On optimal control problem for moving sources for a parabolic equation (Q1743027)

Consider a control system described by a nonhomogeneous heat equation \(u_t = a^2 u_{xx} + h(x,t)\) for the state function \(u=u(x,t)\), \(0 \leq x \leq l, 0 \leq t \leq T\). With control functions \(p(t) = (p_1(t),\dots, p_n(t))\), \(s(t)=(s_1(t),\dots, s_n(t))\), the additive term h(x,t) in the state equation reads \(h(x,t)= \sum_{1 \leq k \leq n} p_k(t)\delta(x-s_k(t))\), where \(\delta\) denotes the Dirac \(\delta-\)function. Given boundary and initial conditions for the state function \(u(x,t)\) and box constraints for the controls \(p(t)\), \(s(t)\), the problem is to minimize a quadratic functional in \(u(x,T)\), \(p(t)\), \(s(t)\) subject to the given state equation and the mentioned conditions for \(u(\cdot,\cdot)\), \(p(\cdot)\), \(s(\cdot)\). The existence and uniqueness of a solution of the control problem and necessary optimality conditions are considered. Furthermore, a numerical example is given.