Numerical Approximation of Controllability of Trajectories for Euler–bernoulli Thermoelastic Plates

Abstract

Euler–Bernoulli thermoelastic plate model with a control function in the thermal equation is considered. This paper is devoted to the analysis and construction of the minimization procedure related to the controllability of its trajectories by applying both penalty and duality arguments. Numerical approximation of the optimality system is carried out through the use of spectral element methods in space and finite difference schemes in time. Numerical results obtained on several test cases are shown.

Null controllability of von Kármán thermoelastic plates under the clamped or free mechanical boundary conditions

“…In addition, there are the papers [8,14] which deal with the null controllability of the thermoelastic variables by means of locally distributed control; in this case of locally distributed control, the observability constants will necessarily obey an exponential rate of blowup, vis-à-vis the rational rates of blowup which are seen in [4,5] and in the present paper (see Lemma 5 below). In general, the problem of finding the sharp observability inequality relative to null controllability, or what is the same, the precise rate of singularity for minimal norm null controllers, is a classical one in control theory.…”

“…In fact, the estimates previously derived in [4] and [5] for the "energy" E(T ) of the adjoint system (6) will be used in this paper to help generate the observability estimate which is associated with the affine thermoelastic problem (6) (i.e., the thermoelastic system (1) with nonlinear term [F(ω), ω] replaced by forcing term f (t)). In addition to the said reverse estimates for E(T ) (posted below in (14) for the clamped case, and in (15) for the free), another necessary ingredient for the proof of the null controllability of the affine problem is the appropriate use of the underlying analyticity for the associated thermoelastic C 0 -semigroup {e At } t 0 . In particular, we will have need to use the following regularity result which is a consequence of said analyticity:…”

“…In this connection, our intent is to show that the integral term in (13) retains the observability constant posted in (14) (for the clamped case) or in (15) (for the free case). The proof of Lemma 5 will be undertaken in the next two subsections.…”

“…Using the observability estimate (14), we then obtain the following pointwise estimate (in norms below that of finite energy):…”

Control problems for Euler-Bernoulli thermoelastic plates without memory and thermoelastic systems with memory

“…In this note we are interest to present two classes of problems: thermoelastic systems of memory-type dealing with 'hyperbolic-like' dynamics, and Euler-Bernoulli thermoelastic plates without memory, where the described model is non-hyperbolic and associated with analyticity of the underlying generator. For these systems we resume our results obtained in the controllability context [8,11,22,23]. We start to recall some definitions about this subject.…”