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Differential duality and applications. (Dualité différentielle et applications.) (French. Abridged English version) Zbl 0863.93016

In ordinary differential control theory the use of the dual system which is controllable (observable) if and only if the given system is observable (controllable), is limited because one cannot avoid introducing the state. In partial differential control theory, introduced by the author in earlier papers, controllability is defined in a formal way which does not depend on the state representation or on the separation of system variables between input and output. The two basic problems to which the paper tries to provide answers, are: 1. Is it possible to test the controllability of a partial differential control system in an effective way, which is compatible with the ordinary differential case and duality? 2. Is it possible to test in an effective way that a given linear differential operator \({\mathcal D}_1\) constitutes all the compatibility conditions of a certain linear differential operator \({\mathcal D}\)? (The operator \({\mathcal D}_1\) is said to be parametrized by \({\mathcal D}\) if this happens.) The main result of the paper states that a linear differential operator is controllable if and only if it is parametrizable, and an effective proof is given. Some applications, including the fact that Einstein equations in vacuum do not possess generic solutions that can be expressed by means of a finite number of arbitrary potentials, complete the paper.

MSC:

93B25 Algebraic methods
93C20 Control/observation systems governed by partial differential equations
93C05 Linear systems in control theory
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