scholarly journals First-Order Necessary Conditions in Optimal Control

2021 ◽  
Author(s):  
David Mayne ◽  
Richard Vinter
Keyword(s):  
Optimal Control ◽  
Optimality Condition ◽  
State Constraints ◽  
Minimum Principle ◽  
Inequality Constraints ◽  
Problem Formulation ◽  
State Constraint ◽  
First Order ◽  
Accumulation Points ◽  
Minimax Condition

AbstractIn an earlier analysis of strong variation algorithms for optimal control problems with endpoint inequality constraints, Mayne and Polak provided conditions under which accumulation points satisfy a condition requiring a certain optimality function, used in the algorithms to generate search directions, to be nonnegative for all controls. The aim of this paper is to clarify the nature of this optimality condition, which we call the first-order minimax condition, and of a related integrated form of the condition, which, also, is implicit in past algorithm convergence analysis. We consider these conditions, separately, when a pathwise state constraint is, and is not, included in the problem formulation. When there are no pathwise state constraints, we show that the integrated first-order minimax condition is equivalent to the minimum principle and that the minimum principle (and equivalent integrated first-order minimax condition) is strictly stronger than the first-order minimax condition. For problems with state constraints, we establish that the integrated first-order minimax condition and the minimum principle are, once again, equivalent. But, in the state constrained context, it is no longer the case that the minimum principle is stronger than the first-order minimax condition, or vice versa. An example confirms the perhaps surprising fact that the first-order minimax condition is a distinct optimality condition that can provide information, for problems with state constraints, in some circumstances when the minimum principle fails to do so.

scholarly journals On geometric duality for state-constrained control problems

1979 ◽  
Vol 20 (2) ◽  
pp. 301-312
Author(s):  
T.R. Jefferson ◽  
C.H. Scott
Keyword(s):  
Optimal Control ◽  
Control Problem ◽  
State Constraints ◽  
Strong Duality ◽  
State Constraint ◽  
Continuous Functions ◽  
Control Problems ◽  
Absolutely Continuous Functions ◽  
Pure State Constraints ◽  
Constrained Control Problems

For convex optimal control problems without explicit pure state constraints, the structure of dual problems is now well known. However, when these constraints are present and active, the theory of duality is not highly developed. The major difficulty is that the dual variables are not absolutely continuous functions as a result of singularities when the state trajectory hits a state constraint. In this paper we recognize this difficulty by formulating the dual probram in the space of measurable functions. A strong duality theorem is derived. This pairs a primal, state constrained convex optimal control problem with a dual convex control problem that is unconstrained with respect to state constraints. In this sense, the dual problem is computationally more attractive than the primal.

scholarly journals KONTROL OPTIMAL MODEL PENYEBARAN VIRUS KOMPUTER DENGAN PENGARUH KOMPUTER EKSTERNAL YANG TERINFEKSI DAN REMOVABLE STORAGE MEDIA

2017 ◽  
Vol 9 (1) ◽  
pp. 113
Author(s):  
Dewi Erla Mahmudah ◽  
Muhammad Zidny Naf’an ◽  
Muh. Sofi’i ◽  
Wika Wika
Keyword(s):  
Optimal Control ◽  
Optimality Condition ◽  
Minimum Principle ◽  
Prevention Strategies ◽  
Optimal Model ◽  
Pontryagin Minimum Principle ◽  
Computer Viruses ◽  
Storage Media ◽  
The Cost

.  In this paper, we discuss an optimal control on the spread of computer viruses under the effects of infected external computers and removable storage media. Prevention Strategies do with ascertaining control prevention to minimize the number of infective computers (Latent and Breakingout) and installing effective antivirus programs in each sub-population. The aim are to derive optimal prevention strategies and minimize the cost associated with the control. The characterization of optimal control is perform analitically by applying Pontryagin Minimum Principle. The obtained optimality system of Hamilton fuction is satistfy the optimality condition.

Guess-Free Pseudospectral Solution for Time Optimal Swing-Up of a Rotary Inverted Pendulum

2017 ◽  
Author(s):  
Paul J. Frontera ◽  
Matthew Feemster ◽  
Michael Hurni ◽  
Mark Karpenko
Keyword(s):  
Optimal Control ◽  
Inverted Pendulum ◽  
Minimum Principle ◽  
Problem Formulation ◽  
Fixed Time ◽  
Time Problem ◽  
Time Optimal ◽  
Switching Structure ◽  
Original Time ◽  
Rotary Inverted Pendulum

Control of the inverted pendulum is a canonical problem in nonlinear and optimal control. Over the years, many workers have developed solutions for inverting the pendulum link (swing-up phase) and for maintaining the pendulum link upright (stabilization/disturbance rejection). In this paper, the time-optimal swing-up of a rotary inverted pendulum is studied. Previous solutions to this problem have required that the original time-optimal problem formulation be transformed to a more computationally tractable form. For example, one transformation is to a fixed-time problem with bounds on the control. Other approaches involve guessing the switching structure in order to construct a candidate solution. Advances in computational optimal control theory, particularly pseudospectral optimal control, allow the original time-optimal problem to be solved directly, and without the need for a guess. One such solution is presented in this paper. It is shown that the result adheres to the conditions of Pontryagin’s minimum principle. An experimental implementation of the solution illustrates its feasibility in practice.

scholarly journals Data-Driven Approximated Optimal Control for Chemical Processes with State and Input Constraints

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Mingfang He
Keyword(s):  
Optimal Control ◽  
Process Model ◽  
State Constraints ◽  
Inequality Constraints ◽  
Chemical Processes ◽  
Adaptive Dynamic Programming ◽  
Model Parameters ◽  
Model Structure ◽  
Input Constraints ◽  
Continuous Stirred Tank Reactor

Input and state constraints widely exist in chemical processes. The optimal control of chemical processes under the coexistence of inequality constraints on input and state is challenging, especially when the process model is only partially known. The objective of this paper is to design an applicable optimal control for chemical processes with known model structure and unknown model parameters. To eliminate the barriers caused by the hybrid constraints and unknown model parameters, the inequality state constraints are first transformed into equality state constraints by using the slack function method. Then, adaptive dynamic programming (ADP) with nonquadratic performance integrand is adopted to handle the augmented system with input constraints. The proposed approach requires only partial knowledge of the system, i.e., the model structure. The value information of the model parameters is not required. The feasibility and performance of the proposed approach are tested using two nonlinear cases including a continuous stirred-tank reactor (CSTR) example.

scholarly journals Optimal Control for the Thin Film Equation: Convergence of a Multi-Parameter Approach to Track State Constraints Avoiding Degeneracies

2016 ◽  
Vol 16 (4) ◽  
pp. 685-702
Author(s):  
Markus Klein ◽  
Andreas Prohl
Keyword(s):  
Thin Film ◽  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
State Constraints ◽  
Necessary Optimality Conditions ◽  
State Constraint ◽  
The State ◽  
Well Posedness ◽  
Thin Film Equation

AbstractWe consider an optimal control problem subject to the thin-film equation. The PDE constraint lacks well-posedness for general right-hand sides due to possible degeneracies; state constraints are used to circumvent this problematic issue and to ensure well-posedness. Necessary optimality conditions for the optimal control problem are then derived. A convergent multi-parameter regularization is considered which addresses both, the possibly degenerate term in the equation and the state constraint. Some computational studies are then reported which evidence the relevant role of the state constraint, and motivate proper scalings of involved regularization and numerical parameters.

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