A Multiplier Rule for a Functional Subject to Certain Integrodifferential Constraints

A problem in the optimal control of a nuclear rocket requires the minimization of a functional subject to an integral equation constraint and an integrodifferential inequality constraint. A theorem giving first-order necessary conditions is derived for this problem in the form of a multiplier rule. The existence of multipliers and the arbitrariness of certain variations is shown. The fundamental lemma of the calculus of variations is applied. A simple example demonstrates the applicability of the theorem.

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Optimal Control and Necessary Optimality Conditions for Nonlinear and Perturbed Dynamic Problems

Journal of Mathematics Research ◽

10.5539/jmr.v10n6p63 ◽

2018 ◽

Vol 10 (6) ◽

pp. 63

Author(s):

Gossan D. Pascal Gershom ◽

Bailly Balè ◽

Yoro Gozo

Keyword(s):

Optimal Control ◽

Optimality Conditions ◽

Variational Formulation ◽

Necessary Conditions ◽

Three Dimensional ◽

Necessary Optimality Conditions ◽

Cancer Cell Proliferation ◽

Dynamic Problems ◽

First Order ◽

Tumor Growth Model

The main goal of this paper is to establish the first order necessary optimality conditions for a tumor growth model that evolves due to cancer cell proliferation. The phenomenon is modeled by a system of three-dimensional partial differential equations. We prove the existence and uniqueness of optimal control and necessary conditions of optimality are established by using the variational formulation.

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Optimal Control in the Presence of Nondifferentiable State Constraints

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3427062 ◽

1976 ◽

Vol 98 (4) ◽

pp. 432-439

Author(s):

E. D. Eyman ◽

D. P. Sudhakar

Keyword(s):

Optimal Control ◽

Integral Equation ◽

Mathematical Programming ◽

State Constraints ◽

Necessary Conditions ◽

Adjoint Equation ◽

Phase Constraint ◽

Maximal Condition ◽

Phase Constraints ◽

Differential Control

Necessary conditions are derived for optimality of differential control processes in the presence of nondifferentiable state (or phase) constraints. The techniques of general Mathematical Programming and the Dubovitskii-Milyutin Theorem are employed. The necessary conditions derived are in the form of an adjoint integral equation and a pointwise maximal condition. It is found that the gradient of the state (or phase) constraint can be replaced by the Gateaux differential of a certain form in the adjoint equation.

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The Erdmann Condition and Hamiltonian Inclusions in Optimal Control and the Calculus of Variations

Canadian Journal of Mathematics ◽

10.4153/cjm-1980-039-9 ◽

1980 ◽

Vol 32 (2) ◽

pp. 494-509 ◽

Cited By ~ 9

Author(s):

Frank H. Clarke

Keyword(s):

Calculus Of Variations ◽

Basic Problem ◽

Necessary Conditions ◽

Legendre Transform ◽

Hamiltonian Mechanics ◽

First Order ◽

Autonomous Case ◽

Necessary Conditions For Optimality ◽

Image Position ◽

Class Of Functions

Consider the basic problem in the calculus of variations, that of minimizing1.1over a class of functions x satisfying certain boundary conditions at 0 and 1. One of the classical first order necessary conditions for optimality is the second Erdmann condition, which asserts, in the case in which L is independent mof t, that1.2along any local solution x. This formula is the customary basis for solving many of the classical problems, such as the brachistochrone. When it is possible to define via the Legendre transform a Hamiltonian H(t, x, p) corresponding to L, the second Erdmann condition, again in the autonomous case, is the assertion that1.3a relation which always evokes classical Hamiltonian mechanics and conservation laws.

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