Some Mathematical Theory of the Penalty Method for Solving Optimum Control Problems

Okamura, Kiyohisa

doi:10.1137/0302026

Cited by 22 publications

(4 citation statements)

References 3 publications

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“…Penalty methods addressing OC problems with inequality constraints such as those presented in Equations ( 13)-( 15) have a long history. 73,74 These methods consist of transforming the constrained OC problem into a single or a sequence of unconstrained problems. This is accomplished by direct adjoining of a penalty function…”

Section: Penalty Methods For State and Control Constraintsmentioning

confidence: 99%

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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In the mid‐1950s, Pontryagin et al. published a principle that became a fundamental concept in optimal control (OC) theory. The principle provides theoretical and practical methods to find the solution of OC problems, in particular, open‐loop control problems. In chemical engineering, the principle has played an important role as a decision making framework for more than 60 years. This study gathers the main contributions on the application of the Pontryagin's principle to the dynamic optimization of chemical processes. A concise overview of the optimality conditions for a wide class of constrained OC problems is provided. Numerical methods to solve the necessary conditions and strategies to address inequality constraints are summarized. The information and illustrative case study presented in this work can be used as a guide to implement the principle in different settings. Opportunities for further application of the principle in relevant chemical engineering problems are also discussed.

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Section: Penalty Methods For State and Control Constraintsmentioning

confidence: 99%

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

2022

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“…The penalty method is a common technique in optimization to approximate constrained optimization problems with a sequence of unconstrained optimization problems [11,14,15,18]. In this section, we apply a penalty approach to (TCR).…”

Section: Approximation With a Penalty Functionmentioning

confidence: 99%

Penalty function method for the minimal time crisis problem

2021

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“…I t is tacitly assumed that any inequality constraints on the control or state variables may be suitabl>-approximated by penalty functions included in the index of performance [15], [16]. I t is also assumed that the functions f, g, and G belong to C2[0, TI, the class of continuous functions n-ith continuous first and second partial derivatives.…”

Section: Theoretical Developmentmentioning

confidence: 99%

Optimal control computation by the Newton-Raphson method and the Riccati transformation

Schley

Lee

1967

IEEE Trans. Automat. Contr.

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A computational procedure is 'described for finding extremal control policies for nonlinear processes. The control problem is formulated as the problem of Bolza in the classical calculus of variations. The method is iterative in nature and prescribes a successive linearization of the Euler-Lagrange equations to obtain the extremal control. The resulting sequence of linear problems is solved such that the successive solutions may, in some well-defined manner, converge to the extremal solution.The linear two-point boundary value problem is decoupled by means of a generalized Riccati transformation. First, the matrix Riccati equation is integrated backwards in time. Then the state equations can be integrated forwards in time. The resulting curves can then be used for the next iteration.It is pointed out that in some instances the method yields a linear feedback control law, which is optimal with respect to initial condition perturbations. Also, the method allows for a suflicieney check. At the termination of the iterations, Jacobi's condition and the strengthened Legendre condition can be checked to see whether or not the control obtained is indeed optimal in the sense of a weak local mini- mum.Numerical results are presented and a brief comparison is made between this method and the First and Second Variation methods.

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Some Mathematical Theory of the Penalty Method for Solving Optimum Control Problems

Cited by 22 publications

References 3 publications

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

Optimal control and the Pontryagin's principle in chemical engineering: History, theory, and challenges

Penalty function method for the minimal time crisis problem

Optimal control computation by the Newton-Raphson method and the Riccati transformation

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