Conjugate Point Theory Research Articles

On Solutions of the Third-Order Ordinary Differential Equations of Emden-Fowler Type

For a linear ordinary differential equation (ODE in short) of the third order, results are presented that supplement the theory of conjugate points and extremal solutions by W. Leighton, Z. Nehari, M. Hanan. It is especially noted the sensitivity of solutions to the initial data, which makes their numerical study difficult. Similar results were obtained for the third-order nonlinear equations of the Emden-Fowler type.

A Straightforward Sufficiency Proof for a Nonparametric Problem of Bolza in the Calculus of Variations

We study a variable end-points calculus of variations problem of Bolza containing inequality and equality constraints. The proof of the principal theorem of the paper has a direct nature since it is independent of some classical sufficiency approaches invoking the Hamiltonian-Jacobi theory, Riccati equations, fields of extremals or the theory of conjugate points. In contrast, the algorithm employed to prove the principal theorem of the article is based on elementary tools of the real analysis.

A proof of instability of some Euler elasticas

Based on the conjugate point theory in calculus of variations, sufficiency of instability of some Euler elasticas with one clamped end and the other clamped in rotation in the post-buckling is presented. The conjugate points of Jacobi equation are investigated by Sturm's qualitative theory of ordinary differential equation. Without numerical methods, we proved that those Euler elasticas with more than one inflexion in small deflection and some Euler elasticas with more than one inflexion in large deflection are unstable.

Extensibility Effects on Euler Elastica’s Stability

Based on the conjugate point theory in calculus of variations, the extensibility effects on the stability of Euler elasticas with one clamped end and the other clamped in rotation in the post-buckling are investigated. For a slender rod, it is shown that: (1) the buckling load is a little bigger than Euler critical load, (2) the addition of extensibility to the elastica does not affect its stability in the post-buckling, in the sense that those Euler elasticas with one inflexion point are stable while those Euler elasticas with more than one inflexion point are unstable.

Optimal Control and Applications to Aerospace: Some Results and Challenges

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.

Conjugate Points Revisited and Neumann–Neumann Problems

The theory of conjugate points in the calculus of variations is reconsidered with a perspective emphasizing the connection to finite-dimensional optimization. The object of central importance is the spectrum of the second-variation operator, analogous to the eigenvalues of the Hessian matrix in finite dimensions. With a few basic properties of this spectrum, one can gain a new perspective on the classic result that “stability requires the lack of conjugate points.” Furthermore, we show how the spectral perspective allows the extension of the conjugate point approach to variants of the classic problems in the literature, such as problems with Neumann–Neumann boundary conditions.

An Extended Conjugate Point Theory with Application to the Stability of Planar Buckling of an Elastic Rod Subject to a Repulsive Self-Potential

The theory of conjugate points in the classic calculus of variations allows, for a certain class of functionals, the characterization of a critical point as stable (i.e., a local minimum) or not. In this work, we generalize this theory to more general functionals, assuming certain generic properties of the second variation operator. The extended conjugate point theory is then applied to a two-dimensional elastic rod subject to pointwise self-repulsion. The critical points are computed by numerically solving first-order integrodifferential equations using a finite difference scheme. The stability of each critical point is then computed by determining conjugate points of the second variation operator. In addition, the generalized theory requires the numerical evaluation of the crossing velocity of the zero eigenvalue of the second variation operator at each conjugate point, a feature not present in the classic case (where the crossing velocity can be shown to always be negative). Results demonstrate that the repulsive potential has a stabilizing influence on some branches of critical points.

Sufficient conditions for stability of Euler elasticas

Based on the conjugate point theory in calculus of variations, sufficient conditions on stability of all Euler elasticas for a column clamped at one end and guided at the other are proposed in this paper. For the Euler elasticas, an initial value problem is solved numerically by the Euler iteration. Sufficient conditions are satisfied that Euler elasticas with one half wave are stable and the others with more than one half wave are unstable in post-buckling. As load is smaller than the Euler critical force, it is sufficient theoretically that straight shape is stable. As load exceeds the Euler critical force, it is sufficient theoretically that straight shape is unstable.

Stability of an elastic rod buckling into a soft wall

The conjugate point theory of the calculus of variations is extended to apply to the buckling of an elastic rod in an external field. We show that the operator approach presented by Manning (Manning et al . 1998 Proc. R. Soc. A 454 , 3047–3074) can be used when the second-variation operator is an integrodifferential operator, rather than a differential operator as in the classical case. The external field is chosen to model two parallel ‘soft’ walls. We consider the examples of two-dimensional buckling under both pinned–pinned and clamped–clamped boundary conditions, as well as the three-dimensional clamped–clamped problem, where we consider the importance of the rod cross-section shape as it ranges from circular to extreme elliptical. For each of these problems, we find that in the appropriate limit, the soft-wall solutions approach a ‘hard-wall’ limit, and thus we make conjectures about these hard-wall contact equilibria and their stability. In the two-dimensional pinned–pinned case, this allows us to assign stability to the configurations reported by Holmes (Holmes et al . 1999 Comput. Methods Appl. Mech. Eng. 170 , 175–207) and reconsider the experimental results discussed therein.

Analysis of conjugate points for constant tridiagonal Hesse matrices of a class of extremal problems

The conjugate point is a global concept in the calculus of variations. It plays an important role in second-order optimality conditions. A conjugate point theory for a minimization problem of a smooth function with $n$ variables was proposed in (H. Kawasaki (2000). Conjugate points for a nonlinear programming problem with constraints. J. Nonlinear Convex Anal., 1, 287-293; H. Kawasaki (2001). A conjugate points theory for a nonlinear programming problem. SIAM J. Control Optim., 40, 54-63.). In those papers, we defined the Jacobi equation and (strict) conjugate points, and derived necessary and sufficient optimality conditions in terms of conjugate points. The aim of this article is to analyze conjugate points for tridiagonal Hesse matrices of a class of extremal problems. We present a variety of examples, which can be regarded as a finite-dimensional analogy to the classical shortest path problem on a surface. The author would like to thank the referees for their valuable comments and helpful suggestions. This research was partially supported by the Grant-in Aid for General Scientific Research form the Japan Society for the Promotion of Science 14340037.

On the geometry of Riemannian cubic polynomials

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

The Bordered Operator and the Index, of a Constrained Critical Point

It is shown that the index of a constrained critical point in the isoperimetric calculus of variations is simply related to the number of negative eigenvalues of a certain bordered operator associated with the second variation. A conjugate point theory for this bordered operator is then established.

Conjugate Points for Variational Problems with Equality and Inequality State Constraints

In this paper, variational problems with equality and inequality state constraints are considered. The theory of conjugate points for these problems is developed, and necessary conditions for weak local optimality are derived in terms of this concept and the Legendre condition. For the case of inequality constraints, the envelope-like effect is taken into consideration in the accessory problem.

The writhing of circular cross–section rods: undersea cables to DNA supercoils

The large deflection theory circular cross–section elastic rods is used to consider the writhing of long straight rods subjected to tension and torque, such as undersea cables, and to closed loops with inserted twist, such as DNA supercoils. The writhed shape of the long straight rod under tension and torque is easily generated by twisting a piece of string with the fingers and consists of three separate parts: a balanced ply region, a free end loop, and two tail regions. The solution for the rod shape in each of the regions is found. The results are then joined together to ensure continuity of the position and tangent vectors of the strand centre line through the introduction of point forces and moments at the points where the strands enter and exit the balanced ply. The results of the model are consistent with simple experiments on long braided rope. The writhed shape of the closed loop with twist inserted between the ends prior to closure is modelled as a balanced ply joined to two end loops. The analysis combines the mechanics solution with the conservation of topological link to provide a simple formula which quantitatively predicts the approximate shape and helix angle of the supercoil. The results are in good agreement with simple experiments on rope and with available data on DNA supercoils.

Constrained Regular LQ-Control Problems

In this paper we give a complete description of how the Jacobi theory of conjugate points can be extended to a regular linear-quadratic control problem where the state end-points are jointly constrained to belong to a subspace of ${\bf R}^n \times {\bf R}^n$ and there is a linear pointwise state-control constraint. We introduce also the definition of semiconjugate point which describes a distinctive feature of these problems and state the corresponding necessary and sufficient conditions for the quadratic form to be nonnegative or coercive. In the case in which the constraints and the costs act separately on the initial and final points we give equivalent characterizations of the coercivity of the quadratic form by means of the solutions of an associated Riccati equation in both the controllable and uncontrollable case.

The Riccati Equation for Optimal Control Problems with Mixed State-Control Constraints: Necessity and Sufficiency

The goal of this paper is to conduct a complete study of second-order conditions for the optimal control problem with mixed state-control constraints. The conjugate point theory is presented and a necessary condition in terms of the corresponding Riccati equation is obtained. Sufficiency criteria are developed in terms of strengthened necessary conditions, including the Riccati equation. The results generalize the known ones for pure control constraints as well as for the mixed state-control constraints.

Solution Branching in Linear Differential Equations

Given a nontrivial y, solution branches of the form z ≡ y + ∑ m i = 1 c iw i are constructed with || w i || = 1, 1 ≤ i ≤ m. Multiple zeros of y "unwind" to nearby simple zeros of the branch z. Key assumptions are 1-dimensionality of the solution space of the boundary value problem, an exactness condition on the zeros of y, and a restriction on multiple zero counts. The branching lemmas and theorems are applied to the theory of conjugate points and extremal solutions, and also to branching of solutions of y ( n) + p( x) y = 0. If l-dimensionality is dropped, then solution branches z exist but the constants c i , may not be small.

Coupled points in optimal control theory

The concept of coupled points is introduced for an optimal control problem where both state endpoints are allowed to vary. This definition leads to the extension of the theory of conjugate points to the optimal-control setting. Under suitable controllability assumptions, weaker than those previously considered, it is shown that the nonexistence of coupled points in the open interval (a, b) is a necessary condition for weak local optimality. This result generalizes the ones of the same kind known from the calculus of variations. In the special case when one or both state endpoints are fixed, the notion of coupled points is more general than those of focal or conjugate points.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Conjugate points and optimal control: counterexamples

The authors provide three counterexamples. One shows that the accessory problem proposed in P. Bernhard (Advance in Hamiltonian Systems, 1983, pp.109-142) is not complete, and the others illustrate that one-sided controllability assumptions are not enough to derive the necessary conditions involving the conjugate point theory in optimal control. Hence, it is concluded that the corresponding theorems by Bernhard and by J. V. Breakwell and Y.C. Ho (1965) are incorrect. >

Quadratic control problems

The present paper is concerned with the study of quadratic control problems on linear spaces. In particular, we are concerned with the conditions under which a quadratic criterion function is positive on certain linear spaces. This involves the elementary theory of conjugate and focal points, the existence of a conjugate system with a nonvanishing determinant, and the existence of extremal fields. The results given are in part a translation into control language of known theory for the problem of Bolza. The method used is based on the Hilbert space techniques developed earlier by the author.