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On the uniqueness of the solution to a minimum problem for a control process with distributed parameters

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Abstract

Let

$$\begin{gathered} z_{xy} + A(x, y)z_x + B(x, y)z_y + C(x, y)z = F(x, y)U(x, y) + G(x, y) \hfill \\ a.e. in[0, a[ \times [0, b[ \hfill \\ \end{gathered} $$

be a control process with distributed parameters. For such a process, a subset of the real line being chosen as target, a problem of optimal control (analogous to that of minimal time for control processes with concentrated parameters) is investigated under suitable assumptions. Some uniqueness theorems are established. To this aim, the notions of positively normal, negatively normal, and normal process are given. The relationships between these notions and the uniqueness results are discussed.

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References

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    Pulvirenti, G., andSantagati, G.,Nuovi Contributi alla Teoria dei Controlli per Sistemi Materiali Retti da una Equazione alle Derivate Parziali di Tipo Iperbolico, Le Matematiche, Vol. 24, pp. 375–391, 1969.

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    Hadamard, J.,Le Problème de Cauchy et les Equations aux Dérivées Partielles Linéaires Hyperboliques, Hermann, Paris, France, 1932.

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    Pulvirenti, G., andSantagati, G.,Un Problema di Minimo Relativo ad un Processo di Controllo con Parametri Distribuiti, Le Matematiche (to appear).

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    Conti, R.,Problemi di Controllo e di Controllo Ottimale, UTET, Torino, Italy, 1974.

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    Marino, M.,Sulla Unicità della Soluzione di Alcuni Problemi di Minimo per un Processo di Controllo con Parametri Distribuiti, Le Matematiche, Vol. 27, pp. 111–132, 1972.

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Additional information

This research was supported by G.N.A.F.A., Consiglio Nazionale delle Ricerche, Rome, Italy.

Communicated by R. Conti

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Pulvirenti, G., Santagati, G. On the uniqueness of the solution to a minimum problem for a control process with distributed parameters. J Optim Theory Appl 28, 109–124 (1979). https://doi.org/10.1007/BF00933604

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Key Words

  • Distributed-control problem
  • minimum problem
  • uniqueness theorems