Generalized solvability and optimization of a parabolic system with a discontinuous solution

We consider the following linear parabolic system in a domain with a thin low-permeable insertion (“imperfect interface”):∂u∂t+q(ξ)u+∇⋅ω→=f(t,ξ),ω→=−K∇u,(t,ξ)∈Q1∪Q2⊂Rn,u|t=0=0,u|ξ∈∂Ω=0,K={kij(ξ)}i,j=1n,[(ω→,n→)Rn]=0,α[u]+limξ→ξ0(ω→,n→)Rn=0,(t,ξ0)∈Q3=Q¯1∩Q¯2. We consider a new formulation of the problem where the unknowns are (u,ω→), and the parabolic problem is converted to a first-order system of partial differential equations with distributional coefficients. We also prove inequalities for negative norms for the parabolic operator with the distributional coefficients and theorems of existence and uniqueness. For optimization problems for the processes we show existence of optimal controls, investigate smoothness of a performance criterion and give a simple condition for controllability of the system. In addition, we consider applications of the obtained results to a pulse control problem and prove convergence of a control mapping regularization procedure.