Exact Cost-Increment Formula for Optimal Control of Semilinear Evolution Equations

Abstract

We address optimal control of semilinear evolution equations on Banach spaces with finitely many control channels, a framework encompassing many infinite-dimensional dynamical systems arising in applications. For this setting, we derive an exact and global formula quantifying the increment of the cost functional with respect to an arbitrary reference control. This identity enables the design of monotone descent algorithms that require no linearization or step-size tuning. We establish the existence of optimal controls and propose a practical sample-and-hold realization of the descent step. The effectiveness of the method is demonstrated on a controlled reaction–diffusion equation.