zbMATH — the first resource for mathematics

Optimal pointwise control of semilinear parabolic equations. (English) Zbl 0939.49005
The paper considers the problem of minimizing the functional \[ I(y,u)= \int_\Omega|y(T)- y_d|^s dx+ \beta\int^T_0 |u(t)|^q dt \] over all pairs \((y,u)\in L_1(0,T; W^{1,1}(\Omega))\times U\), which satisfy \[ \begin{aligned} {\partial y\over\partial t}+ Ay+|y|^{\gamma-1} y= u(t)\delta_{x_0}\quad & \text{in }Q,\\ y|_{t=0}= y_0\quad\text{in }\Omega,\quad{\partial y\over\partial n_A}= 0\quad & \text{on }\Sigma.\end{aligned}\tag{1} \] Here \(\Omega\subset\mathbb{R}^n\) is a bounded domain with a boundary \(\partial\Omega\) of class \(C^2\), \(Q= (0,T)\times \Omega\), \(\Sigma= (0,T)\times\partial \Omega\), \(Ay= -\sum^n_{i,j=1}{\partial\over\partial x_i} a_{ij}(x) y_{x_j}+ a_0(x)y\), \(\delta_{x_0}\) denotes the Dirac measure at \(x_0\in\Omega\) and \(U\subset L_q(0,T)\) is a convex closed set. It is supposed that \(1\leq\gamma< n/(n- 2)\), \(\max\{1; 2\gamma- N\gamma+ N+ 2)\}< q<\infty\).
It is shown that the optimal control problem has a solution, and the corresponding necessary optimality conditions are obtained. To do that the authors obtain interesting results on the solvability and properties of solutions for the linearized state equation (1) with measures as a right-hand side and for the conjugate equation.
Reviewer: U.Raitums (Riga)

49J20 Existence theories for optimal control problems involving partial differential equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
49K20 Optimality conditions for problems involving partial differential equations
Full Text: DOI