Bilinear minimax control problems for a class of parabolic systems with applications to control of nuclear reactors
Consider the following parabolic equation \[ u_t-F(x)u = H(x)u-(f+g)B(x)u \;in \;Q=\Omega \times (0,T), \] subjected to the boundary condition \[ Lu=0 \;in \;\Sigma = \partial \Omega \times (0,T), \] and the initial condition \[ u(0)=u_0 \;in \;\Omega, \] under the pointwise constraints \[ a\leq f(x,t)\leq b, \;c\leq g(x,t)\leq d \;a.e. \;in \;Q. \] Here \(u\) is an \(n\)-dimensional state vector, \(\Omega\) is an open subset of \(\mathbb{R}^N\), \(N\leq 3\), with a \(C^{\infty}\) boundary \(\partial \Omega\), \(0\in [a,b]\cap [c,d]\), and \(B(x)\), \(H(x)\) are matrices with \(L^{\infty}\) entries. In applications \(F\) is an elliptic operator, and the boundary operator \(L\) is of Robin type. Let us define the set of admissible controls, \(U_{ad} = \{f\in L^2(Q); \;a\leq f \leq b \;a.e. \;in \;Q\}\), as well the set of admissible disturbances or noises, \(W_{ad}=\{g\in L^2(Q); \;c\leq g \leq d \;a.e. \;in \;Q\}\). We introduce the cost functional \[ J(f,g) = (1/2){\int}_QK(u-u_{obs})\cdot (u-u_{obs})\, dx\,dt +{\alpha /2}{\| f\| }^2_{L^2(Q)} - {\beta /2}{\| g\| }^2_{L^2(Q)}, \] where \(K\) is symmetric, positive-definite matrix with smooth entries. The paper deals with the following minimax control problem: Find \(f^*\in U_{ad}\), \(g^*\in W_{ad}\) such that \((f^*,g^*)\) is a saddle point of \(J(f,g)\) subject to the above system and constraints. The existence of an optimal solution is proved, and the appropriate necessary conditions for saddle point optimality are derived. Two examples associated with nuclear fission are presented.
scientific article; zbMATH DE number 5083301
scientific article