Numerical solutions of optimal-control problems for multidimensional systems by locally one-dimensional schemes (Q1333718)

The following optimal control problem is considered as a model problem to solve by the proposed approximation procedure: \[ \min\left\{\int_ \Omega g_ 0(x)\phi^ 2(x,T)dx+ \int_ Q [g_ 1(x,t)\phi^ 2(x,t)+ g_ 2(x,t) u^ 2(x,t)]dx dt\right\},\tag{1} \] where \(\Omega\subset \mathbb{R}^ n\), \(Q=\Omega\times [0,T]\), \(g_ 0(x)> 0\), \(g_ 1(x,t)> 0\), \(g_ 2(x,t)> 0\), and the state \(\phi\) and the control \(u\) are linked by the parabolic state equation \[ \begin{cases} \phi_ t= \sum^ n_{i=1} k_ i\phi_{x_ i x_ i}+ u\quad\text{in }Q,\\ \phi(x,0)=0,\quad\phi= 0\quad\text{on }\partial\Omega\times [0,T].\end{cases}\tag{2} \] It is known that the (unique) solution \((\phi,u)\) of (1) satisfies, besides (2), the equations \[ \begin{cases} p_ t= -\sum^ n_{i=1} k_ i p_{x_ i x_ i}+ g_ 1(x,t)\phi,\\ p(x,T)= -g_ 0(x)\phi(x,T),\quad p=0\quad\text{on }\partial\Omega\times [0,T],\\ u(x,t)= g^{-1}_ 2(x,t)p(x,t)\end{cases} \] where \(p\) is the adjoint state. The proposed approximation procedure is \[ \begin{cases} (\phi_ k)_ t= \sum^ n_{i=1} k_ i(\phi_ k)_{x_ i x_ i}+ u_ k\quad\text{in }Q,\\ \phi_ k(x,0)= 0,\quad \phi_ k=0\quad\text{on }\partial \Omega\times [0,T],\\ (p_{k+1})_ t= -\sum^ n_{i=1} k_ i(p_{k+1})_{x_ i x_ i}+ g_ 1(x,t)\phi_ k,\\ p_{k+1}(x,T)= -g_ 0(x)\phi_ k(x,T),\quad p_{k+1}= 0\quad\text{on }\partial\Omega\times [0,T],\\ u_ k= g^{- 1}_ 2 p_ k\end{cases} \] whose convergence is shown under the condition \[ \max g^{-1}_ 2\left\{\max g^ 2_ 0+ {T\over 2\beta}\max g^ 2_ 1\right\}< {2\beta\over T} \] being \(\beta= {4n\min k_ i\over 3(\text{diam }\Omega)^ 2}\).