Abstract
In this paper, we consider the question of necessary conditions for optimality for systems governed by second-order parabolic partial delay-differential equations with first boundary conditions. All the coefficients of the system are assumed bounded measurable and contain controls and delays in their arguments. The second-order parabolic partial delay-differential equation is in divergence form. In Theorem 4.1, we present results on the existence and uniqueness of weak solutions in the sense of Ladyzhenskaya-Solonnikov-Ural'ceva for this class of systems. An integral maximum principle and its point-wise version for the corresponding controlled system are established in Theorem 5.1 and Corollary 5.1, respectively.
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Communicated by G. Leitmann
The authors wish to thank Dr. E. Noussair for his stimulating discussion and valuable comments in the preparation of this paper. Further, they also wish to acknowledge the referee of the paper for his valuable suggestions and comments. The discussion presented in Section 6 is in response to his suggestions.
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Nababan, S., Teo, K.L. Necessary conditions for optimal controls for systems governed by parabolic partial delay-differential equations in divergence form with first boundary conditions. J Optim Theory Appl 36, 565–613 (1982). https://doi.org/10.1007/BF00940547
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DOI: https://doi.org/10.1007/BF00940547