On the optimization formulation of the coefficient inverse problem for a parabolic equation with an additional integral condition

Let a controlled process be described in Q_T = {(x,t)e R : 0 < x < ^,0 < t < T} by the following initial-boundary value problem for a linear parabolic equation: u_t -(k(x,t)u_x) + q(x,t)u = f (x,t), (x,t)e Q_T , u| ₀ = ), 0 < x < I, u*l=0 = u_x\_x=, = f < < . Here i, T > 0 are given numbers f (x,t)e L₂ (Q_T), )eW2 (0,i) are given functions, k(x,t), q(x,t) are unknown coefficients, u(x,t) = (k(x,t),q(x,t)) is a control, u = u(x,t)= u(x,t;u) is the solution to the boundary value problem corresponding to the control u = u(x,t). Let us introduce a set of admissible controls V = {u(x,t) = (k(x,t),q(x,t)) e H = W, (Q_T)x L₂ (Q_T): 0 < v < k(x,t) < |k_x (x,t)| < |₁,|k_t (x,t)| <|₂,0 < q₀ < q(x,t) < q₁ п.в.на Q_T}, where |> v> 0 , |₁, |₂ > 0 , q₁ > q₀ > 0 are given numbers. Let us state the following coefficient inverse problem of optimal control type: among all the admissible controls u(x,t)=(k(x,t),q(x,t))eV, find the controls u₍(x,t)=(k„(x,t),q,,(x,t))eV minimizing the functional t i J (u) = U K (x, t )u (x, t; u) dx - E (t) 0 0 where K(x,t),E(t) are known functions, u = u(x,t) is a control u = u(x,t) = u(x,t;u) is a generalized solution to the boundary value problem from V₂ (Q_t ) corresponding to the control u = u(x, t) is a given set. In the present work, the optimization formulation of the coefficient inverse problem for a parabolic equation with an additional integral condition is considered. The questions of correctness of the optimization formulation of the inverse problem are investigated. The differentiability of the objective functional is proved and the formula for its gradient is found. A necessary condition of optimality is found in the form of a variational inequality.

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