On a class of optimal control problems for parabolic equations (Q1920814)

Let be given the dynamic object \[ {\partial y\over\partial t}+ A(x,t)y= f+B(x) \nu,\;y|_{t=0},\left.\;{\partial y\over \partial x} \right|_{x=0,x=1}=0,\tag{*} \] where \(t\), \(x\) are the time and space coordinate, respectively, \(f(x)\in L_2 (\Omega)\) is a given function, \[ A(x,t)=-{\partial \over\partial t} \left(a(x,t) {\partial\over\partial x} \right),\quad B(x)=-{\partial\over \partial x} \left(b(x) {\partial\over \partial x} \right) \] where \(a(x,t)\) and \(b(x)\) are smooth functions such that \(0<\alpha_0\leq a(x,t)\leq \alpha_1<\infty\), \(0<\beta_0\leq b(x)\leq \beta_1<\infty\). For each \(\nu\in U_g\), where \[ U_g= \bigl\{\nu(x) \in W^2_2(\Omega),\;0<m\leq \nu(x) \leq M<\infty,\;\nu_x|_{x=0,x=1} \bigr\}, \] a state \(y(x,t)\) becomes a solution of (*). The paper deals with the problem of minimisation of the functional \[ I(\nu)=\int^1_0 \bigl[y(x,T; \nu)-y_1(x)\bigr]^2 dx \] for admissible controls taken from the set \(U_g\). The paper shows the way of control selection \(u\in U_g\) such that \(I(u)= \inf_{\nu\in U_g} I(\nu)\).