Conjugate Point Condition Research Articles

Ascent Trajectories of Multistage Launch Vehicles: Numerical Optimization with Second-Order Conditions Verification

Multistage launch vehicles are employed to place spacecraft and satellites in their operational orbits. Trajectory optimization of their ascending path is aimed at defining the maximum payload mass at orbit injection, for specified structural, propulsive, and aerodynamic data. This work describes and applies a method for optimizing the ascending path of the upper stage of a specified launch vehicle through satisfaction of the necessary conditions for optimality. The method at hand utilizes a recently introduced heuristic technique, that is, the particle swarm algorithm, to find the optimal ascent trajectory. This methodology is very intuitive and relatively easy to program. The second-order conditions, that is, the Clebsch-Legendre inequality and the conjugate point condition, are proven to hold, and their fulfillment enforces optimality of the solution. Availability of an optimal solution to the second order is an essential premise for the possible development of an efficient neighboring optimal guidance.

Broccoli curves and the tropical invariance of Welschinger numbers

In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points — for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso–Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane.

Linear-Quadratic Worst-Case Control

This paper gives an applied view of linear-quadratic worst-case control and relates it to linear-quadraticGaussian smoothing. It extends the worst-case control problem formulation to include tracking of desired output histories and nonzero terminal constraintsand separatestheproblem into future, past, and present problems, each of which must satisfy a conjugate-point condition. It includes a e nite time horizon example, namely, a helicopter hover-position change in which the disturbance is horizontal wind velocity. Linear-quadratic best-case controllers can be obtained by using positive instead of negative weights in the quadratic performance index.

Optimum H∞ designs under sampled state measurements

We present the complete solution to the H ∞-optimal control problem when only sampled values of the state are available. For linear time-varying systems the optimum controller is characterized in terms of the solution of a particular generalized Riccati-differential equation, with the optimum performance determined by the conjugate point conditions associated with a family of generalized Riccati differential equations. For the infinite-horizon time-invariant problem, however, the optimum controller is characterized in terms of the solution of a particular generalized algebraic Riccati equation, and the performance is determined in terms of the conjugate-point conditions of a single generalized Riccati equation, defined on the longest sampling interval. If the distribution of the sampling times is also taken as part of the general design, uniform sampling turns out to be optimal for the infinite horizon case, while for the finite horizon problem a nonuniform sampling generally leads to a better performance.

Sufficient conditions for a minimum of the free-final-time optimal control problem

The sufficient conditions for a minimum of the free-final-time optimal control problem are the strengthened Legendre-Clebsch condition and the conjugate point condition. In this paper, a new approach for determining the location of the conjugate point is presented. The sweep method is used to solve the linear two-point boundary-value problem for the neighboring extremal path from a perturbed initial point to the final constraint manifold. The new approach is to solve for the final condition Lagrange multiplier perturbation and the final time perturbation simultaneously. Then, the resulting neighboring extremal control is used to write the second variation as a perfect square and obtain the conjugate point condition. Finally, two example problems are solved to illustrate the application of the sufficient conditions.

The conjugate point condition for smooth control sets

In this paper we present, for optimal control problems with smooth control constraints, second-order necessary conditions stated in terms of conjugate points and a Riccati-type equation. We show explicitly how to construct a set of feasible directions and how to derive the corresponding accessory problem.

On the Jacobi conditions for control problems with input delay

The conjugate-point conditions in linear quadratic optimal-control problems with delayed control vector are studied. The conditions are presented in terms of a differential equation which can be solved. The method is illustrated by a numerical example.

Conjugate-point conditions for variational problems with delayed argument

A variational problem with delayed argument is investigated. The existing necessary conditions are first reviewed. New results, two conjugate-point conditions, are then derived for this problem. The method of proof is similar to that used by Bliss for the classical problem. An example shows that the two conditions are not equivalent, and that the first-order necessary conditions, the strengthened Legendre conditions, and the conjugate-point conditions do not in general constitute a set of sufficient conditions for the delay problem. It is shown that a special case, referred to as the separated-integrand problem, leads to considerable simplification of the results for the general problem.

A conjugate point condition for control problems with delayed state vectors

The new result presented is a conjugate point condition for an optimal control problem in which there is a constant delay in the state vector. The proof is outlined, and the solution method and a differential game application are discussed.

Differential games with open-loop saddle point conditions

Zero-sum differential games are considered in which one or both of the players are restricted to use open-loop control. It is shown that the first-order necessary conditions for such problems are identical to the first-order necessary conditions for the usual form of a differential game, where both players use closed-loop control laws. An investigation of the conjugate point condition for a special class of games shows that this condition is not the same but depends on the type of solution sought. For games where one or both of the players use open-loop control, there are two conjugate point conditions that must be satisfied. This differs from games in which both players use closed-loop control, where there is only one conjugate point necessary condition.

A conjugate-point condition for a class of differential games

A conjugate-point necessary condition is derived for a class of differential games. This is done by considering the conjugate-point condition for the minimum problem and maximum problem associated with a differential game. Two definitions of a conjugate point and two conjugate-point necessary conditions result. These two definitions and necessary conditions are shown to be equivalent and are combined into one definition and one necessary condition.

Optimal control of linear systems with quadratic costs which are not necessarily positive definite

The calculus of variations is applied to the problem of finding the optimal control for linear systems with quadratic costs which are not necessarily positive definite. The conjugate point condition is transformed to an explicit condition upon the parameters of the problem, in the one-dimensional case for arbitrary costs and in the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional case for final costs only. The expressions for the optimal controls are given. A numerical example of a second-order system is solved.

On the conjugate point condition for the control problem

In this paper the conjugate point condition in the classical calculus of variations is derived from a control-theoretic point of view. The relationship of conjugate point condition to relative minimum in optimal control problem is discussed.

Bounds for convex variational programming problems arising in power system scheduling and control

The convex variational programming problem has been found of practical importance to power system scheduling and control. Convex variational programming problems are closely related to the convex programming problems treated by Kuhn and Tucker [8]. Among their positive attributes are the absence of difficulties with weak relative minima and with saddle-point or conjugate-point conditions. As a result it is feasible to develop dual-variational problems and to provide estimates for bracketing the extreme value of the functional of the variational problem when gradient or successive approximations methods are employed for determination of near optimal control schedules. This paper develops necessary and sufficient conditions for the optimal control schedule and a means for numerically determining bounds to the extreme value of the functional. Applications are given to the problem of hydrothermal power system coordination.

Geometric and Numerical Techniques in Optimal Control of Two and Three-Body Problems

The objective of this article is to present geometric and numerical techniques developed to study the orbit transfer between Keplerian elliptic orbits in the two-body problem or between quasi-Keplerian orbits in the Earth-Moon transfer when low propulsion is used. We concentrate our study on the energy minimization problem. From Pontryagin's maximum principle, the optimal solution can be found solving the shooting equation for smooth Hamiltonian dynamics. A first step in the analysis is to find in the Kepler case an analytical solution for the averaged Hamiltonian, which corresponds to a Riemannian metric. This will allow to compute the solution for the original Kepler problem, using a numerical continuation method where the smoothness of the path is related to the conjugate point condition. Similarly, the solution of the Earth-Moon transfer is computed using geometric and numerical continuation techniques.