessarily a manifold but it presents the more general structure of a stratified space. The main ingredients of the method are a reduction of the problem to the orbit space M/G and an analysis of the reachable sets on this space. We give general results relating the stratified structure of the orbit space, and its decomposition into orbit types, with the optimal synthesis. We consider in more detail the case of the so-called K − P problem where the manifold M is itself a Lie group and the group G is determined by a Cartan decomposition of M. In this case, the geodesics can be explicitly calculated and are analytic. As an illustration, we apply our method and results to the complete optimal synthesis on SO(3) .

Siam Journal on Control and Optimization, 1984

In this paper we present a differential geometric approach to the Lagrange problem and the fixed time optimal control problem for nonlinear time-invariant control systems. We restrict attention to first order conditions for optimality and present a generalized Lagrange multiplier rule for restricted variational problems. Our treatment of the optimal control problem uses a recently proposed fibre bundle approach for the definition of nonlinear systems.

1982

A general formalism is introduced for the optimal control problem on manifolds. It is based on a general formulation of Lagrange's multiplier theorem and recent definitions of nonlinear control systems. It is shown that we can give Pontryagin's maximum principle in this formalism. We expect that the problem formulation given in this paper is particularly suitable for application of modern results about controllability etc. in nonlinear control systems.

1984

SIAM Journal on Control and Optimization, 2000

The problem of minimizing the cost functional of an Optimal Control System through the use of constrained Variational Calculus is a generalization of the geodetic problem in Riemannian geometry. In the framework of a geometric formulation of Optimal Control, we define a metric structure associated to the Optimal Control System on the enlarged space of state and time variables, such that the minimal length curves of the metric are the optimal solutions of the system. A twofold generalization of metric structure is applied, considering Finslerian type metrics as well as allowed and forbidden directions (like in sub-Riemannian geometry). Free (null Hamiltonian) or fixed final parameter problems are identified with constant energy leaves, and the restriction of the metric to these leaves gives way to a family of metric structures on the usual state manifold.

Arxiv preprint arXiv:1110.4745, 2011

Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems. Section 1 underlines some science domains where appear multitime optimal control problems. Section 2 (Section 3) recalls the multitime maximum principle for optimal control problems with multiple (curvilinear) integral cost functionals and m-flow type constraint evolution. Section 4 shows that there exists a multitime maximum principle approach of multitime variational calculus. Section 5 (Section 6) proves that the minimal submanifolds (harmonic maps) are optimal solutions of multitime evolution PDEs in an appropriate multitime optimal control problem. Section 7 uses the multitime maximum principle to show that of all solids having a given surface area, the sphere is the one having the greatest volume. Section 8 studies the minimal area of a multitime linear flow as optimal control problem. Section 9 contains commentaries.