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Linear conjugacy offers a new perspective to broaden the scope of stable biochemical reaction networks to the systems linearly conjugated to the well-established complex balanced mass action systems ($\ell$cCBMASs). This paper addresses the challenge posed by time delay, which can disrupt the linear conjugacy relationship and complicate stability analysis for delayed versions of $\ell$cCBMASs (D$\ell$cCBMAS). Firstly, we develop Lyapunov functionals tailored to some D$\ell$cCBMASs by using the persisted parameter relationships under time delays. Subsequently, we redivide the phase space as several invariant sets of trajectories and further investigate the existence and uniqueness of equilibriums in each newly defined invariant set. This enables us to determine the local asymptotic stability of some D$\ell$cCBMASs within an updated framework. Furthermore, illustrative examples are provided to demonstrate the practical implications of our approach.

We propose a dual mode extremum seeking control design technique that achieves real-time optimization of an unknown measured cost function in a prescribed time. The controller is shown to achieve semi-global practical stability of the optimal equilibrium for the state variables and the input variable for a class of nonlinear dynamical control systems with unknown dynamics. The design technique proposes a timescale transformation that enables the use of dither signals with increasing frequencies. The proposed timescale transformation is designed to avoid the singularity occurring at the prescribed time. A simulation study is performed to demonstrate the effectiveness of the proposed technique.

Embedding sequential computations in biochemical environments is challenging because the computations are carried out by chemical reactions, which are inherently disordered. In this paper we apply modular design to specific calculations through chemical reactions and provide a design scheme of biochemical oscillator models in order to generate periodical species for the order regulation of these reaction modules. We take the case of arbitrary multi-module regulation into consideration, analyze the main errors in the regulation process under \textit{mass-action kinetics} and demonstrate our design scheme under existing synthetic biochemical oscillator models.