Optimal Control for Fractional Diffusion Equations with Incomplete Data

Abstract

We are concerned with the optimal control of time-fractional diffusion equations with missing boundary condition. Using the notion of no-regret control and least (or low) regret control developed by Lions, we first prove that the least regret control problem associated with the boundary fractional diffusion equation has a unique solution. Then we show that this solution converges to the no-regret control which we characterize by a singular optimality system.

Averaged No-Regret Control for an Electromagnetic Wave Equation Depending upon a Parameter with Incomplete Initial Conditions

“…In this study, we consider an optimal control problem for electromagnetic wave equation depending upon a parameter and with missing initial conditions. We use the method of no-regret control which was introduced firstly in statistics by Savage [2] and later by Lions [3,4] where he used this concept in optimal control theory, and its related idea is "low-regret" control to apply it to control distributed systems of incomplete data which has the attention of many scholars [5][6][7][8][9][10][11][12], motivated by various applications in ecology, and economics as well [13]. Also, we use the notion of average control because our system depends upon a parameter, Zuazua was the first who introduced this new concept in [14].…”

Hierarchic control of a linear heat equation with missing data

“…Since then, many authors have used this notion to control models with incomplete data. Mophou used this notion to study a time fractional diffusion equation with missing boundary condition. For more literature on control of models with incomplete data, we refer, for instance, to previous studies and the reference therein.…”

Optimal control of averaged state of a parabolic equation with missing boundary condition

“…This notion was then applied to control some model with incomplete data, including model involving fractional derivative in time. We refer for instance to [4,10,11,13,12,14,15]. The averaged control notion was introduced by E. Zuazua [16] to analyse the problem of controlling parameter dependent systems.…”