Optimal actuator location of minimum norm controls for heat equation with general controlled domain

Abstract

In this paper, we study optimal actuator location of the minimum norm controls for a multidimensional heat equation with control defined in the space L p (0, T ; L 2 (Ω)). The actuator domain ω is quite general in the sense that it is required only to have a prescribed Lebesgue measure. A relaxation problem is formulated and is transformed into a two-person zero-sum game problem. By the game theory, we develop a necessary and sufficient condition and the existence of relaxed optimal actuator location for p ∈ [2, +∞], which is characterized by the Nash equilibrium of the associated game problem. An interesting case is for the case of p = 2, for which it is shown that the classical optimal actuator location can be obtained from the relaxed optimal actuator location without additional condition. Finally for p = 2, a sufficient and necessary condition for classical optimal actuator location is presented.

Time optimal sampled-data controls for the heat equation

“…We refer to [15,41] for semi-discrete finite element approximations, and [34,43] for perturbations of equations. About more works on time optimal control problems, we would like to mention [2,10,11,16,17,18,21,22,25,27,30,31,35,37,38,39,40,42,44] and the references therein.…”

On a kind of time optimal control problem of the heat equation

“…For some other interesting work, we refer the reader to [22][23][24]. The approximate controllability of system (1.1) has been studied in much work (see, e.g., [14,[25][26][27][28]). It is clear that, for each ε > 0, we have y(T; y 0 , 0) ≤ ε when T is large enough.…”

Optimal actuator location of the minimum norm controls for stochastic heat equations

“…This generalization facilitates the study of the optimal actuator location problem for a wider class of equations. For example, authors in [8] investigate the optimal actuator location of the minimum norm controls for deterministic heat equations in arbitrary dimensions, while [1] studied the one dimensional case and [9] considered a special class of controlled domains. For other actuator location problems, see [5,17], and related numerical research [14,15,18].…”