Optimization of systems with a moving control (Q2739937)

The paper deals with the point moving control problem for a system described by the linear differential equation \(Lu=\sum_{i=1}^{l} \sum_{j=1}^{n}\delta(\overline x-\overline a_{i}(t))c_{ij}(t)\psi_{j} (\overline y)\), where \(L(\cdot)\) operates from \(D(L)\subset L_2(Q)\) into \(R(L)\subset W^{-s}\); \(Q=[0,T]\times\Omega\), \(\Omega=\Omega_{x}\times \Omega_{y}\), \(\Omega_{x}\subset R^{N_1}\), \(\Omega_{y}\subset R^{N_2}\); \(\overline x\in \Omega_{x}, \overline y\in \Omega_{y}\), \(\overline a_{i}(t)\in (W_2^1(0,T))^{N_1}\), \(c_{ij}(t)\in L_2(0,T)\), \(\psi_{j}(\overline y)\in L_2(\Omega_{y})\); \(W_2^1\) is the Sobolev space; \(h=(a(t),c(t))\) is a control. It is necessary to minimize the quality criterion \(J(u(h))=\sum_{i=1}^{k} \int_{Q} (u-u_{\partial}^{i})dQ\), where \(u_{\partial}^{i}\) is a given element from \(L_2(Q)\). The authors prove that under some conditions there exists an optimal control for the considered system.