In this note, we propose a robust recursive least squares approach for the adaptive estimation of parameters in dynamic regressor systems. The approach provides linear dynamics of the covariance matrix. It is shown that, by choosing the tuning gains of the parameter update and the covariance updates, exponential convergence of the parameters is achieved while the covariance matrix vanishes to the origin at a slower exponential rate. The analysis also demonstrates that exponential convergence of the parameter estimates is preserved in the presence of bounded measurement error. A simulation study is presented to highlight the properties of the proposed technique.

By utilizing the linear conjugacy relationship, the stability of a well-characterized system can be transferred to an unknown one. A key application lies in extending the stability of complex balanced (CB) systems to general systems linearly conjugate to them. However, introducing time delays to certain CB systems and their linearly conjugate counterparts may disrupt this conjugacy relationship, thereby hindering the transfer of stability properties. In this paper, we first establish several sufficient conditions to ensure the Lyapunov stability of the delayed version of systems linearly conjugated to CB systems. To further address the degenerated equilibrium in the stoichiometric compatibility class in such systems, we redefine the invariant sets of trajectories and extend the Lyapunov stability to achieve the local asymptotic stability with respect to the newly defined invariant sets. Illustrative examples, such as the PAK-1 network, are provided to validate the theoretical findings.

In the field of molecular computations, the main challenges is to the consideration of parallelism to enable the molecular computations for composite functions following the layer-by-layer computation rule. This paper proposes to consider the notion of dynamic composability, based on which two (or more) chemical reaction networks (CRNs) can still output the layer-by-layer computation result for the desired composite function even though they are put together and all reactions occur simultaneously. We further leverage the notion of input-to-state stability and some conditions on the network structure to derive several sufficient conditions that suggest dynamic composability. Some examples are presented to demonstrate the validity of our theoretical results. Finally, for the case of birth–death processes, an algorithm is presented for generating the composite CRN for computing the root of a polynomial equation.

In this paper, we are interested in the dynamical analysis and boundary optimal control of counterflow heat exchanger in the case where the dynamics is described by hyperbolic partial differential equations. This topic is addressed by describing the dynamical model of the heat exchanger in an infinite-dimensional state-space, with bounded control and observation operators. First, we review the well-posedness problem and some fundamental properties relating to control theory, such as positivity, stability, reachability, stabilization and observability. These properties are complemented by spectral and pseudospectral analyses. Next, in the view of some results relating to the linear quadratic-optimal control of hyperbolic systems to which the model considered in this paper belongs, we introduce a certain state transformation that allows to put the abstract system in the lower triangular form so as to guarantee the uniqueness of solution of the operator Riccati equation. Finally, the design of an observer-based optimal control law coupled with an integral action is considered. The results are illustrated by means of numerical simulations for the set point tracking, and show the interest of the control approach proposed in this paper.

Embedding efficient calculation instructions into biochemical system has always been a research focus in synthetic biology. One of the key problems is how to sequence the chemical reaction modules that act as units of computation and make them alternate spontaneously. Our work takes the design of chemical clock signals as a solution and presents a 4-dimensional chemical oscillator model based on relaxation oscillation to generate a pair of symmetric clock signals for two-module regulation. We give detailed dynamical analysis of the model and discuss how to control the period and occurrence order of clock signals. We also demonstrate the loop control of molecular computations and provide termination strategy for them. We can expect that our design for module regulation and loop termination will help advance the embedding of more complicate calculations into biochemical environments.

In biological reaction systems, reaction rates may vary over time due to environmental fluctuations, regulation, or coupling with other reaction modules. Input-to-state stability (ISS) provides a useful tool for analyzing the robustness of time-varying chemical reaction networks (CRNs). Existing ISS results for CRNs typically rely on restrictive structural assumptions, such as zero deficiency, a single linkage class, or weak reversibility. This paper makes two main contributions. First, we establish ISS for a broader class of weakly reversible CRNs, allowing nonzero deficiency and multiple linkage classes. Second, we extend the analysis to certain CRNs that are not weakly reversible by using network transformation techniques (linear conjugacy and reconstruction). Together, these results enlarge the class of CRNs for which robustness under time-varying reaction-rate inputs can be certified. Since CRNs are a standard framework for biomolecular computation, our results further enable the stability analysis of parallel molecular computing systems, an important problem in biomolecular computation where multiple CRN-based computing modules operate simultaneously and perturb one another through time-varying effective reaction rates.

Embedding sequential computations in biochemical environments is challenging because the computations are carried out through 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 mass-action kinetics and illustrate our design scheme under existing synthetic biochemical oscillator models. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Complex balanced mass-action systems (CBMASs) are of great importance in the filed of biochemical reaction networks. However analyzing the persistence of these networks with high dimensions and time delays poses significant challenges. To tackle this, we propose a novel approach that combines 1-dimensional (1d) or 2d delayed CBMASs (DeCBMASs) and introduces inheritable combination methods based on the relationship between semilocking sets and intersecting species. These methods account for various scenarios, including cases where the set of intersecting species is empty, or there are no common species in non-trivial semilocking sets and the intersecting species set, or when special forms are present. By utilizing these combination methods, we derive sufficient conditions for the persistence of high-dimensional DeCBMASs. This significantly expands the known class of delayed chemical reaction network systems that exhibit persistence. The effectiveness of our proposed approach is also demonstrated through several examples, highlighting its practical applicability in real-world scenarios. This research contributes to advancing the understanding of high-dimensional DeCBMASs and offers insights into their persistent behavior.

This paper proposes a non-adaptive control solution framework to the practical output regulation problem (PORP) for a class of nonlinear systems with uncertain parameters, unknown control directions and uncertain exosystem dynamics. The concurrence of the unknown control directions and uncertainties in both the system dynamics and the exosystem pose a significant challenge to the problem. In light of a nonlinear internal model approach, we first convert the robust PORP into a robust non-adaptive stabilization problem for the augmented system with integral Input-to-State Stable (iISS) inverse dynamics. By employing an extremum-seeking control (ESC) approach, the construction of our solution method avoids the use of Nussbaum-type gain techniques to address the robust PORP subject to unknown control directions with time-varying coefficients. The stability of the non-adaptive output regulation design is proven via a Lie bracket averaging technique where uniform ultimate boundedness of the closed-loop signals is guaranteed. As a result, both the estimation and tracking errors converge to zero exponentially, provided that the frequency of the dither signal goes to infinity. Finally, a simulation example with unknown coefficients is provided to exemplify the validity of the proposed control solution frameworks.

Molecular computation based on chemical reaction networks (CRNs) has emerged as a promising paradigm for designing programmable biochemical systems. However, the implementation of complex computations still requires excessively large and intricate network structures, largely due to the limited understanding of composability, that is, how multiple subsystems can be coupled while preserving computational functionality. Existing composability frameworks primarily focus on rate-independent CRNs, whose computational capabilities are severely restricted. This article aims to establish a systematic framework for composable CRNs governed by mass-action kinetics, a common type of rate-dependent CRNs. Drawing upon the concepts of composable rate-independent CRNs, we introduce the notions of mass-action chemical reaction computers (msCRCs), dynamic computation and dynamic composability to establish a rigorous mathematical framework for composing two or more msCRCs to achieve layer-by-layer computation of composite functions. Further, we derive several sufficient conditions based on the notions of input-to-state stability (ISS) to characterize msCRCs that can be composed to implement desired molecular computations, thereby providing theoretical support for this framework. Some examples are presented to illustrate the efficiency of our method. Finally, comparative results demonstrate that the proposed method exhibits notable advantages in both computational ability and accuracy over the state-of-the-art methods.

We design a predictive flow rate and concentration controller for wastewater transport and treatment networks. It manages flow rates to avoid overflows during times of high flow, and maximizes treatment efficiency when the system is within capacity limits. The underlying optimization is nonlinear due to the microbial growth kinetics and bilinear mass flows. Using a second-order cone relaxation of the microbial growth constraints and the alternating direction method of multipliers, we break down the problem into second-order cone and quadratic programs. This allows us to solve the problem at large scales in real-time. In a case study based on the wastewater transport and treatment system in the City of Paris, our controller outperforms the conventional flowrate-based controller by removing 13.7% more pollutant mass while treating the same amount of wastewater.

Information processing relying on biochemical interactions in the cellular environment is essential for biological organisms. The implementation of molecular computational systems holds significant interest and potential in the fields of synthetic biology and molecular computation. This two-part article aims to introduce a programmable biochemical reaction network (BCRN) system endowed with mass action kinetics that realizes the fully connected neural network (FCNN) and has the potential to act automatically in vivo. In part I, the feedforward propagation computation, the backpropagation component, and all bridging processes of FCNN are ingeniously designed as specific BCRN modules based on their dynamics. This approach addresses a design gap in the biochemical assignment module and judgment termination module and provides a novel precise and robust realization of bi-molecular reactions for the learning process. Through equilibrium approaching, we demonstrate that the designed BCRN system achieves FCNN functionality with exponential convergence to target computational results, thereby enhancing the theoretical support for such work. Finally, the performance of this construction is further evaluated on two typical logic classification problems.

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 prescribed-time 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 illustrate the effectiveness of the proposed technique.

In the field of molecular computation based on chemical reaction networks (CRNs), leveraging parallelism to enable coupled mass-action systems (MASs) to retain predefined computational functionality has been a research focus. MASs exhibiting this property are termed composable. This paper investigates the structural conditions under which two MASs are composable. By leveraging input-to-state stability (ISS) property, we identify a specific class of CRN architectures that guarantee composability with other networks. A concrete example demonstrates the validity of this conclusion and illustrates the application of composability in computing composite functions.

Neural network related machine learning algorithms, inspired by biological neuron interaction mechanisms, are advancing rapidly in the field of computing. This development may be leveraged in reverse to advance synthetic biology progress. A challenging exploration is to implement neural network functionalities through biochemical reaction networks (BCRNs), a language that is inherently compatible with in vivo, with difficulties specifically in constructing an appropriate BCRN that respects computation and information processing steps involved in neural networks. By addressing these difficulties, this paired article manages to develop a biochemical fully connected neural network (BFCNN) that has the potential to behave like a neural network independently in vivo. In Part I, BFCNN is designed according to five different modules with concrete theoretical support on the stability of every module. In this article, we establish a systematic framework for error analysis of the designed BFCNN. We introduce current error evaluating error generation within an individual computational module by measuring the deviation between finite-time concentration and steady-state concentration, and the accumulative error characterizing error propagation across different modules connected via oscillatory signals and within the repeated modules due to multiple iterations. We further derive the formula for the total error upper bound about the iteration number and illustrate its exponential convergence order about non-zero duration time of an oscillatory signal concentration. Ultimately, the numerical experiments on two classification examples are made to exhibit the change tendency of total error bounds about the non-zero duration time and iteration number.

The dynamics of irreversible thermodynamic systems have been expressed in terms of conservative contact systems where contact vector fields are generated by contact Hamiltonian functions defined on the Thermodynamic Phase Space (TPS). In this paper, we first emphasize the importance of both the Gibbs relation and the Gibbs-Duhem relation of the entropy or energy contact form in developing a first-order invariance constraint that every contact Hamiltonian function must satisfy. This novel insight is then considered together with the zero-order invariance constraint to infer solutions, thereby yielding a generalized family of contact Hamiltonian functions generating non-strict or strict contact vector fields which are equal on the associated Legendre submanifold on which the dynamics of the thermodynamic system is living. Finally, we show sufficient conditions under which the inverse images of zero by the contact Hamiltonian functions or the Legendre submanifold are globally attractive when lifting the system dynamics to the complete TPS. A simulated example is given to support the theoretical developments and to discuss the difference of the dynamic behaviours between the generated strict and non-strict contact vector fields. (c) 2025 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

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 (ℓcCBMASs). This paper addresses the challenge posed by time delay, which can disrupt the linear conjugacy relationship and complicate stability analysis for delayed versions of ℓcCBMASs (DℓcCBMAS). Firstly, we develop Lyapunov functionals tailored to some DℓcCBMASs 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 DℓcCBMASs within an updated framework. Furthermore, illustrative examples are provided to demonstrate the practical implications of our approach.

Polyhedral models of metabolic networks are computationally tractable and can predict some cellular functions. A longstanding challenge is incorporating metabolites without losing tractability. In this paper, we do so using a new second-order cone representation of the Michaelis-Menten kinetics. The resulting model consists of linear stoichiometric constraints alongside second-order cone constraints that couple the reaction fluxes to metabolite concentrations. We formulate several new problems around this model: conic flux balance analysis, which augments flux balance analysis with metabolite concentrations; dynamic conic flux balance analysis; and finding minimal cut sets of networks with both reactions and metabolites. Solving these problems yields information about both fluxes and metabolite concentrations. They are second-order cone or mixed-integer second-order cone programs, which, while not as tractable as their linear counterparts, can nonetheless be solved at practical scales using existing software.

Download This Paper Open PDF in Browser Add Paper to My Library Share: Permalink Using these links will ensure access to this page indefinitely Copy URL Copy DOI

This paper deals with the application of second order come convex optimization to the real time management of the wastewater treatment plants and sewer network of Paris and of its suburbs. It presents preliminary results applied on a simple case study composed of (simple) validated models of three wastewater treatment plants (Seine Aval (SAV), Seine Centre (SEC) and Seine Gresillons (SEG)) and their (inter)connections to the sewer network modelled by transport delays. Copyright (c) 2024 The Authors. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/)