Optimality Conditions For Control Problems Research Articles

Two-Aircraft Acoustic Optimal Control Problem: SQP algorithms

This contribution aims to develop an acoustic optimization model of flight paths minimizing two-aircraft perceived noise on the ground. It is about minimizing the noise taking into account all the constraints of flight without conflict. The flight dynamics associated with a cost function generate a non-linear optimal control problem governed by ordinary non-linear differential equations. To solve this problem, the theory of necessary conditions for optimal control problems with instantaneous constraints is well used. This characterizes the optimal solution as a local one when the newtonian approach has been used alongside the optimality conditions of Karush-Kuhn-Tucker and the trust region sequential quadratic programming. The SQP methods are suggested as an option by commercial KNITRO solver under AMPL programming language. Among several possible solution, it was shown that there is an optimal trajectory (for each aircraft) leading to a reduction of noise levels on the ground. Cette contribution vise à développer un modèle mathématique d’optimisation acoustique des trajectoires de vol de deux avions en approche et sans conflit, en minimisant le bruit perçu au sol. Toutes les contraintes de vol des deux avions sont considérées. La dynamique de vol associée au coût génère un problème de contrôle optimal régis par des équations différentielles ordinaires non-linéaires. Pour résoudre ce problème, la théorie des conditions nécessaires d’optimalité pour des problèmes de commande optimale avec contraintes instanées est bien développée. Ceci se caractérise par une solution optimale locale lorsque l’approche newtonienne est utilisée en tenant compte des conditions d’optimalité de Karush-Kuhn-Tucker et la programmation quadratique séquentielle globalisée par région de confiance. Les méthodes SQP sont proposées comme option par KNITRO sous le langage de programmation AMPL. Parmi plusieurs solutions admissibles, il est retenu une trajectoire optimale menant à une réduction du niveau de bruit au sol.

Second-order sufficient optimality conditions for control problems with linearly independent gradients of control constraints

Second-order sufficient conditions of a bounded strong minimum are derived for optimal control problems of ordinary differential equations with initial-final state constraints of equality and inequality type and control constraints of inequality type. The conditions are stated in terms of quadratic forms associated with certain tuples of Lagrange multipliers. Under the assumption of linear independence of gradients of active control constraints they guarantee the bounded strong quadratic growth of the so-called “violation function”. Together with corresponding necessary conditions they constitute a no-gap pair of conditions.

Necessary conditions for optimal control problems with and without state constraints

It is commonly accepted that optimal control theory was born with the publication of a seminal paper by Pontryagin and collaborates last century, at the end of 50's. Since then optimal control theo...

Non‐smooth monotonicity constraints in optimal control problems: Some economic applications

AbstractThis paper presents a theorem on necessary conditions for optimal control problems containing monotonicity constraints that bear on a joint function of the control variable, the state variable, and the time. These constraints are often found, under continuity and piecewise smoothness assumptions for the endogenous variables of the problem, in various economic fields that include monopoly regulation, non‐uniform pricing, implicit contracts, and optimal taxation. After applying our theorem to a general incentive provision model, we show its usefulness in relaxing the standard continuity and smoothness assumptions, for the case of two screening problems among those that have received more attention in the literature. Copyright © 2010 John Wiley & Sons, Ltd.

Quadratic weak-minimum conditions for optimal control problems with intermediate constraints

Optimal control problems with constraints at intermediate trajectory points are considered. By using a certain natural method (of reproduction of state and control variables), these problems reduce to the standard optimal control problem of Pontryagin type, which allows one to obtain quadratic weak-minimum conditions for them.

On a Variational Approach to Optimization of Hybrid Mechanical Systems

This paper deals with multiobjective optimization techniques for a class of hybrid optimal control problems in mechanical systems. We deal with general nonlinear hybrid control systems described by boundary-value problems associated with hybrid-type Euler-Lagrange or Hamilton equations. The variational structure of the corresponding solutions makes it possible to reduce the original “mechanical” problem to an auxiliary multiobjective programming reformulation. This approach motivates possible applications of theoretical and computational results from multiobjective optimization related to the original dynamical optimization problem. We consider first order optimality conditions for optimal control problems governed by hybrid mechanical systems and also discuss some conceptual algorithms.

Sufficient Optimality Conditions for Control Problem of Random Structure Systems with Infinite Aftereffect under the Presence of Markovian Parameters

Sufficient Optimality Conditions for Control Problem of Random Structure Systems with Infinite Aftereffect under the Presence of Markovian Parameters

Optimal control and properties of nonlinear multistage dynamical system for planning horizontal well paths

The goal of planning a horizontal well path is to obtain a trajectory that arrives at a given target subject to various constraints. In this paper, the optimal control problem subject to a nonlinear multistage dynamical system (NMDS) for horizontal well paths is investigated. Some properties of the multistage system are proved. In order to derive the optimality conditions, we transform the optimal control problem into one with control constraints and inequality-constrained trajectories by defining some functions. The properties of these functions are then discussed and optimality conditions for optimal control problem are also given. Finally, an improved simplex method is developed and applied to the optimal design for well Ci-16-Cp146 in Oil Field of Liaohe, and the numerical results illustrate the validity of both the model and the algorithm.

How to state necessary optimality conditions for control problems with deviating arguments?

The aim of this paper is to give a general idea to state optimality conditions of control problems in the following form: , (1) where is a set of admissible controls and is the solution of the following equation: ; . (2). The results are nonlocal and new.

Optimal control of Volterra integral equations in two independent variables

We prove an extremality condition for optimal control problems for systems of integral equations of Volterra type with two independent variables. These systems generalize both, the hyperbolic control problems in two independent variables, and the optimal control of ordinary (i.e. with one independent variable) Volterra integral equations.

Necessary Conditions for Constrained Problems under Mangasarian–Fromowitz Conditions

The focus of this paper is on first order necessary conditions for optimal control problems with mixed state-control equality and inequality constraints. We consider the case when the cost and dynamics are nonsmooth, and the constraints satisfy Mangasarian-Fromowitz-type assumptions that weaken the commonly used hypothesis that the Jacobian of the active constraints, with respect to the free variable, is of full rank. The results are formulated as unmaximized Hamiltonian inclusion-type conditions involving not the customary product of partial subdifferentials but the joint subdifferential with respect to the state and control variables.

Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints

The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal control is Lipschitz in the whole domain. Necessary and sufficient second order conditions are proved with a minimal gap between them.

No-gap second-order optimality conditions for optimal control problems with a single state constraint and control

The paper deals with optimal control problems with only one control variable and one state constraint, of arbitrary order. We consider the case of finitely many boundary arcs and touch times. We obtain a no-gap theory of second-order conditions, allowing to characterize second-order quadratic growth.

SECOND-ORDER OPTIMALITY CONDITIONS FOR OPTIMAL CONTROL PROBLEMS GOVERNED BY 3-DIMENSIONAL NEVIER-STOKES EQUATIONS

This article is concerned with second-order necessary and sufficient optimality conditions for optimal control problems governed by 3-dimensional Navier-Stokes equations. The periodic state constraint is considered.

Cooperative optimal control: broadening the reach of bio-inspiration

Inspired by the process by which ants gradually optimize their foraging trails, this paper investigates the cooperative solution of a class of free final time, partially constrained final state optimal control problems by a group of dynamical systems. We propose an iterative, pursuit-based algorithm which generalizes previously proposed models and converges to an optimal solution by iteratively optimizing an initial feasible trajectory/control pair. The proposed algorithm requires only short-range, limited interactions between group members, avoids the need for a ‘global map’ of the environment in which the group evolves, and solves an optimal control problem in ‘small’ pieces, in a manner which will be made precise. The performance of the algorithm is illustrated in a series of simulations and laboratory experiments.

Sufficient conditions for optimal control problems with mixed constraints

AbstractIn this paper we derive sufficient conditions for optimal control problems with mixed control and state constraints by applying a dual approach to the dynamic programming. These conditions guarantee that a relative minimum is achieved. We seek an optimal pair in the class of those admissible pairs for which graphs of trajectories belong to a set defined by a function satisfying a dual equation of dynamic programming. This set usually differs from the classical neighbourhood of an optimal trajectory. We prove a verification theorem and introduce an optimal dual feedback control. Copyright © 2005 John Wiley & Sons, Ltd.

On multi-item inventory

The paper gives a new approach towards a two––item inventory model for deteriorating items with a linear stock––dependent demand rate. In fact, for the first time, the interacting terms showing the mutual increase in the demand of one commodity due to the presence of the other is accommodated in the model. Again, from the linear demand rate, it follows that more is the inventory, more is the demand. So a control parameter is introduced, such that it maintains the continuous supply to the inventory. Next an objective function is formed to calculate the net profit with respect to all possible profits and all possible loss (taken with negative sign). The paper obtains a necessary criterion for the steady state optimal control problem for optimizing the objective function subjected to the constraints given by the ordinary differential equations of the inventory. It also considers a particular choice of parameters satisfying the above necessary conditions. Under this choice, the optimal values of control parameters are calculated; also the optimal amount of inventories is found out. Finally, with respect to these optimal values of control parameters and those of the optimal inventories, the optimal value of the objective function is determined. Next another choice of parameters is considered for which the aforesaid necessary conditions do not hold. Obviously, in that case the steady state solution is non-optimal. In such a case a suboptimal problem is considered corresponding to the more profitable inventory. It is shown that such suboptimal steady state solution fails to exist in this case.

On necessary optimality conditions for optimal control problems governed by elliptic systems

The article considers an optimal control problem for the linear elliptic system div for the case where the coefficient matrix A plays the role of control and belongs to a nonconvex set and the cost functional is a quadratic form with respect to . By transforming the original problem to a more suitable one and by using ideas from the homogenization theory a necessary optimality condition is derived.

Minimax Optimal Control

This paper provides a framework for deriving necessary conditions, in the form of a maximum principle, for minimax optimal control problems. The distinguishing feature of these problems is that the data depends on a vector $\alpha$ of unknown parameters, and "optimality" is defined on a worst case basis, as $\alpha$ ranges over the parameter set ${\cal A}$. The centerpiece, a minimax maximum principle, is a set of optimality conditions for such problems. Here, the parameter set ${\cal A}$ is taken to be an arbitrary compact metric space and the hypotheses imposed on the dynamics and endpoint constraints are of an unrestrictive nature. The minimax maximum principle captures as special cases necessary conditions for optimal control problems with minimax costs, for problems involving "semi-infinite" endpoint constraints, and also a maximum principle for state constrained optimal control problems.

OPTIMAL CONTROL THROUGH BIOLOGICALLY-INSPIRED PURSUIT

Inspired by the process by which ants gradually optimize their foraging trails, this paper investigates the cooperative solution of a class of free-final time, partially-constrained final state optimal control problems by a group of dynamic systems. A cooperative, pursuit-based algorithm is proposed for finding optimal solutions by iteratively optimizing an initial feasible control. The proposed algorithm requires only short-range, limited interactions between group members, and avoids the need for a “global map” of the environment on which the group evolves. The performance of the algorithm is illustrated in a series of numerical experiments.