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A maximum principle for nonconservative self-adjoint systems. (English) Zbl 0704.73065

The present paper states and proves a maximum principle that is readily adaptable to a class of control problems for a damped distributed parameter system. The system is governed by a set of linear partial differential equations with homogeneous boundary conditions. The governing equations of this system could describe vibrations of a Timoshenko beam or a Bernoulli plate as well as composite plates or shells on the basis of the effective modulus theory of composites. In the first part of the paper, precise conditions are given to ensure uniqueness and existence in solutions of the governing equations; the proof is based of Gronwall’s inequality and a compactness argument. Also, the existence of the optimal control is studied and the maximum principle is shown to be a sufficient condition for the determination of the optimal control. In the second part, the solution of the homogeneous adjoint problem is considered, and sample examples are provided.
This paper contains some new results which seem to have fruitful applications for nonconservative adjoint systems.
Reviewer: M.C.Dökmeci

MSC:

74M05 Control, switches and devices (“smart materials”) in solid mechanics
74P99 Optimization problems in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
49J20 Existence theories for optimal control problems involving partial differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74K20 Plates
74E30 Composite and mixture properties
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