Constraint Algorithm for Extremals in Optimal Control Problems
Maria Barbero-Liñan, Miguel C. Muñoz-Lecanda
备注：
15 pages. The introduction and the abstract have been modified. The paper has been significantly shortened. The previous examples have been substituted for a new example
A geometric method is described to characterize the different kinds of extremals in optimal control theory. This comes from the use of a presymplectic constraint algorithm starting from the necessary conditions given by Pontryagin's Maximum Principle. Apart from the design of this general algorithm useful for any optimal control problem, it is showed how it works to split the set of extremals and, in particular, to characterize the strict abnormality. An example of strict abnormal extremal for a particular control-affine system is also given.
In this paper we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k- dimensional closed ball. We prove the optimality of broken normal extremals when n = 3 and the controllable vector fields form a contact distribution, and when the Lie algebra of the controllable fields is locally orthogonal to the singular locus and the drift does not belong to it. Moreover, if k = 2, we show the optimality of any broken extremal even abnormal when the controllable fields do not form a contact distribution in the poin…
We obtain a nonsmooth higher-order extension of Noether's symmetry theorem for variational isoperimetric problems with delayed arguments. The result is proved to be valid in the class of Lipschitz functions, as long as the delayed higher-order Euler--Lagrange extremals are restricted to those that satisfy the delayed higher-order DuBois--Reymond necessary optimality condition. The important case of delayed isoperimetric optimal control problems is considered as well.
In this paper we are concerned with generalised L 1-minimisation problems, i.e. Bolza problems involving the absolute value of the control with a control-affine dynamics. We establish sufficient conditions for the strong local optimality of extremals given by the concatenation of bang, singular and inactive (zero) arcs. The sufficiency of such conditions is proven by means of Hamiltonian methods. As a byproduct of the result, we provide an explicit invariant formula for the second variation along the singular arc.
Consider a general nonlinear optimal control problem in finite dimension, with constant state and/or control delays. By the Pontryagin Maximum Principle, any optimal trajectory is the projection of a Pontryagin extremal. We establish that, under appropriate assumptions, Pontryagin extremals depend continuously on the parameter delays, for adequate topologies. The proof of the continuity of the trajectory and of the control is quite easy, however, for the adjoint vector, the proof requires a much finer analysis. The continuity property of the adjoint with respect to the parameter delay opens a …
In this paper we give sufficient conditions for a Pontryagin extremal trajectory, consisting of two bang arcs followed by a singular one, to be a strong local minimizer for a Mayer problem. The problem is defined on a manifold $M$ and the end-points constraints are of fixed-free type. We use a Hamiltonian approach and its connection with the second order conditions in the form of an accessory problem on the tangent space to $M$ at the final point of the trajectory. Two examples are proposed.
This paper presents a novel neighboring extremal approach to establish the neighboring optimal guidance (NOG) strategy for fixed-time low-thrust multi-burn orbital transfer problems. Unlike the classical variational methods which define and solve an accessory minimum problem (AMP) to design the NOG, the core of the proposed method is to construct a parameterized family of neighboring extremals around a nominal one. A geometric analysis on the projection behavior of the parameterized neighboring extremals shows that it is impossible to establish the NOG unless not only the typical Jacobi condit…
In this paper, we analyse control affine optimal control problems with a cost functional involving the absolute value of the control. The Pontryagin extremals associated with such systems are given by (possible) concatenations of bang arcs with singular arcs and with inactivated arcs, that is, arcs where the control is identically zero. Here we consider Pontryagin extremals given by a bang-inactive-bang concatenation. We establish sufficient optimality conditions for such extremals, in terms of some regularity conditions and of the coercivity of a suitable finite-dimensional second variation.
While the Pontryagin Maximum Principle can be used to calculate candidate extremals for optimal orbital transfer problems, these candidates cannot be guaranteed to be at least locally optimal unless sufficient optimality conditions are satisfied. In this paper, through constructing a parameterized family of extremals around a reference extremal, some second-order necessary and sufficient conditions for the strong-local optimality of the free-time multi-burn fuel-optimal transfer are established under certain regularity assumptions. Moreover, the numerical procedure for computing these optimali…
For any quantity of interest in a system governed by ordinary differential equations, it is natural to seek the largest (or smallest) long-time average among solution trajectories, as well as the extremal trajectories themselves. Upper bounds on time averages can be proved a priori using auxiliary functions, the optimal choice of which is a convex optimization problem. We prove that the problems of finding maximal trajectories and minimal auxiliary functions are strongly dual. Thus, auxiliary functions provide arbitrarily sharp upper bounds on time averages. Moreover, any nearly minimal auxili…
In the work, the property of the second-order subdifferential is studied and second-order optimality conditions are obtained for the minimization problem. We also obtained necessary and sufficient conditions for an extremum for the extremal problem of operator inclusions.
Necessary conditions for existence of normal extremals in optimal control of systems subject to nonholonomic constraints are derived as solutions of a constrained second order variational problems. In this work, a geometric interpretation of the derivation is studied from the theory of Lie algebroids. We employ such a framework to describe the problem into a unifying formalism for normal extremals in optimal control of nonholonomic systems and including situations that have not been considered before in the literature from this perspective. We show that necessary conditions for existence of ex…
In 1934, Reinhardt conjectured that the shape of the centrally symmetric convex body in the plane whose densest lattice packing has the smallest density is a smoothed octagon. This conjecture is still open. We formulate the Reinhardt Conjecture as a problem in optimal control theory. The smoothed octagon is a Pontryagin extremal trajectory with bang-bang control. More generally, the smoothed regular $6k+2$-gon is a Pontryagin extremal with bang-bang control. The smoothed octagon is a strict (micro) local minimum to the optimal control problem. The optimal solution to the Reinhardt problem is a…
We develop a computational method for extremal Steklov eigenvalue problems and apply it to study the problem of maximizing the $p$-th Steklov eigenvalue as a function of the domain with a volume constraint. In contrast to the optimal domains for several other extremal Dirichlet- and Neumann-Laplacian eigenvalue problems, computational results suggest that the optimal domains for this problem are very structured. We reach the conjecture that the domain maximizing the $p$-th Steklov eigenvalue is unique (up to dilations and rigid transformations), has p-fold symmetry, and an axis of symmetry. Th…
Quadratic order conditions for bang-singular extremals
CONICET CIFASIS（阿根廷国家科学研究委员会科学和技术研究常设委员会）；INRIA Saclay - CMAP Ecole Polytechnique（法国国家信息与自动化研究所萨克雷校区 - 巴黎综合理工学院数学与建模中心）；Russian Academy of Sciences - CEMI（俄罗斯科学院计算数学与控制问题研究所）；Moscow State University（莫斯科国立大学）；CONICET PLADEMA - Univ. Nacional de Centro de la Prov. de Buenos Aires（阿根廷国家科学研究委员会PLADEMA项目 - 布宜诺斯艾利斯省中部国立大学）
On curvature and feedback classification of two-dimensional optimal control systems
本文的目的是利用Cartan的活动标架方法将二维黎曼曲面的高斯曲率的经典概念推广到具有标量输入的二维最优控制系统。这个概念已经由A. A. Agrachev和R. V.Gamkrelidze使用纯变分方法的更一般的控制系统。然后我们将看到，类似于高斯曲率的“控制”反映了极值流的类似的内在性质。特别是当曲率为负时，任意长的极值段都是局部最优的。最后，我们将定义和刻画平坦控制系统。
We consider the motion planning of an object in a Riemannian manifold where the object is steered from an initial point to a final point utilizing optimal control. Considering Pontryagin Minimization Principle we compute the Optimal Controls needed for steering the object from initial to final point. The Optimal Controls were solved with respect to time t and shown to have norm 1 which should be the case when the extremal trajectories, which are the solutions of Pontryagin Principle, are arc length parametrized. The extremal trajectories are supposed to be the geodesics on the Riemannian manif…
The spectrum of a Schrödinger operator with periodic potential generally consists of bands and gaps. In this paper, for fixed m, we consider the problem of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class of potentials which have fixed periodicity and are pointwise bounded above and below. We prove that the potential maximizing the m-th gap-to-midgap ratio exists. In one dimension, we prove that the optimal potential attains the pointwise bounds almost everywhere in the domain and is a step-function attaining the imposed minimum and maximum values on exactly m interv…
The benefits of a recently proposed method to approximate hard optimization problems are demonstrated on the graph partitioning problem. The performance of this new method, called Extremal Optimization, is compared to Simulated Annealing in extensive numerical simulations. While generally a complex (NP-hard) problem, the optimization of the graph partitions is particularly difficult for sparse graphs with average connectivities near the percolation threshold. At this threshold, the relative error of Simulated Annealing for large graphs is found to diverge relative to Extremal Optimization at e…
Installation of capacitors in distribution networks is one of the most used procedure to compensate reactive power generated by loads and, consequently, to reduce technical losses. So, the problem consists in identifying the optimal placement and sizing of capacitors. This problem is known in the literature as optimal capacitor placement problem. Neverthless, depending on the location and size of the capacitor, it may become a harmonic source, allowing capacitor to enter into resonance with the distribution network, causing several undesired side effects. In this work we propose a parsimonious…
In this paper we discuss a general framework based on symplectic geometry for the study of second order conditions in constrained variational problems on curves. Using the notion of L-derivatives we construct Jacobi curves, which represent a generalization of Jacobi fields from the classical calculus of variations, but which also works for non-smooth extremals. This construction includes in particular the previously known constructions for specific types of extremals. We state and prove Morse-type theorems that connect the negative inertia index of the Hessian of the problem to some symplectic…
