Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations

Abstract We analyze the numerical approximation of a control problem governed by non-monotone and non-coercive semilinear elliptic equation. The lack monotonicity coercivity is due to presence convection term. First, we study finite element partial differential While can prove existence solution for discrete equation when discretization parameter small enough, uniqueness an open us if nonlinearity not globally Lipschitz. Nevertheless, sequence solutions bounded in $$L^\infty (\varOmega )$$ L ∞ ( Ω ) converging continuous problem. Error estimates these are obtained. Next, discretize Existence optimal controls proved, as well their convergence analysis error quite involved possible non-uniqueness state given control. To overcome this difficulty define appropriate control-to-state mapping neighbourhood strict This allows introduce reduced functional obtain first order optimality conditions estimates. Some experiments included illustrate theoretical results.