The Green function of a parabolic boundary-value problem and an optimization problem (Q2722240)

Let \(D\subset\mathbb R^n\) be a convex bounded domain, \(Q:=(0,T)\times D\), \(\psi_i(t,x)\in C^\alpha(Q)\), \(V:=\{p(t,x)\in C^\alpha(Q):\psi_1(t,x)\leq p(t,x)\leq \psi_2(t,x)\}\).NEWLINENEWLINENEWLINEThe author studies the problem of finding a pair of functions \((u,p)\) with the following properties: 1) \(p\in V\); 2) \(u(t,x,p)\) is a solution of a parabolic \((2b)\)th order equation \(\mathcal{L}u=f_0(t,x,p)\) and satisfies certain initial and boundary conditions; 3) the pair \((u,p)\) minimizes the functional \(I(p)=\int_{0}^{T} dt\int_{D}F(t,x,\bar u) dx\) where \(\bar u=(u,D_xu,\ldots,D_x^{2b-1}u,p):=(u_0,\ldots,u_{2b})\) and \(F\) is a sufficiently smooth function.NEWLINENEWLINENEWLINEUsing the Green function \(E(t,x,\tau,\xi)\) of the corresponding homogeneous parabolic problem the following functions are introduced: NEWLINE\[NEWLINE\begin{aligned} \omega(t,x)&=\sum_{i=0}^{2b-1}\int_{t}^{T} d\tau \int_{D} D_x^iE(\tau,\xi,t,x)D_{u_i} F(t,x,u) d\xi;\\ H(\bar u,\omega)& =F(t,x,\bar u)+\omega(t,x)f_0(t,x,u_{2b}).\end{aligned} NEWLINE\]NEWLINE It is shown that if \(H\) is monotonically increasing (decreasing) with respect to \(u_{2b}\in V,\) then the optimal solution is \((u(t,x,\psi_1(t,x)),\psi_1(t,x))\) (\((u(t,x,\psi_2(t,x)),\psi_2(t,x))\)).NEWLINENEWLINENEWLINEAn optimality criterion in the case of non-monotone function \(H\) is also obtained.