Infinite Horizon Control Problems for Semilinear Parabolic Equations with Pointwise State Constraints

Abstract We study an optimal control problem for semilinear parabolic equations with infinite horizon, pointwise state contraints and two different types of control constraints (pointwise in space and time, and pointwise in time and $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> in space). First we prove first-order necessary conditions, then we provide a second-order sufficient condition for local optimality. The second-order condition is formulated using an extended cone that considers the infinite horizon and the control and state constraints. This condition is sufficient for strict local optimality in the $$L^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -sense. Finally, we address the approximation of the infinite horizon control problem by finite horizon problems. We analyze the convergence of these approximations and provide error estimates.