On an optimal control problem for a parabolic equation with an integral condition and controls in coefficients

In this paper, an optimal control problem for a parabolic equation with an integral boundary condition and controls in coefficients is considered. Let it be required to minimize the functional i J() = f I м(,T; и) - У(x) | dx 0 on the solutions м = м( x, t) = м( x, t; o) of the boundary value problem u_t - (k (x, t )m_x )_x + q( x, t )m = f (x, t), (x, t) е Q_T ={(x,t):0 < x < 1, 0 < t < T} м (x,0) = ф(x), 0 < x < I, i m_x(0,t) = 0, k(l,t)u_x(I,t) = f H(x)m_x(x,t)dx + g(t), 0 < t < T, 0 corresponding to all allowable controls о = o(x,t) = (k(x,t), q(x,t)) from the set V = {u( x, t) = (k (x, t), q( x, t)) е H = W₂(Q_T) x L₂(Q_T) :0 < v < k (x, t )< ц, |k_x (x, t)<^₁, |k_t (x, t) <ц₂ |q (x, t)| <ц₃ a.e. on Q_T}. Here, l,T,v,ц,ц₁, ц₂, >0 are given numbers and y(x),ф(x)е №^(0,1), H(x)е W2(0,1), f (x, t )е L₂(Q_t ), and g (t )е№₂'(0,Г) are known functions. The work deals with problems of correctness in formulating the considered optimal control problem in the weak topology of the space H = W2^(Q_T) x L₂(Q_T). Examples showing that this problem is incorrect in the general case in the strong topology of the space H are presented. The objective functional is proved to be continuously Frechet differentiable and a formula for its gradient is found. A necessary condition of optimality is established in the form of a variational inequality.

Keywords

Authors

References