113 research outputs found
Introduction to optimal control theory
No full textThese are lecture notes of the introductory course in Optimal Control theory treated from the geometric point of view. Optimal Control Problem is reduced to the study of controls (and corresponding trajectories) leading to the boundary of attainable sets. We discuss Pontryagin Maximum Principle, basic existence results, and apply these tools to concrete simple optimal control problems. Special sections are devoted to the general theory of linear time-optimal problems and linear-quadratic problems
Lie-algebraic conditions for exponential stability of switched systems
No full textIt has recently been shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference
Continuity of optimal control costs and its application to weak KAM theory
No full textWe prove continuity of certain cost functions arising from optimal control of affine control systems. We give sharp sufficient conditions for this continuity. As an application, we prove a version of weak KAM theorem and consider the Aubry-Mather problems corresponding to these system
Smooth optimal synthesis for infinite horizon variational problems
No full textWe study Hamiltonian systems which generate extremal flows of regular variational problems on smooth manifolds and demonstrate that negativity of the generalized curvature of such a system implies the existence of a global smooth optimal synthesis for the infinite horizon problem. We also show that in the Euclidean case negativity of the generalized curvature is a consequence of the convexity of the Lagrangian with respect to the pair of arguments. Finally, we give a generic classification for 1-dimensional problems
An estimation of the controllability time for single-input systems on compact Lie groups
No full textGeometric control theory and Riemannian techniques are used to describe the reachable
set at time t of left invariant single-input control systems on semi-simple compact Lie groups and to
estimate the minimal time needed to reach any point from identity. This method provides an effective
way to give an upper and a lower bound for the minimal time needed to transfer a controlled quantum
system with a drift from a given initial position to a given final position. The bounds include diameters
of the flag manifolds; the latter are also explicitly computed in the paper
Jacobi fields in optimal control: one-dimensional variations
No full textWe study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piece-wise analytic in this case but may have much more singularities than the control. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighborhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps
Normal forms for the endpoint map near nice singular curves for rank-two distributions
No full textGiven a rank-two sub-Riemannian structure (M, Δ) and a point x0 ∈ M, a singular curve is a critical point of the endpoint map F : γ ↦ γ (1) defined on the space of horizontal curves starting at x0. The typical least degenerate singular curves of these structures are called regular singular curves; they are nice if their endpoint is not conjugate along γ. The main goal of this paper is to show that locally around a nice singular curve γ, once we choose a suitable topology on the control space we can find a normal form for the endpoint map, in which F writes essentially as a sum of a linear map and a quadratic form. This is a preparation for a forthcoming generalization of the Morse theory to rank-two sub-Riemannian structures
Switching in Time-Optimal Problem with control in a ball
No full textIn this paper we analyse local regularity of time-optimal controls and trajectories for an n-dimensional affine control system with a control parameter, taking values in a k -dimensional closed ball. In the case of k = n − 1, we give sufficient conditions in terms of Lie bracket relations for all optimal controls to be smooth or to have only isolated jump discontinuities
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