Approximation theory for generalized Young measures.

*(English)*Zbl 0854.65051The Young measures are a mathematical tool to obtain a certain “limit” information about rapid oscillations in nonlinear problems arising in optimal control theory, variational calculus, partial differential equations, game theory, etc. Young measures can be understood as certain linear continuous functionals on a space of Caratheodory integrands, which gives a basis for their various generalizations. Then the typical technique of the approximation of Young measures will rely on the adjoint operators to suitable linear continuous operator acting on these integrands. In each case, a certain discretization of the problem in order to solve it approximately on computers is needed. There are now only few attempts to develop a satisfactory theory of approximation of Young measures.

The purpose of this paper is to develop a general and simple theory which would cover all already known techniques. At the same time, the proposed theory will apply also to various generalizations of Young measures that have recently appeared in the literature. Typically, these generalizations can handle oscillation effects simultaneously with the concentration ones, which is especially desirable for nonconvex optimal control problems, for nonconvex variational problems, and for nonlinear partial differential equations. Illustrative applications to a relaxed optimization problem are given to compare the various techniques.

The purpose of this paper is to develop a general and simple theory which would cover all already known techniques. At the same time, the proposed theory will apply also to various generalizations of Young measures that have recently appeared in the literature. Typically, these generalizations can handle oscillation effects simultaneously with the concentration ones, which is especially desirable for nonconvex optimal control problems, for nonconvex variational problems, and for nonlinear partial differential equations. Illustrative applications to a relaxed optimization problem are given to compare the various techniques.

Reviewer: G.S.Stavrakakis (Chania)

##### MSC:

65K10 | Numerical optimization and variational techniques |

49J40 | Variational inequalities |

49M15 | Newton-type methods |

##### Keywords:

finite element method; relaxed optimal control problems; Young measures; rapid oscillations; optimal control; Caratheodory integrands; nonconvex variational problems
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\textit{T. Roubíček}, Numer. Funct. Anal. Optim. 16, No. 9--10, 1233--1253 (1995; Zbl 0854.65051)

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