Optimal control in parabolic singular perturbated problem with obstacle (Q2735860)

One considers the optimal control problem (P) of the following form. Find \((u,y)\) such that NEWLINE\[NEWLINEI(u)= \min_{v\in U}I(v),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEI(v)= \tfrac 12 \Biggl( \int_0^T \int_\Omega (y(x,t)- z(x))^2+\nu \int_0^T v^2(t) dt\Biggr)NEWLINE\]NEWLINE and \(y\) is the solution of the parabolic variational inequality: NEWLINE\[NEWLINE(y_t- \varepsilon^2 \Delta^2 y-g(x)v(t)y) (y-\psi(x))= 0\text{ a.e. }\Omega\times (0,T);NEWLINE\]NEWLINE NEWLINE\[NEWLINEy_t- \varepsilon^2 \Delta y-g(x) v(t)y\geq 0; \quad y\geq \psi(x)\text{ a.e. in }\Omega;NEWLINE\]NEWLINE NEWLINE\[NEWLINEy(x,0)= y_0(x),\;x\in \Omega; \quad y(x,t)=0,\;x\in \partial\Omega\times (0,T).NEWLINE\]NEWLINE The main results of this paper concern with the construction and the justification in respected norms of the inner and outer asymptotic expansion when \(\varepsilon\) tends to zero for \(u^\varepsilon\), \(y^\varepsilon\), \(p^\varepsilon\), and \(\tau^\varepsilon\). Here \((u^\varepsilon, y^\varepsilon)\) is the optimal pair for the problem (P) with fixed \(\varepsilon> 0\), \(p^\varepsilon\), \(\tau^\varepsilon\) are conjugate states and according for the control switching moment. The specific solved problem here is to find the first-order approximation for the sets \(Q_0^\varepsilon= \{(x,t): y^\varepsilon(x,t)= \psi(x)\}\), \(Q_+^\varepsilon= \{(x,t): y^\varepsilon(x,t)> \psi(x)\}\).