The uniform controllability property of semidiscrete approximations for the parabolic distributed parameter systems in Banach spaces

TL;DR

This paper extends the uniform controllability results for semidiscrete approximations of parabolic systems to L^q norms with q > 2, even when the control operator's unboundedness exceeds previous limits, and provides a minimization method with an application to the heat equation.

Contribution

It demonstrates that uniform controllability holds in L^q norms for q > 2 under broader conditions and introduces a minimization procedure for computing approximate controls.

Findings

Uniform controllability in L^q for q > 2 is established.

The control operator's unboundedness condition is relaxed beyond 1/2.

A practical example with the 1D heat equation demonstrates the approach.

Abstract

The problem we consider in this work is to minimize the L^q-norm (q > 2) of the semidiscrete controls. As shown in [LT06], under the main approximation assumptions that the discretized semigroup is uniformly analytic and that the degree of unboundedness of control operator is lower than 1/2, the uniform controllability property of semidiscrete approximations for the parabolic systems is achieved in L^2. In the present paper, we show that the uniform controllability property still continue to be asserted in L^q. (q > 2) even with the con- dition that the degree of unboundedness of control operator is greater than 1/2. Moreover, the minimization procedure to compute the ap- proximation controls is provided. An example of application is imple- mented for the one dimensional heat equation with Dirichlet boundary control.