Low-regret control for a fractional wave equation with incomplete data

Abstract

We investigate in this manuscript an optimal control problem for a fractional wave equation involving the fractional Riemann-Liouville derivative and with missing initial condition. For this purpose, we use the concept of no-regret and low-regret controls. Assuming that the missing datum belongs to a certain space we show the existence and the uniqueness of the low-regret control. Besides, its convergence to the no-regret control is discussed together with the optimality system describing the no-regret control.

Optimal Control of Diffusion Equation with Fractional Time Derivative with Nonlocal and Nonsingular Mittag-Leffler Kernel

“…Therefore we have that y m ⇀ y in L 2 (0, T ); H 1 0 (Ω) , y m ⇀ y in C [0, T ]; L 2 (Ω) . Now proceeding as in [23], one can prove by interpretation that the solution y is equivalent to the initial problem (1.1). Furthermore, setting Λ 1 = max (C 1 (α, λ 1 , T ), C 2 (α, λ 1 , T )) and Λ 2 = max (C 3 (α, λ 1 , T ), C 4 (α, λ 1 , T )), an using the estimates obtained at the beginning of the proof, we have that y L 2 ((0,T );H 1 0 (Ω)) = ˆT 0 y H 1 0 (Ω) dt…”

“…Remark. We would like to draw the attention of the reader that in the case for the Caputo fractionaltime derivative, the existence and uniqueness of solutions for the fractional diffusion equation of type (1.1), were obtained under the conditions that α ∈ (1/2, 1) [16,17,23]. But here in our case using the Atangana-Baleanu fractional-time derivative, we got the existence and uniqueness of solutions for the fractional diffusion equation of type (1.1) for all α ∈ (0, 1).…”

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.…”

Controllability of Fractional Differential Equation of Order α∈(1,2] With Non‐Instantaneous Impulses

“…Baleanu et al . investigated an optimal control problem governed by a fractional wave equation with missing initial condition using the concept of no‐regret and low‐regret controls. Recently, Shukla et al .…”

“…For more details on the fractional order initial value problem, see [10,11]. Baleanu et al [12] investigated an optimal control problem governed by a fractional wave equation with missing initial condition using the concept of no-regret and low-regret controls. Recently, Shukla et al [13] studied the approximate controllability of semilinear fractional control systems of order ∈ (1, 2] using the strongly continuous -order cosine family.…”