Algorithms for Nonlinear Optimal Control Problems Based on the First and Second Order Necessary Conditions (original) (raw)
A second order constraint qualification for certain classes of optimal control problems
2016
In this paper we introduce a second order constraint qualification which yields necessary conditions for optimal control problems with inequality and equality constraints in the time variable and the control functions. The second order conditions are shown to hold on a cone of critical directions which properly contains not only the cone obtained by considering only active constraints but also the usual one which depends on the sign of the corresponding Lagrange multipliers. Key–Words: Optimal control, second order necessary conditions, extremals, normality
Numerical solution of nonlinear optimal control problems using nonlinear programming
Applied Mathematics and Computation, 2007
To solving nonlinear control problems and especially nonlinear optimal control problems (NOCP), classical methods are not usually efficient. In this paper we introduce a new approach for solving this class of problems by using Nonlinear Programming Problem (NLPP). First, we transfer the original problem to a new problem in form of calculus of variations. Then we discretize the new problem and solve it by using NLPP packages. The solution of the NLPP is used to obtain the optimal control and states, which are the exact solution of the original problem (NOCP). What is more, a NLPP is transferred to a Linear Programming Problem (LPP) which empower us to use powerful LP softwares. The degree of desirability is described for suboptimal approximate solutions. Also the nonlinear approximate solution and the optimal control are shown as a combination of polynomial functions or periodic functions. Finally, efficiency of our approach is confirmed by some numerical examples.
Optimal Control of Nonlinear Systems with Constraints
DAAAM International scientific book 2011, 2011
The contribution presents defined problems of a control of nonlinear systems, processes with constraints and limitations. Optimality conditions for a control and iterative gradient methods of nonlinear systems optimization have been given. The defined problem with the Lagrangian criterion function is converted to an optimization problem of the theory of games. In order to solve such a problem, the search for a saddle point of a functional, an algorithm of an iterative method has been designed. There are also presented algorithms designed based on gradient methods (e.g. Generalized Method of Reduced Gradient) using the projection principles to solve mathematical models of the tasks of processes optimal control, where constraining conditions and limitations to state and control variables of the process are fulfilled.
On Optimality Conditions for Control Problems with Constraints
IFAC Proceedings Volumes, 2003
\\"p report Oil optimalit~• cOllditiollS for control proble ms with mixed state control constraints alld pllfe state cOllstraints. Our maill goal is to unify preyiousl\' ciPwloped \\'ork ano to illu st rate the different approaches used. \\'e describe lH'CeSsary conditions that illclude all Euler-Lagmllge illdusioll for a \yeak minimizer as well as lleCeSSary conditions in t hp form of a lllaxilllulll principle for a strong minimizer. The sufficiency of the same conditions for a certain class of these problems is also anal~'sed.
Necessary Conditions for Optimal Control Problems with State Constraints: Theory and Applications
paginas.fe.up.pt
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 theory has played a relevant role not only in the dynamic optimization but also in the control and system engineering. Another crucial moment in this theory is closely related with the development of nonsmooth analysis during the 70's and 80's. Nonsmooth analysis has triggered a renew interest in optimal control problems and brought new solutions to old problems.
Numerical Solution of a Class of Nonlinear Optimal Control Problems
2015
In this article, a numerical approach for solving a class of nonlinear optimal control problems is presented. This approach is a combination of a spectral collocation method and the parametric iteration method. As will be shown, the proposed indirect strategy provides good approximations of all variables i.e. control, state and costate as opposed to the many direct methods. Several examples are considered to assess the accuracy and features of the presented method.
This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre-Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control. Résumé Dans cet article on s'intéresse au problème de commande optimale d'une équation différentielle ordinaire avec plusieures contraintes pures sur l'état, d'ordres quelconques, et des contraintes mixtes sur la commande et sur l'état. On suppose que (i) la commande est continue et la condition forte de Legendre-Clebsch satisfaite, et (ii) une condition d'indépendance linéaire des contraintes actives est satisfaite. Des résultats de régularité des solutions et multiplicateurs et des conditions de jonction sont don-nés. Lorsqu'il y a un nombre fini de points de jonction, on obtient des conditions d'optimalité du second-ordre nécessaires ou suffisantes, ainsi qu'une caractérisation du caractère bien posé de l'algorithme de tir. Ces résultats généralisent les résultats obtenus dans le cas d'une contrainte sur l'état et d'une commande scalaires.