Boundary control for the heat equation with nonlinear boundary condition. (English) Zbl 0679.49002
This paper treats boundary control of the heat equation subject to a boundary condition which prescribes the normal derivative of the surface temperature as a function of the surface temperature and the ambient temperature, the latter being treated as the control. The existence and uniqueness of weak solutions is established; this permits one to prove the existence of optimal solutions to certain approximation problems. Finally a “bang-bang” principle is proved for a problem in 1 space dimension. This depends in a delicate way on unique backwards continuation as well as on results concerning strict total positivity and the heat equation due to the author [ibid. 62, 275–298 (1986; Zbl 0549.35052)].
Reviewer: E. T. Schmidt (Budapest)
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35D30 | Weak solutions to PDEs |
| 35K60 | Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations |
| 49J30 | Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 80A19 | Diffusive and convective heat and mass transfer, heat flow |
| 35K05 | Heat equation |
Keywords:
bang-bang principle; nonlinear boundary condition; boundary control; heat equation; strict total positivityCitations:
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