Some Sufficiency Conditions for Pareto-Optimal Control

1973 ◽  
Vol 95 (4) ◽  
pp. 356-361 ◽  
Author(s):  
G. Leitmann ◽  
W. Schmitendorf
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Pareto Optimality ◽  
Optimal Control Theory ◽  
Cooperative Games ◽  
Sufficient Conditions ◽  
Pareto Optimal ◽  
Vector Valued ◽  
Sufficiency Conditions

We consider the optimal control problem with vector-valued criterion (including cooperative games) and seek Pareto-optimal (noninferior) solutions. Scalarization results, together with modified sufficiency theorems from optimal control theory, are used to deduce sufficient conditions for Pareto-optimality. The utilization of these conditions is illustrated by various examples.

scholarly journals Dynamic Analysis of a Two-Language Competitive Model with Control Strategies

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Lin-Fei Nie ◽  
Zhi-Dong Teng ◽  
Juan J. Nieto ◽  
Il Hyo Jung
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Numerical Simulations ◽  
Control Strategy ◽  
Sufficient Conditions ◽  
Positive Order ◽  
State Dependent ◽  
Competitive Model ◽  
The Impact

The dynamic behavior of a two-language competitive model is analyzed systemically in this paper. By the linearization and the Bendixson-Dulac theorem on dynamical system, some sufficient conditions on the globally asymptotical stability of the trivial equilibria and the existence and the stability of the positive equilibrium of this model are presented. Nextly, in order to protect the endangered language, an optimal control problem relative to this model is explored. We derive some necessary conditions to solve the optimal control problem and present some numerical simulations using a Runge-Kutta fourth-order method. Finally, the languages competitive model is extended to this model assessing the impact of state-dependent pulse control strategy. Using the Poincaré map, differential inequality, and method of qualitative analysis, we prove the existence and stability of positive order-1 periodic solution for this control model. Numerical simulations are carried out to illustrate the main results and the feasibility of state-dependent impulsive control strategy.

scholarly journals CONSTRAINT ALGORITHM FOR EXTREMALS IN OPTIMAL CONTROL PROBLEMS

2009 ◽  
Vol 06 (07) ◽  
pp. 1221-1233 ◽  
Author(s):  
MARÍA BARBERO-LIÑÁN ◽  
MIGUEL C. MUÑOZ-LECANDA
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Theory ◽  
Control Problem ◽  
Optimal Control Theory ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Control Problems ◽  
Affine System

A geometric method is described to characterize the different kinds of extremals in optimal control theory. This comes from the use of a presymplectic constraint algorithm starting from the necessary conditions given by Pontryagin's Maximum Principle. The algorithm must be run twice so as to obtain suitable sets that once projected must be compared. Apart from the design of this general algorithm useful for any optimal control problem, it is shown how to classify the set of extremals and, in particular, how to characterize the strict abnormality. An example of strict abnormal extremal for a particular control-affine system is also given.

An optimal control problem for a system of linear hyperbolic differential equations with constant coefficients

2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Hamlet F. Guliyev ◽  
Vera B. Nazarova
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Fourth Order ◽  
Hyperbolic Equations ◽  
Sufficient Conditions ◽  
Constant Coefficients ◽  
Gradient Of The Functional ◽  
Necessary And Sufficient ◽  
Hyperbolic Differential Equations

AbstractIn this paper, an optimal control problem is considered for a system of fourth order hyperbolic equations with constant coefficients. The gradient of the functional is calculated and the necessary and sufficient conditions of optimality in the form of an integral inequality are derived.

scholarly journals Optimal control theory with general constraints

1981 ◽  
Vol 23 (2) ◽  
pp. 115-126 ◽  
Author(s):  
John M. Blatt
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Control Theory ◽  
Time Dependent ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
General Constraints ◽  
Elementary Methods ◽  
And Control

AbstractWe consider an optimal control problem with, possibly time-dependent, constraints on state and control variables, jointly. Using only elementary methods, we derive a sufficient condition for optimality. Although phrased in terms reminiscent of the necessary condition of Pontryagin, the sufficient condition is logically independent, as can be shown by a simple example.

scholarly journals On an optimal control problem involving second order hyperbolic systems with boundary controls

1983 ◽  
Vol 27 (1) ◽  
pp. 139-148 ◽  
Author(s):  
K.G. Choo ◽  
K.L. Teo ◽  
Z.S. Wu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Hyperbolic Systems ◽  
Sufficient Conditions ◽  
Second Order ◽  
Optimal Controls ◽  
Convergence Properties ◽  
Boundary Controls ◽  
Necessary And Sufficient

In this paper, we consider an optimal control problem involving second-order hyperbolic systems with boundary controls. Necessary and sufficient conditions are derived and a result on the existence of optimal controls is obtained. Also, a computational algorithm which generated minimizing sequences of controls is devised and the convergence properties of the algorithm are investigated.

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