Constrained bilinear optimal control of reactive evolution equations

We consider constrained bilinear optimal control of second-order linear evolution partial differential equations (PDEs) with a reaction term on the half line, where control arises as a time-dependent reaction coefficient and constraints are imposed on the state and control variables. These PDEs represent a wide range of physical phenomena in fluid flow, heat, and mass transfer. Existing computational methods for this type of control problems only consider constraints on the control variable and lack global convergence guarantee. In this paper, we propose a novel optimize-then-discretize framework for computing constrained bilinear optimal control with both state and control constraints. Unlike existing methods that derive optimality conditions directly from the PDE constraint, this framework first replaces the PDE constraint with an equivalent integral representation of the PDE solution and then derives optimality conditions for the reformulated problem. The integral representation, derived from the unified transform method, does not involve differential operators, and thus explicit expressions for necessary conditions of optimality can be derived using the Karush-Kuhn-Tucker conditions for infinite-dimensional optimization. Discretizing the optimality conditions results in a system of finite-dimensional smooth nonlinear equations, which can be efficiently solved using existing algorithms with guaranteed global convergence at a quadratic rate. This is in contrast with discretize-then-optimize methods that discretize the PDE first and then solve the optimality conditions of the approximated finite-dimensional problem. Computational results for two applications, namely nuclear reactivity control and water quality treatment in a reactor, are presented to illustrate the effectiveness of the proposed framework.