Monotone operator method in problems of control of distributed systems of elliptic type with discontinuous nonlinearity (Q1901962)

Let \(Y\) and \(Y_1\) be real Banach spaces with \(Y_1\subset Y\). Suppose the natural imbedding of \(Y_1\) into \(Y\) is continuous. Let \(U\) be a Banach space. Let \(L: D(L)\subset Y_1\to Y^*\) be a linear operator, let \(T: Y_1\to Y^*\) be a locally bounded (may be discontinuous) operator and let \(B: U\to Y^*\) be a linear bounded operator. Consider the controlled system described by the equation \(Lw+ Tw= Bv\). Denote by \(U_{ad}\subset U\) the set of all admissible controls. A ``control-state'' pair \((v, w)\) is said to be admissible for the system if \(v\in U_{ad}\) and \(w\) is a solution of the equation for \(v\). Denote by \(D\) set of all admissible ``control-state'' pairs for the system. Define a cost function \(J(v, w)= |w- w_0|^1_Z+ A|v|^\mu_U\), where \(Z\) is a Banach space containing \(D(L)\), \(w_0\in Z\) and 1, \(\mu\) and \(A\) are positive constants. Under the assumption that \(T\) is monotone and with some further assumptions for the spaces, the author shows that there is a pair \((u, z)\in D\) such that \(J(u, z)= \inf_D J(v, w)\). Applications to the problems of control of distributed systems described by the equation of elliptic type with Dirichlet boundary value condition are illustrated.