PDE-ODE System Research Articles

Well-posedness of an evaporation model for a spherical droplet exposed to an air flow

In this paper, we address the well-posedness of an evaporation model for a spherical liquid droplet taking into account the convective impact of an air flow in the ambient gas phase. From a mathematical perspective, we are dealing with a coupled ODE–PDE system for the droplet radius, the temperature distribution, and the vapor concentration. The nonlinear coupling arises from the evaporation rate modeled by the Hertz–Knudsen equation. Under physically meaningful assumptions, we prove existence and uniqueness of a weak solution until the droplet has evaporated completely. Numerical simulations are performed to illustrate how different air flows affect the evaporation process.

Observer-Based Adaptive Control for a Coupled PDE–ODE System of a Flexible Wing

Observer-Based Adaptive Control for a Coupled PDE–ODE System of a Flexible Wing

Optimization-based control strategy for a large-scale polyhydroxyalkanoates production in a fed-batch bioreactor using a coupled PDE–ODE system

Abstract Control strategy development for fed-batch bioreactor (FBBR) plays an important role in the improvement of polyhydroxyalkanoate (PHA) production. To develop a feeding strategy for PHA production in a large-scale FBBR, an optimization-based control scheme that considers nutrient dispersion is proposed in this work. A coupled partial differential equations and ordinary differential equation model is proposed to describe the axial-dispersed nutrient and well-dispersed microbial dynamics with process constraints. An analytical model predictive control (AMPC) method that applies integrated variables of nutrients is employed to develop the real-time control system. The control objective is to regulate the PHA concentration at the updated set points by adjusting the nutrient feed rates; a process disturbance is introduced to evaluate the control robustness. Simulation experiments of a fed-batch operation are conducted to investigate the performance of the developed controller; the controlled output is designed to track the updated set points corresponding to the biomass concentration. Results of closed-loop and regulatory systems showed that the proposed control strategy could provide more productivity (33–38%) compared to the applied PI controller. The performance test demonstrates that the developed control system could apply the biomass concentration for updating set points, provide the optimal control actions that promote PHB accumulation and handle the disturbance effectively.

Multiscale modeling of collective cell migration elucidates the mechanism underlying tumor–stromal interactions in different spatiotemporal scales

Metastasis is the pathogenic spread of cancer cells from a primary tumor to a secondary site which happens at the late stages of cancer. It is caused by a variety of biological, chemical, and physical processes, such as molecular interactions, intercellular communications, and tissue-level activities. Complex interactions of cancer cells with their microenvironment components such as cancer associated fibroblasts (CAFs) and extracellular matrix (ECM) cause them to adopt an invasive phenotype that promotes tumor growth and migration. This paper presents a multiscale model for integrating a wide range of time and space interactions at the molecular, cellular, and tissue levels in a three-dimensional domain. The modeling procedure starts with presenting nonlinear dynamics of cancer cells and CAFs using ordinary differential equations based on TGFβ, CXCL12, and LIF signaling pathways. Unknown kinetic parameters in these models are estimated using hybrid unscented Kalman filter and the models are validated using experimental data. Then, the principal role of CAFs on metastasis is revealed by spatial–temporal modeling of circulating signals throughout the TME. At this stage, the model has evolved into a coupled ODE–PDE system that is capable of determining cancer cells’ status in one of the quiescent, proliferating or migratory conditions due to certain metastasis factors and ECM characteristics. At the tissue level, we consider a force-based framework to model the cancer cell proliferation and migration as the final step towards cancer cell metastasis. The ability of the multiscale model to depict cancer cells’ behavior in different levels of modeling is confirmed by comparing its outputs with the results of RT PCR and wound scratch assay techniques. Performance evaluation of the model indicates that the proposed multiscale model can pave the way for improving the efficiency of therapeutic methods in metastasis prevention.

Finite‐time estimation of second‐order linear time‐invariant systems in the presence of delayed measurement

In this article, we design a nonlinear observer for second-order linear time-invariant (LTI) systems in the observable form with measurement delay. The design guarantees the convergence of the error state to the origin within a finite-time that depends on the initial conditions. To accomplish this, we reformulate the original system into a cascade ODE–PDE system where the PDE part is a transport equation that models the effect of the delay on the output. We construct the nonlinear gains in a way that ensures the error system to be finite-time stable (FTS). To prove this, we use an invertible backstepping transformation to convert the error system into a target system which is shown to be finite-time stable using Lyapunov-based analysis and homogeneity tools. We use the inverse transformation to transfer this property to the error system.

Cardiac electro-mechanical activity in a deforming human cardiac tissue: modeling, existence-uniqueness, finite element computation and application to multiple ischemic disease.

In this study, the cardiac electro-mechanical model in a deforming domain is taken with the addition of mechanical feedback and stretch-activated channel current coupled with the ten Tusscher human ventricular cell level model that results in a coupled PDE-ODE system. The existence and uniqueness of such a coupled system in a deforming domain is proved. At first, the existence of a solution is proved in the deformed domain. The local existence of the solution is proved using the regularization and the Faedo-Galerkin technique. Then, the global existence is proved using the energy estimates in appropriate Banach spaces, Gronwall lemma, and the compactness procedure. The existence of the solution in an undeformed domain is proved using the lower semi-continuity of the norms. Uniqueness is proved using Young's inequality, Gronwall lemma, and the Cauchy-Schwartz inequality. For the application purpose, this model is applied to understand the electro-mechanical activity in ischemic cardiac tissue. It also takes care of the development of active tension, conductive, convective, and ionic feedback. The Second Piola-Kirchoff stress tensor arising in Lagrangian mapping between reference and moving frames is taken as a combination of active, passive, and volumetric components. We investigated the effect of varying strength of hyperkalemia and hypoxia, in the ischemic subregions of human cardiac tissue with local multiple ischemic subregions, on the electro-mechanical activity of healthy and ischemic zones. This system is solved numerically using the [Formula: see text] finite element method in space and the implicit-explicit Euler method in time. Discontinuities arising with the modeled multiple ischemic regions are treated to the desired order of accuracy by a simple regularization technique using the interpolating polynomials. We examined the cardiac electro-mechanical activity for several cases in multiple hyperkalemic and hypoxic human cardiac tissue. We concluded that local multiple ischemic subregions severely affect the cardiac electro-mechanical activity more, in terms of action potential (v) and mechanical parameters, intracellular calcium ion concentration [Formula: see text], active tension ([Formula: see text]), stretch ([Formula: see text]) and stretch rate ([Formula: see text]), of a healthy cell in its vicinity, compared to a single Hyperkalemic or Hypoxic subregion. The four moderate hypoxically generated ischemic subregions affect the waveform of the stretch along the fiber and the stretch rate more than a single severe ischemic subregion.

Delay-compensated event-triggered boundary control of hyperbolic PDEs for deep-sea construction

In this paper, we present delay-compensated event-triggered boundary control of heterodirectional coupled hyperbolic PDE–ODE systems, with the purpose of employing a piecewise-constant control input which makes the control law more user-friendly in deep-sea construction vessels where the actuator, i.e., the ship-mounted crane, is massive, and compensating delays that exist in transmission of sensing signals from the seabed to the vessel on the ocean surface through a set of acoustics devices. After treating the time delay of arbitrary length as a transport PDE whose boundary connects with the plant, a state observer is built to estimate the states of the overall system of heterodirectional coupled hyperbolic PDEs–ODE–transport PDE, only using the measurements at the right boundary of the last transport PDE. An observer-based output-feedback continuous-in-time controller is designed to stabilize the overall system using the backstepping method, followed by an observer-based event-triggering mechanism, which is designed based on the evolution of the overall PDE–ODE–PDE system, to determine the updating times of the actuation signal. The absence of a Zeno behavior and exponential convergence in the event-based closed-loop system are proved. In the simulation, the obtained theoretical result is verified in control of a deep-sea construction vessel to place the equipment in the target area on the seabed. Performance deterioration under extreme and unmodeled disturbances is also illustrated.

Fault diagnosis for linear heterodirectional hyperbolic ODE–PDE systems using backstepping-based trajectory planning

This paper is concerned with the fault diagnosis problem for general linear heterodirectional hyperbolic ODE–PDE systems. A systematic solution is presented for additive time-varying actuator, process and sensor faults in the presence of disturbances. The faults and disturbances are represented by the solutions of finite-dimensional ODEs, which allow to take a large class of signals into account. For disturbances, that are only bounded, a threshold for secured fault diagnosis is derived. By applying integral transformations to the system an algebraic fault detection equation to detect faults in finite time is obtained. The corresponding integral kernels result from the realization of a finite-time transition between a non-equilibrium initial state and a vanishing final state of a hyperbolic ODE–PDE system. For this new challenging problem, a systematic trajectory planning approach is presented. In particular, this problem is facilitated by mapping the kernel equations into backstepping coordinates and tracing the solution of the transition problem back to a simple trajectory planning. The fault diagnosis for a 4 × 4 heterodirectional hyperbolic system coupled with a second order ODE demonstrates the results of the paper.

Output Feedback Control of Coupled Linear Parabolic ODE–PDE–ODE Systems

This article deals with the backstepping design of observer-based compensators for parabolic ODE–PDE–ODE systems. The latter consist of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\boldsymbol {n}$</tex-math></inline-formula> coupled parabolic PDEs with distinct diffusion coefficients and spatially-varying coefficients, that are bidirectionally coupled to ODEs at both boundaries. The actuation and sensing appears through these ODEs resulting in a challenging control problem. For this setup a systematic backstepping approach is proposed, in order to determine a state feedback controller and an observer. In particular, the state feedback loop and the observer error dynamics are mapped into stable ODE–PDE–ODE cascades by making use of a sequence of transformations. With this, the design can be traced back to the solution of kernel equations already found in the literature as well as initial and boundary value problems, that can be solved numerically. Exponential stability of the closed-loop system is verified, wherein the decay rate can be directly specified in the design. The results of the article are illustrated by the output feedback control of an unstable ODE–PDE–ODE system with two coupled parabolic PDEs.

Backstepping boundary control design for a cascaded viscous Hamilton–Jacobi PDE–ODE system

Backstepping boundary control design for a cascaded viscous Hamilton–Jacobi PDE–ODE system

Boundary Output Feedback Stabilization for a Cascaded-Wave PDE-ODE System with Velocity Recirculation

This paper considers the output feedback stabilization for a cascaded-wave PDE-ODE system with velocity recirculation by boundary control. First, we choose a well-known exponentially stable system as its target system and find a backstepping transformation to design a state feedback controller for the original system. Second, we attempt to give an output feedback controller for the original system by introducing the observer. The resulting closed-loop system admits a unique solution which is proved to be exponentially stable. Finally, we give some numerical examples to prove the validity for the theoretical results.

Qualitative Analysis of a Three-Species Reaction-Diffusion Model with Modified Leslie-Gower Scheme

The qualitative analysis of a three-species reaction-diffusion model with a modified Leslie-Gower scheme under the Neumann boundary condition is obtained. The existence and the stability of the constant solutions for the ODE system and PDE system are discussed, respectively. And then, the priori estimates of positive steady states are given by the maximum principle and Harnack inequality. Moreover, the nonexistence of nonconstant positive steady states is derived by using Poincaré inequality. Finally, the existence of nonconstant positive steady states is established based on the Leray-Schauder degree theory.

Theoretical analysis for a PDE–ODE system related to a glioblastoma tumor with vasculature

In this paper we study a PDE-ODE system as a simplification of a Glioblastoma model. Mainly, we prove the existence and uniqueness of global in time classical solution using a fixed point argument. Moreover, we show some stability results of the solution depending on some conditions on the parameters.

Dynamics of epidemic spreading on connected graphs.

We propose a new model that describes the dynamics of epidemic spreading on connected graphs. Our model consists in a PDE-ODE system where at each vertex of the graph we have a standard SIR model and connections between vertices are given by heat equations on the edges supplemented with Robin like boundary conditions at the vertices modeling exchanges between incident edges and the associated vertex. We describe the main properties of the system, and also derive the final total population of infected individuals. We present a semi-implicit in time numerical scheme based on finite differences in space which preserves the main properties of the continuous model such as the uniqueness and positivity of solutions and the conservation of the total population. We also illustrate our results with a collection of numerical simulations for a selection of connected graphs.

Towards a replicator dynamics model of age structured populations

We present a new modelling framework combining replicator dynamics, the standard model of frequency dependent selection, with an age-structured population model. The new framework allows for the modelling of populations consisting of competing strategies carried by individuals who change across their life cycle. Firstly the discretization of the McKendrick von Foerster model is derived. We show that the Euler–Lotka equation is satisfied when the new model reaches a steady state (i.e. stable frequencies between the age classes). This discretization consists of unit age classes where the timescale is chosen so that only a fraction of individuals play a single game round. This implies a linear dynamics and individuals not killed during the round are moved to the next age class; linearity means that the system is equivalent to a large Bernadelli–Lewis–Leslie matrix. Then we use the methodology of multipopulation games to derive two, mutually equivalent systems of equations. The first contains equations describing the evolution of the strategy frequencies in the whole population, completed by subsystems of equations describing the evolution of the age structure for each strategy. The second contains equations describing the changes of the general population’s age structure, completed with subsystems of equations describing the selection of the strategies within each age class. We then present the obtained system of replicator dynamics in the form of the mixed ODE-PDE system which is independent of the chosen timescale, and much simpler. The obtained results are illustrated by the example of the sex ratio model which shows that when different mortalities of the sexes are assumed, the sex ratio of 0.5 is obtained but that Fisher’s mechanism, driven by the reproductive value of the different sexes, is not in equilibrium.

Adaptive stabilization for a reaction–diffusion equation with uncertain nonlinear actuator dynamics

This paper is devoted to the stabilization of a reaction–diffusion equation with uncertain nonlinear actuator dynamics described by a cascaded parabolic PDE–ODE system. Essentially different from the systems in the related literature, control input appears in the ODE subsystem rather than in the PDE one, and moreover, the actuator dynamics are nonlinear and possess uncertainties rather than linear without uncertainty. Note that the existing control schemes so far are ineffective to handle such systems, and hence the stabilization problem for these systems is still unsolved. For this, backstepping approaches in both infinite and finite-dimension are smartly combined with the adaptive dynamic compensation technique in order to solve the control problem. Specifically, by using an infinite-dimensional backstepping transformation and its inverse, the original PDE subsystem is first changed to a new one from which control design is much convenient. Then, based on the new system together with the actuator dynamics, an adaptive state-feedback controller is explicitly constructed by using a couple of finite-dimensional backstepping transformations, which guarantees that all the states of the resulting closed-loop systems are bounded. Moreover, both the control input and the states of the original system converge to zero ultimately. Finally, a simulation example is provided to validate the effectiveness of the established theoretical results.

Synchronous oscillations for a coupled cell-bulk ODE–PDE model with localized cells on $${\mathbb {R}}^2$$

In many micro- and macro-scale systems, collective dynamics occurs from the coupling of small spatially segregated, but dynamically active, units through a bulk diffusion field. This bulk diffusion field, which is both produced and sensed by the active units, can trigger and then synchronize oscillatory dynamics associated with the individual units. In this context, we analyze diffusion-induced synchrony for a class of cell-bulk ODE–PDE system in $${\mathbb {R}}^2$$ that has two spatially segregated dynamically active circular cells of small radius. By using strong localized perturbation theory in the limit of small cell radius, we calculate the steady-state solution and formulate the linear stability problem. For Sel’kov intracellular reaction kinetics, we analyze how the effect of bulk diffusion can trigger, via a Hopf bifurcation, either in-phase or anti-phase intracellular oscillations that would otherwise not occur for cells that are uncoupled from the bulk medium. Phase diagrams in parameter space where these oscillations occur are presented, and the theoretical results from the linear stability theory are validated from full numerical simulations of the ODE–PDE system. In addition, the two-cell case is extended to study the onset of synchronous oscillatory instabilities associated with an infinite hexagonal arrangement of small identical cells in $${\mathbb {R}}^2$$ with Sel’kov intracellular kinetics. This analysis for the hexagonal cell pattern relies on determining a new, computationally efficient, explicit formula for the regular part of a certain periodic reduced-wave Green’s function.

Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension

The paper is concerned with the finite-time stabilization of a coupled PDE–ODE system describing the motion of an overhead crane with a flexible cable. The dynamics of the flexible cable is described by the wave equation with a variable coefficient which is an affine function of the curvilinear abscissa along the cable. Using several changes of variables, a backstepping transformation, and a finite-time stable second-order ODE for the dynamics of a conveniently chosen variable, we prove that a global finite-time stabilization occurs for the full system constituted of the platform and the cable. The kernel equations and the finite-time stable ODE are numerically solved in order to compute the nonlinear feedback law, and numerical simulations validating our finite-time stabilization approach are presented.

Localised signalling compartments in 2D coupled by a bulk diffusion field: Quorum sensing and synchronous oscillations in the well-mixed limit

We analyse oscillatory instabilities for a coupled partial-ordinary differential equation (PDE-ODE) system modelling the communication between localised spatially segregated dynamically active signalling compartments that are coupled through a passive extracellular bulk diffusion field in a bounded 2D domain. Each signalling compartment is assumed to secrete a chemical into the extracellular medium (bulk region), and it can also sense the concentration of this chemical in the region around its boundary. This feedback from the bulk region, resulting from the entire collection of cells, in turn modifies the intracellular dynamics within each cell. In the limit where the signalling compartments are circular discs with a small common radius ɛ ≪ 1 and where the bulk diffusivity is asymptotically large, a matched asymptotic analysis is used to reduce the dimensionless PDE–ODE system into a nonlinear ODE system with global coupling. For Sel’kov reaction kinetics, this ODE system for the intracellular dynamics and the spatial average of the bulk diffusion field are then used to investigate oscillatory instabilities in the dynamics of the cells that are triggered due to the global coupling. In particular, numerical bifurcation software on the ODEs is used to study the overall effect of coupling defective cells (cells that behave differently from the remaining cells) to a group of identical cells. Moreover, when the number of cells is large, the Kuramoto order parameter is computed to predict the degree of phase synchronisation of the intracellular dynamics. Quorum sensing behaviour, characterised by the onset of collective behaviour in the intracellular dynamics as the number of cells increases above a threshold, is also studied. Our analysis shows that the cell population density plays a dual role of triggering and then quenching synchronous oscillations in the intracellular dynamics.

Mathematical Analysis of a Non-Local Mixed ODE-PDE Model for Tumor Invasion and Chemotherapy

We present a new mathematical model for acid-mediated tumor invasion encompassing chemotherapy treatment. The model consists of a mixed ODE-PDE system with four differential equations, describing the spatio-temporal dynamics of normal cells, tumor cells, lactic acid concentration, and chemotherapy drug concentration. The model assumes non-local diffusion coefficients for tumor cells. We provide an analysis on the existence and uniqueness of model solutions. We also provide numerical simulations illustrating the model behavior, showing the invasion and the treatment phases, and comparing the model solutions with the case of constant diffusion coefficients instead of the non-local terms.