Non-local parabolic boundary value optimization problem with bounded limit control (Q2768796)

Let \(D\) be a bounded convex domain in \(R^{n}\) with the boundary \(\partial D\). Let us consider in \(Q=[0,T)\times D\) the problem of finding functions \((u,p)\) which minimize the functional NEWLINE\[NEWLINE I(p)=\int_{0}^{T} dt\int_{D}F_1(t,x,\vec u) dx+\int_{0}^{T} dt\int_{\partial D}F_2(t,x,\vec \omega) d_{x}SNEWLINE\]NEWLINE on the class of functions \(V=\{p(t,x):\;p(t,x)\in C^{(2b-r+\alpha)}(\Gamma), r=\min_{i}r_{i}, \psi_{1}(t,x)\leq p(t,x)\leq\psi_2(t,x)\}\), where \(u(t,x,p)\) is a solution of the problem NEWLINE\[NEWLINE\biggl(D_{t}-\sum_{|k|\leq 2b}A_{k}(t,x)D_{x}^{k} \biggr)u= f_0(t,x),\;u(0,x,p)+\sum_{j=1}^{m}\sum_{|k|\leq\beta_{j}}a_{k}(t_{j},x)D_{x}^{k}u(t_{j},x,p)=\phi(x),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\biggl(\sum_{|k|\leq r _{i}}b_{k}^{(i)}(t,x)D_{x}^{k} \biggr)u|_{\Gamma}=f_{i}(t,x,p), NEWLINE\]NEWLINE \(0\leq r_{i}\leq 2b-1\), \(i=1,\ldots,b\), \(\vec u=(u,D_{x}u,\ldots,D_{x}^{r-1}u)\), \(\vec\omega=(u|_{\Gamma},D_{x}u|_{\Gamma},\ldots,D_{x}^{r-1}u|_{\Gamma},p)\); \(\Gamma=(0,T)\times \partial D\), \(D_{x}^{k}=D_{x_1}^{k_1}\ldots D_{x_{n}}^{k_{n}}\), \(k=k_1+\ldots+k_{n}\), \(0<t_1<\ldots<t_{m}\leq T\). The author obtains the necessary and sufficient conditions of optimality in the considered problem.