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

Abstract

We consider the optimal control of general heat governed by an operator depend on an unknown parameter and with missing boundary condition. Using the notion of no-regret and low-regret control we prove that we can bring the average of the state of our model to a desired state. Then by means of Euler-Lagrange first order optimality condition, we expressed the optimal control in term of average of an appropriate adjoint state that we characterize by an optimality system. The main tools are the Lebesgue dominated convergence theorem and an appropriate Hilbert space endowed with a norm containing the average of the state.

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].…”

Optimal Control of Averaged State of a Population Dynamics Model

“…In this notion, the aim is to find a control, independent of the unknown parameter, so that the average of the state is controlled. For more literature on the topic, we refer for instance to Lohac and Zuazua [6], Lazar and Zuazua [5], Hafdallah and Ayadi [2] and LU and Zuazua [7], G. Mophou et al [8] and the reference therein. In this paper, we are concerned with the control of a parameter dependent age structured population dynamics system.…”

Hierarchic control of a linear heat equation with missing data

“…It can also be obtained if possible (since it is not always the case) as a limit of the low‐regret control. These notions of low‐regret and no‐regret controls were combined to the notion of average control to address a problem with missing boundary condition for variable parameter in the coefficient of diffusion. We also refer to Hafdallah and Ayadi, where such a control has been considered for electromagnetic wave displacement with an unknown velocity of propagation.…”