scholarly journals Necessary optimality conditions for a problem with costs of rapid variation of control

1984 ◽  
Vol 26 (1) ◽  
pp. 45-55 ◽  
Author(s):  
K. Kibalczyc ◽  
S. Walczak
Keyword(s):  
Control Problem ◽  
Optimality Conditions ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Necessary Optimality Conditions ◽  
Rapid Variation ◽  
Cost Functional

AbstractIn this paper a control problem with a cost functional depending on the number of switchings and on the speed of alterations of control is considered. Necessary conditions for the existence of an optimal solution are given.

scholarly journals Necessary optimality condition for the minimal time crisis relaxing transverse condition via regularization

2021 ◽  
Author(s):  
Térence Bayen ◽  
Kenza Boumaza ◽  
Alain Rapaport
Keyword(s):  
Maximum Principle ◽  
Optimality Conditions ◽  
Optimal Control Problems ◽  
Necessary Conditions ◽  
Optimal Solution ◽  
Necessary Optimality Conditions ◽  
Convergence Result ◽  
General Hypothesis ◽  
Constraint Set ◽  
The Pontryagin Maximum Principle

We derive necessary optimality conditions for the time of crisis problem under a more general hypothesis than the usual one encountered in the hybrid setting, which requires that any optimal solution should cross the boundary of the constraint set transversely. Doing so, we apply the Pontryagin Maximum Principle to a sequence of regular optimal control problems whose integral cost approximates the time of crisis. Optimality conditions are derived by passing to the limit in the Hamiltonian system (without the use of the hybrid maximum principle). This convergence result essentially relies on the boundedness of the sequence of adjoint vectors in L∞. Our main contribution is to relate this property to the boundedness in L1 of a suitable sequence which allows to avoid the use of the transverse hypothesis on optimal paths. An example with non-transverse trajectories for which necessary conditions are derived highlights the use of this new condition.

scholarly journals Necessary Optimality Conditions for a Class of Impulsive and Switching Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Lihua Li ◽  
Yan Gao ◽  
Gexia Wang
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Necessary Optimality Conditions ◽  
Switching Systems ◽  
Jump Conditions ◽  
Fréchet Subdifferential ◽  
Extended State ◽  
Adjoint Variables

An optimal control problem for a class of hybrid impulsive and switching systems is considered. By defining switching times as part of extended state, we get the necessary optimality conditions for this problem. It is shown that the adjoint variables satisfy certain jump conditions and the Hamiltonian are continuous at switching instants. In addition, necessary optimality conditions of Fréchet subdifferential form are presented in this paper.

scholarly journals Optimal Control with Time Delays via the Penalty Method

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Mohammed Benharrat ◽  
Delfim F. M. Torres
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Calculus Of Variations ◽  
Convergence Theorem ◽  
Unknown Function ◽  
Time Delays ◽  
Penalty Method ◽  
Necessary Optimality Conditions

We prove necessary optimality conditions of Euler-Lagrange type for a problem of the calculus of variations with time delays, where the delay in the unknown function is different from the delay in its derivative. Then, a more general optimal control problem with time delays is considered. Main result gives a convergence theorem, allowing us to obtain a solution to the delayed optimal control problem by considering a sequence of delayed problems of the calculus of variations.

scholarly journals Maximum Principle for Stochastic Recursive Optimal Control Problems Involving Impulse Controls

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zhen Wu ◽  
Feng Zhang
Keyword(s):  
Optimal Control ◽  
Maximum Principle ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimality Conditions ◽  
Control Variable ◽  
Necessary Optimality Conditions ◽  
Sufficient Optimality Conditions ◽  
Linear Quadratic ◽  
Regular Control

We consider a stochastic recursive optimal control problem in which the control variable has two components: the regular control and the impulse control. The control variable does not enter the diffusion coefficient, and the domain of the regular controls is not necessarily convex. We establish necessary optimality conditions, of the Pontryagin maximum principle type, for this stochastic optimal control problem. Sufficient optimality conditions are also given. The optimal control is obtained for an example of linear quadratic optimization problem to illustrate the applications of the theoretical results.

NECESSARY OPTIMALITY CONDITIONS FOR SUPER MINIMIZERS IN STRUCTURAL PROBLEMS OF MULTIOBJECTIVE OPTIMIZATION

2009 ◽  
Vol 02 (02) ◽  
pp. 321-358 ◽  
Author(s):  
Lu-Chuan Zeng
Keyword(s):  
Multiobjective Optimization ◽  
Optimality Conditions ◽  
Variational Analysis ◽  
Necessary Conditions ◽  
Necessary Optimality Conditions ◽  
Generalized Differentiation ◽  
Infinite Dimensional ◽  
Structural Problems ◽  
Constrained Problems ◽  
Finite Dimensional

The primary goal of this paper is to study the so-called super minimizers of the sum F1 + F2 related to the concept of super efficiency in constrained problems of multiobjective optimization, where each cost mapping Fi is generally set-valued for i = 1, 2. We will derive necessary conditions (of the subdifferential type) for super minimizers of the sum F1 + F2 on the basis of advanced tools of variational analysis and generalized differentiation that are new in both finite-dimensional and infinite-dimensional settings for problems with single-valued and set-valued objectives.

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