Maximum principle for the optimal control of a hyperbolic equation in two space dimensions. (English) Zbl 0949.49500
Summary: A direct method based on normal forms is proposed for constructing the nonlinear normal modes of continuous systems. The proposed method is compared with the method of multiple scales and the methods of Shaw and Pierre and of King and Vakakis by applying them to three conservative systems with cubic nonlinearities: (a) a hinged-hinged beam resting on a nonlinear elastic foundation, (b) a model of a relief valve (linear elastic spring attached to a nonlinear spring with a mass), and (c) a simply supported linear beam with nonlinear torsional springs at both ends. In the absence of internal resonance, the nonlinear modes constructed by means of the methods are the same. The method of multiple scales seems to be the simplest and the least computationally demanding. The methods of multiple scales and normal forms are applicable to problems with and without internal resonances, whereas the present forms of the methods of Shaw and Pierre and of King and Vakakis are not applicable to problems with internal resonances.
MSC:
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 74M05 | Control, switches and devices (“smart materials”) in solid mechanics |
| 93C20 | Control/observation systems governed by partial differential equations |
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