Optimal control problem for the modified Swift–Hohenberg equation in 3D case

Abstract

Communicated by P. SacksIn this paper, for 3D modified Swift-Hohenberg equation, the optimal control problem is considered, the existence of optimal solution is proved, and the optimality system is established.

Statistical solutions for a nonautonomous modified Swift–Hohenberg equation

“…Later, it has also played a valuable role extensively in the study of plasma confinement in toroidal devices, 5 viscous film flow, lasers, 6 and pattern formation. 7 In the previous work, most attention was paid to the existence of attractors (global attractor, 8,9 uniform attractor, 10 pullback attractor, 11,12 and random attractor [12][13][14] ), bifurcations (dynamical bifurcations 15,16 and nontrivial-solution bifurcations 17 ), and optimal control [18][19][20][21] of different types of modified Swift-Hohenberg equations. Wang et al presented in their work 22 a lower number of recurrent solutions for the nonautonomous case by topological methods (see more in other works [23][24][25][26] ).…”

Recurrent Solutions of a Nonautonomous Modified Swift-Hohenberg Equation

“…In the previous work, most attention was paid to the existence of attractors (global attractor [16,23], pullback attractor [15,27], uniform attractor [29] and random attractor [9,27]), bifurcations (dynamical bifurcations [5,6], nontrivial-solution bifurcations [28]) and optimal control ( [8,24,30,31]) of different types of modified Swift-Hohenberg equations. Xiao and Gao in [28] gave specific nontrivial bifurcation solutions that bifurcate from the trivial solution for the modified Swift-Hohenberg equations in rectangular domain in R 2 with periodic boundary value.…”