On the necessary conditions for optimal control of retarded systems

Angell, T. S.; Kirsch, Andreas

doi:10.1007/bf01447323

Cited by 12 publications

(3 citation statements)

References 14 publications

(22 reference statements)

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“…x, u, , ) := L(t, x, u)+ * f (t, x, u)+ * C(t, x, u) (9) Here and in the sequel, * denotes the transposition. The extension of the classical Pontryagin's minimum principle to the mixed control-state constraints (6) requires a regularity condition or constraint qualification.…”

Section: Optimal Control Problems With Delays In State and Controlmentioning

confidence: 99%

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Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Göllmann

Kern

Maurer

2008

Optim Control Appl Methods

228

190

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Optimal control problems with delays in state and control variables are studied. Constraints are imposed as mixed control-state inequality constraints. Necessary optimality conditions in the form of Pontryagin's minimum principle are established. The proof proceeds by augmenting the delayed control problem to a nondelayed problem with mixed terminal boundary conditions to which Pontryagin's minimum principle is applicable. Discretization methods are discussed by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. It is shown that the Lagrange multipliers associated with the programming problem provide a consistent discretization of the advanced adjoint equation for the delayed control problem. An analytical example and numerical examples from chemical engineering and economics illustrate the results. problems with a constant state delay. In [3], he gave similar results for control problems with pure control delays. Halany [4] proves a maximum principle for optimal control problems with multiple constant delays in state and control variables that, however, are chosen to be equal for state and control. Similar results were obtained by Ray and Soliman [5]. Guinn [6] sketches a simple method for obtaining necessary conditions for control problems with a constant delay in the state variable. He suggests to augment the delayed control problem that yields a higher-dimensional undelayed control problem to which the standard maximum principle is applicable. Banks [7] derives a maximum principle for control systems with a time-dependent delay in the state variable. Delays in the control are admitted for systems linear in the control variable. Colonius and Hinrichsen [8] provide a unified approach to control problems with delays in the state variable by applying the theory of necessary conditions for optimization problems in function spaces. All articles mentioned so far do not consider general control or state inequality constraints.Angell and Kirsch [9] treat functional differential equations with function-space state inequality constraints. However, they do not discuss the regularity of the multiplier associated with the state constraint and do not provide a numerical example with a pure state space constraint. To our knowledge, optimal control problems with constant delays in state and control variables and mixed control-state inequality constraints have not yet been considered in the literature. The first goal in this paper is to derive a Pontryagin-type minimum (maximum) principle for this class of delayed control problems. Concerning the development of numerical methods and the numerical treatment of practical examples, our impression is that this topic has not yet been adequately addressed in the literature. Bader [10] uses collocation methods to solve the boundary value problem for the retarded state variable and advanced adjoint variable. He successfully solves several academic examples, but his method does not give accurate results for the more difficult CSTR...

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Section: Optimal Control Problems With Delays In State and Controlmentioning

confidence: 99%

“…The fixed starting profiles (3) and (4) (8) and (9) for the nondelayed augmented control problem are given by…”

Section: Proofmentioning

confidence: 99%

Optimal control problems with delays in state and control variables subject to mixed control–state constraints

Göllmann

Kern

Maurer

2008

Optim Control Appl Methods

228

190

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“…The mainstream in the study of optimal control problems for nondelayed differential inclusions consists in obtaining necessary conditions for optimality; we refer the reader to Clarke [4], Ioffe [11], Loewen and Rockafellar [14], Mordukhovich [20], Sussmann [25], Vinter and Zheng [26], Zhu [29], and the bibliographies therein. The first results on necessary optimality conditions for control problems governed by smooth delay-differential equations ẋ(t) = g x(t), x(t − ), u(t), t , u(t) ∈ U, were obtained by Kharatishvili [13] and then were developed in many publications; see, e.g., Angell and Kirsch [1], Banks and Manitius [3], Mordukhovich [18], Warga [27,28], and their references. However, not much has been done for dynamic optimization problems governed by delay-differential inclusions.…”

Section: Introductionmentioning

confidence: 99%

Untitled

Mordukhovich

Trubnik

2001

Annals of Operations Research

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This paper is devoted to the study of optimization problems for dynamical systems governed by constrained delay-differential inclusions with generally nonsmooth and nonconvex data. We provide a variational analysis of the dynamic optimization problems based on their data perturbations that involve finite-difference approximations of time-derivatives matched with the corresponding perturbations of endpoint constraints. The key issue of such an analysis is the justification of an appropriate strong stability of optimal solutions under finite-dimensional discrete approximations. We establish the required pointwise convergence of optimal solutions and obtain necessary optimality conditions for delay-differential inclusions in intrinsic Euler-Lagrange and Hamiltonian forms involving nonconvex-valued subdifferentials and coderivatives of the initial data.

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