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11.
In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and evolution operators that are subdifferentials of a proper convex lower semicontinuous function depending on time. Along with the original inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are subdifferentials of the Moreau–Yosida regularizations of the original function. We show that the attainable set of the original inclusion, regarded as a multivalued function of time, is the uniform (in time) limit in the Hausdorff metric of the sequence of attainable sets of the approximating inclusions. As an application we consider an example of a control system with discontinuous nonlinearity. 相似文献
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In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and the evolution operators that are the compositions of a linear operator and the subdifferentials of a time-dependent proper convex lower semicontinuous function. Alongside the initial inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are the compositions of the same linear operator and the subdifferentials of the Moreau–Yosida regularizations of the initial function. We demonstrate that the attainable set of the initial inclusion as a multivalued function of time is the time uniform limit of a sequence of the attainable sets of the approximating inclusions in the Hausdorff metric. We obtain similar results for evolution control systems of subdifferential type with mixed constraints on control. As application we consider an example of a control system with discontinuous nonlinearities containing some linear functions of the state variables of the system. 相似文献
15.
D. A. Tolstonogov 《Mathematical Notes》1999,65(1):109-119
We prove an abstract existence theorem for the minimum of the functional
where the mappingG(y) is concave and the functionh(x, u) is nonconvex inu, under constraints of inequality type imposed on solutions of systems described by linear elliptic operators. This theorem
is further specified for some problems in calculus of variations and optimal control theory.
Translated fromMatematicheskie Zametki, Vol. 65, No. 1, pp. 130–142, January, 1999. 相似文献
16.
A. A. Tolstonogov 《Siberian Mathematical Journal》2002,43(5):926-941
We continue the research of the first part of the article. We mainly study codensity for the set of admissible trajectory-control pairs of a system with nonconvex constraints in the set of admissible trajectory-control pairs of the system with convexified constraints. We state necessary and sufficient conditions for the set of admissible trajectory-control pairs of a system with nonconvex constraints to be closed in the corresponding function spaces. Using an example of a control hyperbolic system, we give an interpretation of the abstract results obtained. As application we consider the minimization problem for an integral functional on solutions of a control system. 相似文献
17.
A. A. Tolstonogov 《Journal of Mathematical Sciences》2007,140(6):850-872
We consider the minimization problem of an integral functional in a separable Hilbert space with integrand not convex in the
control defined on solutions of the control system described by nonlinear evolutionary equations with mixed nonconvex constraints.
The evolutionary operator of the system is the subdifferential of a proper, convex, lower semicontinuous function depending
on time.
Along with the initial problem, the author considers the relaxed problem with the convexicated control constraint and the
integrand convexicated with respect to the control.
Under sufficiently general assumptions, it is proved that the relaxed problem has an optimal solution, and for any optimal
solution, there exists a minimizing sequence of the initial problem converging to the optimal solution with respect to trajectories
and the functional. An example of a controlled parabolic variational inequality with obstacle is considered in detail.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear
Dynamics, 2005. 相似文献
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A. N. Volobuev A. P. Tolstonogov 《Journal of Surface Investigation: X-ray, Synchrotron and Neutron Techniques》2013,7(4):762-773
Light propagation through a polarizer-analyzer system is investigated on the basis of quantum concepts about the nature of light. It is shown that Malus’ law, which is based on principles of classical electrodynamics, does not fully take into account all effects that can arise under light propagation through a polarizer-analyzer system. In particular, the phenomenon of a possible change in the light frequency is not considered, e.g., in the case of X-ray radiation. The derivation of Malus’ law based on quantum principles is presented. For comparison, the differential effective cross section of the interaction between a photon and an electron is found taking into account rotation of the photon polarization plane in the Compton effect. 相似文献