On optimal control problem for the heat equation with integral boundary condition. (Russian. English summary) Zbl 1413.49002
Summary: In this paper we consider the optimal control problem for the heat equation with an integral boundary condition. Control functions are the free term and the coefficient of the equation of state and the free term of the integral boundary condition. The coefficients and the constant term of the equation of state are elements of a Lebesgue space and the free term of the integral condition is an element of Sobolev space. The questions of correct setting of optimal control problem in the weak topology of controls space are studied. We prove that in this problem there exist at least one optimal control. The set of optimal controls is weakly compact in the space of controls and any minimizing sequence of controls of a functional of goal converges weakly to the set of optimal controls. Fréchet differentiability of the functional of purpose on the set of admissible controls is proved. Formulas for the differential of the gradient of the purpose functional are obtained. A necessary optimality condition is established in the form of variational inequality.
MSC:
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 35K05 | Heat equation |
| 49J50 | Fréchet and Gateaux differentiability in optimization |
| 49J40 | Variational inequalities |
References:
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