Reduced basis method and error estimation for parametrized optimal control problems with control constraints (Q421339)

The author formulates the reduced basis (RB) method for the solution of parametrized optimal control problems with control constraints (\(\mu\)-cOCPs) in the linear-quadratic case, where the primal equation is parabolic linear time invariant (LTI) and the cost functional is quadratic in both the control and primal variables. Here the RB approximation of \(\mu\)-cOCP on the basis of the \(<\text{truth}>\) finite element approximation one is formulated, and the author provides an RB error estimate for \(\mu\)-cOCP and proposes a heuristic RB error indicator which is used at the offline step (the offline-online decomposition based on the affine decomposition hypothesis) for the choice of the RB space and at the online step to evaluate the error associated with the RB optimal solution of the \(\mu\)-cOCP for any parameter \(\mu\). The parametrized optimal control problem in the control constrained case (\(\mu\)-cOCP) reads: Find \(u(t, \mu) \in U_{ad}\), \(u(t, \mu)=\arg \min (J(v(t, \mu), u (t, \mu), \mu))\) for any given \(\mu \in D\), where \(u(t, \mu) \in U_{ad}\) is the control variable, \(J(\cdot, \cdot, \mu)\) is the cost functional, while the primal variable \(v (t, \mu) \in \nu\) is a solution of the well-posed parabolic linear time invariant \(\mu\)-partial differential equation (the primal equation): \(m(\partial v (t, \mu)/\partial t, \phi, \mu) + a (v(t, \mu) \phi, \mu) = b (\phi, \mu) u(t, \mu) \quad\forall \phi \in Z,~ t\in (0,T)\) with \(v(0, \mu) = v_0(\mu)\). Main result: An error estimate is derived and an heuristic indicator is proposed to evaluate the RB error on the optimal control problem at the online step. The indicator is used at the offline step in a greedy algorithm to build the RB space. Finally in this paper numerical tests in the two-dimensional case with applications to heat conduction and environmental optimal control problems are presented.