2024/12/29 by Alberto Domínguez Corella, Corella, Alberto Domínguez, Nicolai Jork +3
Earth and Planetary Sciences · Mathematics · #35K58 #49K40 #49L99 #Aquatic and Environmental Studies #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2412.20310
openalex publication_date 2024/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Along the optimal trajectory of an optimal control problem constrained by a semilinear parabolic partial differential equation, we prove the differentiability of the value function with respect to the initial condition and, under additional assumptions on the solution of the state equation, the differentiability of the value function with respect to the time variable. In our proof, we rely on local growth assumptions commonly associated with the study of second-order sufficient conditions. These assumptions are generally applicable to a wide range of problems, including, for instance, certain tracking-type problems. Finally, we discuss the differentiability of the value function in a neighborhood of the optimal trajectory when a growth condition for optimal controls is used.