The uniform controllability property of semidiscrete approximations for the parabolic distributed parameter systems in Banach spaces
For researchers in control theory, this work generalizes existing results to broader function spaces and weaker conditions, though it is an incremental extension of prior work.
This paper extends the uniform controllability property of semidiscrete approximations for parabolic systems from L^2 to L^q (q>2) spaces, even when the control operator's unboundedness exceeds 1/2, and provides a minimization procedure for computing approximate controls.
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.