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Pseudospectral Knotting Methods for Solving Optimal Control Problems
 Journal of Guidance, Control, and Dynamics
"... A class of computational methods for solving a wide variety of optimal control problems is presented; these problems include nonsmooth, nonlinear, switched optimal control problems, as well as standard multiphase problems. Methods are based on pseudospectral approximations of the differential constr ..."
Abstract

Cited by 14 (4 self)
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A class of computational methods for solving a wide variety of optimal control problems is presented; these problems include nonsmooth, nonlinear, switched optimal control problems, as well as standard multiphase problems. Methods are based on pseudospectral approximations of the differential constraints that are assumed to be given in the form of controlled differential inclusions including the usual vector field and differentialalgebraic forms. Discontinuities and switches in states, controls, cost functional, dynamic constraints, and various other mappings associated with the generalized Bolza problem are allowed by the concept of pseudospectral (PS) knots. Information across switches and corners is passed in the form of discrete event conditions localized at the PS knots. The optimal control problem is approximated to a structured sparse mathematical programming problem. The discretized problem is solved using offtheshelf solvers that include sequential quadratic programming and interior point methods. Two examples that demonstrate the concept of hard and soft knots are presented.
Geometry and Optimal Control
 J. Appl. Math. Physics
, 1998
"... Optimal control has strongly influenced geometry since the early days of both subjects. In particular, it played a crucial role in the birth of differential geometry in the nineteenth century through the revolutionary ideas of redefining the notion of "straight line" (now renamed "geodesic") by mean ..."
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Cited by 7 (3 self)
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Optimal control has strongly influenced geometry since the early days of both subjects. In particular, it played a crucial role in the birth of differential geometry in the nineteenth century through the revolutionary ideas of redefining the notion of "straight line" (now renamed "geodesic") by means of a curve minimization problem, and of emphasizing general invariance and covariance conditions. More recently, modern control theory has been heavily influenced by geometry. One aspect of this influence is the geometrization of the necessary conditions for optimality, which are recast as geometric conditions about reachable sets, thus becoming special cases of the broader question of the structure and properties of these sets. Recently, this has led to a new general version of the finitedimensional maximum principle, stated here in full detail for the first time. A second aspectin which Roger Brockett's ideas have played a crucial roleis the use in control theory of concepts and t...
Multidifferential Calculus: Chain Rule, Open Mapping and Transversal Intersection Theorems
, 1997
"... this paper (Theorem 4.3.3). For this reason, we now believe that the more restrictive definition proposed here ought to supersede that of [15]. Since we still feel that the word used in [15] is the most adequate name for the concept, we have taken the liberty of retaining the word while slightly cha ..."
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this paper (Theorem 4.3.3). For this reason, we now believe that the more restrictive definition proposed here ought to supersede that of [15]. Since we still feel that the word used in [15] is the most adequate name for the concept, we have taken the liberty of retaining the word while slightly changing its meaning, rather than creating a new word. The proof of the open mapping theorem is an elaboration of ideas originally due to S. / Lojasiewicz Jr., and is based on a method introduced by Wazewski (cf. [21]) for the study of implicitly defined mappings. The multidifferentials at a point (¯x; ¯ y) 2 X \Theta Y of a setvalued map
Warga Derivate Containers and Other Generalized Differentials
, 2002
"... This is the first of two papers devoted to recent ideas on the theory of generalized differentials with good open mapping properties. Here we will discuss "generalized differentiation theories" (abbr. GDTs), with special emphasis on the series of developments initiated by Jack Warga's pioneering wor ..."
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This is the first of two papers devoted to recent ideas on the theory of generalized differentials with good open mapping properties. Here we will discuss "generalized differentiation theories" (abbr. GDTs), with special emphasis on the series of developments initiated by Jack Warga's pioneering work on derivate containers. In the second paper, we will focus on the most recent theory, of "pathintegral generalized differentials," and prove that it has the crucial properties required for a version of the Pontryagin Maximum Principle (abbr. PMP) to exist, namely, the chain rule and the directional open mapping property. Our work continues the study of general smooth, nonsmooth, highorder, and hybrid versions of the PMP for finitedimensional deterministic optimal control problems without state space constraints by means of a method developed by us in recent years. As explained in [11, 12, 13, 14], such versions can be derived in a unified way, by using a modified version of...
www.elsevier.com/locate/mcm Issues in the realtime computation of optimal control
, 2005
"... Under appropriate conditions, the dynamics of a control system governed by ordinary differential equations can be formulated in several ways: differential inclusion, control parametrization, flatness parametrization, higherorder inclusions and so on. A plethora of techniques have been proposed for ..."
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Under appropriate conditions, the dynamics of a control system governed by ordinary differential equations can be formulated in several ways: differential inclusion, control parametrization, flatness parametrization, higherorder inclusions and so on. A plethora of techniques have been proposed for each of these formulations but they are typically not portable across equivalent mathematical formulations. Further complications arise as a result of configuration and control constraints such as those imposed by obstacle avoidance or control saturation. In this paper, we present a unified framework for handling the computation of optimal controls where the description of the governing equations or that of the path constraint is not a limitation. In fact, our method exploits the advantages offered by coordinate transformations and harnesses any inherent smoothness present in the optimal system trajectories. We demonstrate how our computational framework can easily and efficiently handle different cost formulations, control sets and path constraints. We illustrate our ideas by formulating a robotics problem in eight different ways, including a differentially flat formulation subject to control saturation. This example establishes the loss of convexity in the flat formulation as well as its ramifications for computation and optimality. In addition, a numerical comparison of our unified approach to a recent technique tailored for controlaffine systems reveals that we get about 30 % improvement in the performance index. c ○ 2005 Elsevier Ltd. All rights reserved. 1.