Miroslav Krstić

This short note shows that the Deadzone-Adapted Disturbance Suppression (DADS) adaptive control scheme is applicable to systems with unknown input coefficients. We study time-invariant, control-affine systems that satisfy the matching condition for which no bounds for the disturbance and the unknown parameters are known. The input coefficients can be time-varying as well as the unknown parameters. The only thing assumed for the input coefficients is their sign. The adaptive control design is Lyapunov-based and can be accomplished for every system for which a smooth globally stabilising feedback exists when the disturbances are absent and all unknown parameters are known. The design is given by simple, explicit formulas. The proposed controllers guarantee an attenuation of the plant state to an assignable small level, despite unknown bounds on the parameters and disturbance, without a drift of the gain, state, and input.

Abstract In this work, we study an age-structured chemostat model with a renewal boundary condition and a coupled substrate equation. The model is nonlinear and consists of a hyperbolic partial differential equation and an ordinary differential equation with nonlinear, nonlocal terms appearing both in the ordinary differential equation and the boundary condition. Both differential equations contain a non-negative control input, while the states of the model are required to be positive. Under an appropriate weak solution framework, we determine the state space and the input space for this model. We prove global existence and uniqueness of solutions for all admissible initial conditions and all allowable control inputs. To this purpose, we employ a combination of Banach’s fixed-point theorem with implicit solution formulas and useful solution estimates. Finally, we show that the age-structured chemostat model gives a well-defined control system on a metric space.

One of Zhong-Ping Jiang's early-career seminal results have been on the trajectory tracking feedback design for chained systems, and for the nonholonomic unicycle in particular. However, for equilibrium stabilisation (parking), globally stabilising feedback laws are exceedingly rare. Just as rare are closed-form optimal controllers for nonlinear systems, which avoid the need to solve – and store the solutions of – Hamilton-Jacobi-Bellman PDEs. In fact, until our very recent work, for the unicycle stabilisation problem, closed-form optimal controllers, even of the inverse optimal kind, have been nonexistent. Another subject in which Zhong-Ping has made pioneering contributions, over the last decade and a half, is adaptive optimal control, also referred to by the moniker reinforcement learning. An interest from my early career has commuted adjectives – optimal adaptive control, where optimality holds over the entire infinite time horizon, rather than just as the time approaches infinity. In this short paper I celebrate Zhong-Ping with a result that pulls together his various interests in nonholonomic, adaptive, and optimal control with my interests in these subjects, spanning three decades. For unicycles with large parametric uncertainties on both inputs, I present designs that are both adaptive and infinite-horizon optimal, while also being globally stabilising. This is achieved by constructing strict control Lyapunov functions, using backstepping, forwarding, or passivity, and then taking adaptive LgV controllers, which are inverse optimal.

With the rapid growth of missions in cislunar space, spacecraft must operate safely and autonomously in a strongly nonlinear gravitational environment where traditional linearized relative-motion models are insufficient. This paper develops a safety critical control framework that directly uses the fully nonlinear relative dynamics for both the Circular Restricted Three-Body Problem (CR3BP) and a high fidelity ephemeris model. A minimally intrusive Control Barrier Function (CBF)-based quadratic program (QP) enforces a Euclidean distance safety constraint between spacecraft, with the high relative degree of positional constraints addressed through CBF backstepping. After validating the safety filter under CR3BP dynamics, we extend it to the ephemeris model by incorporating SPICE-derived Earth–Moon states, a time varying rotating frame, and realistic angular rate and acceleration computations. For formation flying, we design a Lyapunov-based nominal controller that achieves a constant-radius orbit around the chief by steering the out-of-plane component to zero, driving the relative motion into a user-defined invariant circle set, and maintain a constant orbit. Simulations for Earth and Moon orbiting chief–deputy configurations show that the safety filter maintains the deputy on the boundary of the safe region before guiding it to the desired orbit, even under strong nonlinear coupling and near-constraint conditions. These results demonstrate that safety critical nonlinear control can be rigorously integrated with high fidelity cislunar dynamics to enable autonomous and reliable multi-spacecraft operations.

In this work, we propose a rigorous method for implementing predictor feedback controllers in nonlinear systems with unknown and arbitrarily long actuator delays. To address the analytically intractable nature of the predictor, we approximate it using a learned neural operator mapping. This mapping is trained once, offline, and then deployed online, leveraging the fast inference capabilities of neural networks. We provide a theoretical stability analysis based on the universal approximation theorem of neural operators and the transport partial differential equation (PDE) representation of the delay. We then prove, via a Lyapunov-Krasovskii functional, semi-global practical convergence of the dynamical system dependent on the approximation error of the predictor and delay bounds. Finally, we validate our theoretical results using a biological activator/repressor system, demonstrating speedups of 15 times compared to traditional numerical methods.

Age-structured models represent the dynamic behaviors of populations over time and result in integro-partial differential equations (IPDEs). Such models describe countless processes in biotechnology, economics, or demography. Age-structured population models with more than one species, leading to coupled IPDEs, are especially relevant for epidemics or ecology, but have received little attention thus far. We consider here an exponentially unstable model of two competing predator populations. If one were to use an input that simultaneously harvests both predator species, one would have control over only the product of the densities of the species, not over their ratio. Therefore, it is necessary to design a control input that directly harvests only one of the two predator species, while indirectly influencing the other via a backstepping approach. The model is transformed into a system of two coupled ordinary differential equations (ODEs), of which only one is actuated, and two autonomous, exponentially stable integral delay equations (IDEs) that enter the ODEs as nonlinear disturbances. We stabilize the ODEs globally with backstepping and provide an estimate of the region of attraction of the asymptotically stabilized equilibrium of the full IPDE system, under a positivity restriction on control.

This paper introduces an event-triggered source seeking control (ET-SSC) for autonomous vehicles modeled as the nonholonomic unicycle. The classical source seeking control is enhanced with static-triggering conditions to enable aperiodic and less frequent updates of the system’s input signals, offering a resource-aware control design. Our convergence analysis is based on time-scaling combined with Lyapunov and averaging theories for systems with discontinuous right-hand sides. ET-SSC ensures exponentially stable behavior for the resulting average system, leading to practical asymptotic convergence to a small neighborhood of the source point. We guarantee the avoidance of Zeno behavior by establishing a minimum dwell time to prevent infinitely fast switching. The performance optimization is aligned with classical continuous-time source seeking algorithms while balancing system performance with actuation resource consumption. Our ET-SSC algorithm, the first of its kind, allows for arbitrarily large inter-sampling times, overcoming the limitations of classical sampled-data implementations for source seeking control.

This paper presents a safe output regulation control strategy for a class of systems modeled by a coupled $2\times 2$ hyperbolic PDE-ODE structure, subject to fully distributed disturbances throughout the system. A state-feedback controller is developed by the {nonovershooting backstepping} method to simultaneously achieve exponential output regulation and enforce safety constraints on the regulated output that is the state furthest from the control input. To handle unmeasurable states and external disturbances, a state observer and a disturbance estimator are designed. Explicit bounds on the estimation errors are derived and used to construct a robust safe regulator that accounts for the uncertainties. The proposed control scheme guarantees that: 1) If the regulated output is initially within the safe region, it remains there; otherwise, it will be rescued to the safety within a prescribed time; 2) The output tracking error converges to zero exponentially; 3) The observer accurately estimates both the distributed states and external disturbances, with estimation errors converging to zero exponentially; 4) All signals in the closed-loop system remain bounded. The effectiveness of the proposed method is demonstrated through a UAV delivery scenario with a cable-suspended payload, where the payload is regulated to track a desired reference while avoiding collisions with barriers.

This paper introduces a novel Lyapunov-based small-gain methodology for establishing fixed-time stability (FxTS) guarantees in interconnected dynamical systems. Specifically, we consider interconnections in which each subsystem admits an individual fixed-time input-to-state stability (ISS) Lyapunov function that certifies FxT-ISS. We then show that if a nonlinear small-gain condition is satisfied, then the entire interconnected system is FxTS. Our results are analogous to existing Lyapunov-based small-gain theorems developed for asymptotic and finite-time stability, thereby filling an important gap in the stability analysis of interconnected dynamical systems. The proposed theoretical tools are further illustrated through analytical and numerical examples, including the first result on fixed-time feedback optimization of dynamical systems without time-scale separation between the plant and the controller.

This paper addresses the design and analysis of an extremum-seeking (ES) controller for scalar static maps in the context of infinite-dimensional dynamics governed by complex-valued partial differential equations (PDEs) of Schrodinger type. The system is actuated at one boundary, and the map input is defined as a real-valued quadratic functional corresponding to the squared norm of the complex state at the uncontrolled boundary. An isomorphism between the complex Hilbert space and its two-dimensional real-valued representation is established to enable the use of the standard multivariable Newton-based ES method. To compensate for the PDE actuation dynamics, a boundary control strategy based on a two-step backstepping procedure is employed. With a perturbation-based estimate of the Hessian inverse, the local exponential stability to a small neighborhood of the unknown extremum point is proved. A numerical example illustrates the effectiveness of the proposed extremum-seeking methodology.

This paper addresses the multivariable gradient-based extremum seeking control (ESC) subject to saturation. Two distinct saturation scenarios are investigated here: saturation acting on the input of the function to be optimized, which is addressed using an anti-windup compensation strategy, and saturation affecting the gradient estimate. In both cases, the unknown Hessian matrix is represented using a polytopic uncertainty description, and sufficient conditions in the form of linear matrix inequalities (LMIs) are derived to design a stabilizing control gain. The proposed conditions guarantee exponential stability of the origin for the average closed-loop system under saturation constraints. With the proposed design conditions, non-diagonal control gain matrices can be obtained, generalizing conventional ESC designs that typically rely on diagonal structures. Stability and convergence are rigorously proven using the Averaging Theory for dynamical systems with Lipschitz continuous right-hand sides. Numerical simulations illustrate the effectiveness of the proposed ESC algorithms, confirming the convergence even in the presence of saturation.

Abstract The title of this semi-tutorial, expository paper might be applicable to the remarkable body of research by Professor Tamer Başar with his students Didinski and Pan in the 1990s on robustification of parameter identifiers and adaptive controllers through game-theoretic methods. The author’s inspiration indeed fuses their work with his results from that period on inverse optimal adaptive stabilization and inverse optimal disturbance attenuation in a differential game formulation. Inverse optimal adaptive and minimax techniques are merged in this paper’s first half to obtain solutions to adaptive differential games between the adaptive controller and a disturbance. The article’s second half introduces a recent advance in disturbance-robustification of adaptive control for persistent disturbances (merely bounded, rather than square-integrable), by Iasson Karafyllis and this article’s author. While not with a minimax capability, this robustification is first in forty years to attain regulation with a bias that is arbitrarily low and independent of both the unknown parameter and the persistent disturbance.

This paper presents a safe stabilization of the Stefan PDE model with a moving boundary governed by a high-order dynamics. We consider a parabolic PDE with a time-varying domain governed by a second-order response with respect to the Neumann boundary value of the PDE state at the moving boundary. The objective is to design a boundary heat flux control to stabilize the moving boundary at a desired setpoint, with satisfying the required conditions of the model on PDE state and the moving boundary. We apply a PDE backstepping method for the control design with considering a constraint on the control law. The PDE and moving boundary constraints are shown to be satisfied by applying the maximum principle for parabolic PDEs. Then the closed-loop system is shown to be globally exponentially stable by performing Lyapunov analysis. The proposed control is implemented in numerical simulation, which illustrates the desired performance in safety and stability. An outline of the extension to third-order moving boundary dynamics is also presented. Code is released at https://github.com/shumon0423/HighOrderStefan_CDC2025.git.

This paper presents novel extremum seeking (ES) strategies for maximum power point tracking (MPPT) in photovoltaic (PV) systems that ensure unbiased convergence and prescribed-time performance. Conventional ES methods suffer from steady-state bias due to persistent dither signal. We introduce two novel ES algorithms: the exponential unbiased ES (uES), which guarantees exponential convergence to the maximum power point (MPP) without steady-state oscillation bias, and the unbiased prescribed-time ES (uPT-ES), which ensures convergence within a user-defined time horizon. Both methods leverage time-varying perturbation amplitudes and demodulation gains, with uPT-ES additionally utilizing chirp signals to enhance excitation over finite-time intervals. Experimental results on a hardware-in-the-loop testbed validate the proposed algorithms, demonstrating improved convergence speed and tracking accuracy compared to classical ES, under both static and time-varying environmental conditions.

This work establishes the first rigorous stability guarantees for approximate predictors in delay-adaptive control of nonlinear systems, addressing a key challenge in practical implementations where exact predictors are unavailable. We analyze two scenarios: (i) when the actuated input is directly measurable, and (ii) when it is estimated online. For the measurable input case, we prove semi-global practical asymptotic stability with an explicit bound proportional to the approximation error $ε$. For the unmeasured input case, we demonstrate local practical asymptotic stability, with the region of attraction explicitly dependent on both the initial delay estimate and the predictor approximation error. To bridge theory and practice, we show that neural operators-a flexible class of neural network-based approximators-can achieve arbitrarily small approximation errors, thus satisfying the conditions of our stability theorems. Numerical experiments on two nonlinear benchmark systems-a biological protein activator/repressor model and a micro-organism growth Chemostat model-validate our theoretical results. In particular, our numerical simulations confirm stability under approximate predictors, highlight the strong generalization capabilities of neural operators, and demonstrate a substantial computational speedup of up to 15x compared to a baseline fixed-point method.

Since the mid-1990s, it has been known that, unlike in Cartesian form where Brockett's condition rules out static feedback stabilization, the unicycle is globally asymptotically stabilizable by smooth feedback in polar coordinates. In this note, we introduce a modular framework for designing smooth feedback laws that achieve global asymptotic stabilization in polar coordinates. These laws are bidirectional, enabling efficient parking maneuvers, and are paired with families of strict control Lyapunov functions (CLFs) constructed in a modular fashion. The resulting CLFs guarantee global asymptotic stability with explicit convergence rates and include barrier variants that yield "almost global" stabilization, excluding only zero-measure subsets of the rotation manifolds. The strictness of the CLFs is further leveraged in our companion paper, where we develop inverse-optimal redesigns with meaningful cost functions and infinite gain margins.

This paper presents a framework for stabilizing the Dubins vehicle model to zero in finite time (deadbeat parking) by interpreting distance as a time-like variable. We develop control laws that bring the system to a desired position and orientation even when the forward velocity cannot be directly actuated. While the controllers employ inverse-distance gains, we show that the control input remains bounded for all time. In addition to basic deadbeat parking, we incorporate safety considerations by proposing algorithms that prevent the vehicle from crossing in front of the target, enforce deceleration as it approaches the target, and guarantee parking without curb violations. The resulting methods are well-suited for missile guidance and fixed-wing pursuit, but are broadly applicable to physical systems that are represented by the Dubins vehicle model.

A transport PDE with a spatial integral and recirculation with constant delay has been a benchmark for neural operator approximations of PDE backstepping controllers. Introducing a spatially-varying delay into the model gives rise to a gain operator defined through integral equations which the operator's input—the varying delay function—enters in previously unencountered manners, including in the limits of integration and as the inverse of the ‘delayED time’ function. This, in turn, introduces novel mathematical challenges in estimating the operator's Lipschitz constant. The backstepping kernel function having two branches endows the feedback law with a two-branch structure, where only one of the two feedback branches depends on both of the kernel branches. For this rich feedback structure, we propose a neural operator approximation of such a two-branch feedback law and prove the approximator to be semiglobally practically stabilizing. With numerical results we illustrate the training of the neural operator and its stabilizing capability.

We present Assignably Safe Extremum Seeking (ASfES), an algorithm designed to minimize a measured, static objective function while maintaining a measured, static metric of safety (a control barrier function or CBF) to be positive in a practical sense. We ensure that for trajectories with safe initial conditions, the violation of safety can be made arbitrarily small through appropriately chosen design constants. We also guarantee an assignable ‘‘attractivity’’ rate: from unsafe initial conditions, the trajectories approach the safe set, in the sense of the measured CBF, at a rate no slower than a user-assigned rate. Similarly, from safe initial conditions, the trajectories approach the unsafe set, in the sense of the CBF, no faster than the assigned attractivity rate. The feature of assignable attractivity is not present in the semiglobal version of safe extremum seeking, where the semiglobality of convergence is achieved by slowing the adaptation. We also demonstrate local convergence of the parameter to a neighborhood of the minimum of a quadratic objective function constrained to the safe set with a linear CBF. The ASfES algorithm and analysis are multivariable, but we also extend the algorithm to a Newton-Based ASfES scheme which we show is only useful in the scalar case. The proven properties of the designs are illustrated through simulation examples.

This paper considers the problem of adaptively overriding unsafe actions of a nominal controller in the presence of high-relative-degree nonovershooting constraints and parametric uncertainties. To prevent the design from being coupled with high-order derivatives of the parameter estimation error, we adopt a modular design approach in which the controller and the parameter identifier are designed separately. The controller module ensures that any safety violations caused by parametric uncertainties remain bounded, provided that the parameter estimation error and its first-order derivative are either bounded or square-integrable. The identifier module, in turn, guarantees that these requirements on the parameter estimation error are satisfied. Both theoretical analysis and simulation results demonstrate that the closed-loop safety violation is bounded by a tunable function of the initial estimation error. Moreover, as time increases, the parameter estimate converges to the true value, and the amount of safety violation decreases accordingly.

In this work, we present the first stability results for approximate predictors in multi-input non-linear systems with distinct actuation delays. We show that if the predictor approximation satisfies a uniform (in time) error bound, semi-global practical stability is correspondingly achieved. For such approximators, the required uniform error bound depends on the desired region of attraction and the number of control inputs in the system. The result is achieved through transforming the delay into a transport PDE and conducting analysis on the coupled ODE-PDE cascade. To highlight the viability of such error bounds, we demonstrate our results on a class of approximators - neural operators - showcasing sufficiency for satisfying such a universal bound both theoretically and in simulation on a mobile robot experiment.

Systems modeled by partial differential equations (PDEs) are at least as ubiquitous as systems that are by nature finite-dimensional and modeled by ordinary differential equations (ODEs). And yet, systematic and readily usable methodologies, for such a significant portion of real systems, have been historically scarce. Around the year 2000, the backstepping approach to PDE control began to offer not only a less abstract alternative to PDE control techniques replicating optimal and spectrum assignment techniques of the 1960s, but also enabled the methodologies of adaptive and nonlinear control, matured in the 1980s and 1990s, to be extended from ODEs to PDEs, allowing feedback synthesis for physical and engineering systems that are uncertain, nonlinear, and infinite-dimensional. The PDE backstepping literature has grown in its nearly a quarter century of development to many hundreds of papers and nearly a dozen books. This survey aims to facilitate the entry, for a new researcher, into this thriving area of overwhelming size and topical diversity. Designs of controllers and observers, for parabolic, hyperbolic, and other classes of PDEs, in one and more dimensions (in box and spherical geometries), with nonlinear, adaptive, sampled-data, and event-triggered extensions, are covered in the survey. The lifeblood of control are technology and physics. The survey places a particular emphasis on applications that have motivated the development of the theory and which have benefited from the theory and designs: applications involving flows, flexible structures, materials, thermal and chemically reacting dynamics, energy (from oil drilling to batteries and magnetic confinement fusions), and vehicles.

In this work, we propose a rigorous method for implementing predictor feedback controllers in nonlinear systems with unknown and arbitrarily long actuator delays. To address the analytically intractable nature of the predictor, we approximate it using a learned neural operator mapping. This mapping is trained once, offline, and then deployed online, leveraging the fast inference capabilities of neural networks. We provide a theoretical stability analysis based on the universal approximation theorem of neural operators and the transport partial differential equation (PDE) representation of the delay. We then prove, via a Lyapunov-Krasovskii functional, semi-global practical convergence of the dynamical system dependent on the approximation error of the predictor and delay bounds. Finally, we validate our theoretical results using a biological activator/repressor system, demonstrating speedups of 15 times compared to traditional numerical methods.

For a map that is strictly but not strongly convex, model-based gradient extremum seeking has an eigenvalue of zero at the extremum, i.e., it fails at exponential convergence. Interestingly, perturbation-based model-free extremum seeking has a negative Jacobian, in the average, meaning that its (practical) convergence is exponential, even though the map’s Hessian is zero at the extremum. Although these observations for gradient-based extremum seeking control (GESC) are not trivial, in this paper we focus on an even more nontrivial study of the same phenomenon for Newton-based extremum seeking control (NESC). NESC is a second-order method which corrects for the unknown Hessian of the unknown map, not only in order to speed up parameter convergence, but also (1) to make the convergence rate user-assignable in spite of the unknown Hessian, and (2) to equalize the convergence rates in different directions for multivariable maps. Previous NESC work established stability only for maps whose Hessians are strictly positive definite everywhere, so the Hessian is invertible everywhere. For a scalar map, we establish the rather unexpected property that, even when the map is strictly convex but not strongly convex, i.e., when the Hessian may be zero, NESC guarantees practical asymptotic stability, semiglobally. While a model-based Newton-based algorithm would run into non-invertibility of the Hessian, the perturbation-based NESC, surprisingly, avoids this challenge by leveraging the fact that the average of the perturbation-based Hessian estimate is always positive, even though the actual Hessian may be zero. However, these stability results for the NESC, and even for the GESC, do not hold for multivariable maps. We show that these ESCs can be locally destabilized for certain symmetric maps by highly asymmetric dither choices. Thus, we present unexpected robustness of NESC on maps that are scalar and strictly but not necessarily strongly convex.

This paper presents novel methods for achieving stable and efficient convergence in multivariable extremum seeking control (ESC) using sliding mode techniques. Drawing inspiration from both classical sliding mode control and more recent developments in finite-time and fixed-time control, we propose a new framework that integrates these concepts into Gradient- and Newton-based ESC schemes based on sinusoidal perturbation signals. The key innovation lies in the use of discontinuous "relay-type" control components, replacing traditional proportional feedback to estimate the gradient of unknown quadratic nonlinear performance maps with Unit Vector Control (UVC). This represents the first attempt to address real-time, model-free optimization using sliding modes within the classical extremum seeking paradigm. In the Gradient-based approach, the convergence rate is influenced by the unknown Hessian of the objective function. In contrast, the Newton-based method overcomes this limitation by employing a dynamic estimator for the inverse of the Hessian, implemented via a Riccati equation filter. We establish finite-time convergence of the closed-loop average system to the extremum point for both methods by leveraging Lyapunov-based analysis and averaging theory tailored to systems with discontinuous right-hand sides. Numerical simulations validate the proposed method, illustrating significantly faster convergence and improved robustness compared to conventional ESC strategies, which typically guarantee only exponential stability. The results also demonstrate that the Gradient-based method exhibits slower convergence and higher transients since the gradient trajectory follows the curved and steepest-descent path, whereas the Newton-based method achieves faster convergence and improved overall performance going straightly to the extremum.

Age-structured models capture the dynamic behavior of populations over time and result in nonlinear integro-partial differential equations (IPDEs). These processes arise in various fields such as biotechnology, economics, or demography. While coupled age-structured IPDEs modeling two or more interacting species occur naturally in epidemiology and ecology, they remain relatively underexplored. Prior work has primarily addressed stable and marginally stable dynamics. In constrast, this work considers an exponentially unstable model of two competing predator populations, formally referred to in the literature as ``competition'' dynamics. If one were to apply an input that simultaneously harvests both predator species, one would have control over only the product of the densities of the species, not over their ratio. Therefore, it is necessary to design a control input that directly harvests only one of the two predator species, while indirectly influencing the other via a backstepping approach. The model is transformed into a system of two coupled ordinary differential equations (ODEs), of which only one is actuated, and two autonomous, exponentially stable integral delay equations (IDEs) which enter the ODEs as nonlinear disturbances. The ODEs are globally stabilized with backstepping and an estimate of the region of attraction of the asymptotically stabilized equilibrium of the full IPDE system is provided, under a positivity restriction on control. Additionally, the full IPDE system is also shown to be local exponential stable. Such generalizations of competition dynamics open exciting possibilities for future research directions for systems with more than two species.

We extend the Deadzone-Adapted Disturbance Suppression (DADS) control to time-invariant systems with dynamic uncertainties that satisfy the matching condition and for which no bounds for the disturbance and the unknown parameters are known. This problem is equivalent to partial-state adaptive feedback, where the states modeling the dynamic uncertainty are unmeasured. We show that the DADS controller can bypass small-gain conditions and achieve robust regulation for systems in spite of the fact that the strength of the interconnections has no known bound. Moreover, no gain and state drift arise, regardless of the size of the disturbances and unknown parameters. Finally, the paper provides the detailed analysis of a control system where the unmeasured state (or the dynamic uncertainty) is infinite-dimensional and described by a reaction-diffusion Partial Differential Equation, where the diffusion coefficient and the reaction term are unknown. It is shown that even in the infinite-dimensional case, a DADS controller can be designed and guarantees robust regulation of the plant state.

Populations (in ecology, epidemics, biotechnology, economics, social processes) do not only interact over time but also age over time. It is therefore common to model them as “age-structured” partial differential equations (PDEs), where age is the ‘space variable.' Since the models also involve integrals over age, both in the birth process and in the interaction among species, they are in fact integro-partial differential equations (IPDEs) with positive states. To regulate the population densities to desired profiles, harvesting is used as input. But non-discriminating harvesting, where wanting to repress one (overpopulated) species will inevitably repress the other (near-extinct) species as well, the positivity restriction on the input (no insertion of population, only removal), and the multiplicative (nonlinear) nature of harvesting, makes control challenging even for ordinary differential equation (ODE) versions of such dynamics, let alone for their IPDE versions, on an infinite-dimensional nonnegative state space. With this paper, we introduce a design for a benchmark version of such a problem: a two-population predator-prey setup. The model is equivalent to two coupled ODEs, actuated by harvesting which must not drop below zero, and strongly (“exponentially”) disturbed by two autonomous but exponentially stable integral delay equations (IDEs). We develop two control designs. With a modified Volterra-like control Lyapunov function, we design a simple feedback which employs possibly negative harvesting for global stabilization of the ODE model, while guaranteeing regional regulation with positive harvesting. With a more sophisticated, restrained controller we achieve regulation for the ODE model globally, with positive harvesting. For the full IPDE model, with the IDE dynamics acting as large disturbances, for both the simple and saturated feedback laws we provide explicit estimates of the regions of attraction. Simulations illustrate the nonlinear infinite-dimensional solutions under the two feedbacks. The paper charts a new pathway for control designs for infinite-dimensional multi-species dynamics and for nonlinear positive systems with positive controls.

Age-structured models describe the dynamic behaviors of populations over time and result in integro-partial differential equations (IPDEs). These models are useful to represent a multitude of processes in biotechnology or economics. Single population models are, for example, used to control the harvest rate of a chemostat in order to maximize the yield of a process. Age-structured population models with more than one species, leading to coupled IPDEs, are relevant for epidemics or ecology, but have received little attention in the literature. In this work, we present a model for two interacting populations in a predator-prey setup, whose input is the inevitable harvesting of both species, which represents a challenge for stabilization. The model is transformed to a system of two coupled ordinary differential equations (ODEs) actuated by the input, and two autonomous but exponentially stable integral delay equations (IDEs). The controllable ODE is stabilized through a weighted Control Lyapunov Function (CLF) feedback. We establish that the CLF-based control law exclusively derived from the ODE system dynamics, locally stabilizes the transformed ODE-IDE system.

In safety-critical control, managing safety constraints with high relative degrees and uncertain obstacle dynamics pose significant challenges in guaranteeing safety performance. Robust Control Barrier Functions (RCBFs) offer a potential solution, but the non-smoothness of the standard RCBF definition can pose a challenge when dealing with multiple derivatives in high relative degree problems. As a result, the definition was extended to the marginally more conservative smooth Robust Control Barrier Functions (sRCBF). Then, by extending the sRCBF framework to the CBF backstepping method, this paper offers a novel approach to these problems. Treating obstacle dynamics as disturbances, our approach reduces the requirement for precise state estimations of the obstacle to an upper bound on the disturbance, which simplifies implementation and enhances the robustness and applicability of CBFs in dynamic and uncertain environments. Then, we validate our technique through an example problem in which an agent, modeled using a kinematic unicycle model, aims to avoid an unknown moving obstacle. The demonstration shows that the standard CBF backstepping method is not sufficient in the presence of a moving obstacle, especially with unknown dynamics. In contrast, the proposed method successfully prevents the agent from colliding with the obstacle, proving its effectiveness.

Extremum seeking is an online, model-free optimization algorithm traditionally known for its practical stability. This paper introduces an extremum seeking algorithm designed for unbiased convergence to the extremum asymptotically, allowing users to define the convergence rate. Unlike conventional extremum seeking approaches utilizing constant gains, our algorithms employ time-varying parameters. These parameters reduce perturbation amplitudes towards zero in an asymptotic manner, while incorporating asymptotically growing controller gains. The stability analysis is based on state transformation, achieved through the multiplication of the input state by asymptotic growth function, and Lie bracket averaging applied to the transformed system. The averaging ensures the practical stability of the transformed system, which, in turn, leads to the asymptotic stability of the original system. Moreover, for strongly convex maps, we achieve exponentially fast convergence. The numerical simulations validate the feasibility of the introduced designs.

Deception is a key tactic for agents in adversarial environments, used to mislead opponents into adopting unaware strategies. In cyber-physical systems, for instance, deception can conceal attacks against critical infrastructure. This tutorial highlights the usefulness of deception for attacking and protecting systems against adversaries, but also as a tool to increase payoff in general game-theoretic and data-driven settings. It presents several state-of-the-art techniques for control-theoretic deception, including deception in defensive cyber-physical security, game-theoretic reinforcement learning, general multi-agent learning systems, Nash equilibrium seeking, and data-driven control. Although showcased in specific contexts, the underlying concepts and ideas that we study should be generalizable by researchers to settings beyond the scope of this tutorial.

We consider the nonlinear Kuramoto-Sivashinsky equation and its linear part on a finite interval subject to periodic boundary conditions. The linear part can have a finite number of unstable eigenvalues so we assume that there are point actuators that allow a linear feedback to move all the unstable eigenvalues into the open left half plane. Such a linear feedback law is found by the well-known technique of Linear Quadratic Regulation (LQR). This leads to a new Riccati partial differetial equation for quadratic Fredholm kernel of the optimal cost. From this quadratic kernel we obtain the linear kernel of the optimal feedback. We prove that this feedback moves all the unstable eigenvalues into the open left half plane. But it has little effect on the open loop eigenvalues that were already stable. This linear feedback locally stabilizes the nonlinear Kuramoto-Sivashinsky equation but nonLinear nonQuadratic Regulation (nLnQR), which we discuss in this paper, can be used to find a cubic feedback that stabilizes it faster and/or with less control energy.

In socio-technical multi-agent systems, deception exploits privileged information to induce false beliefs in “victims,” keeping them oblivious and leading to outcomes detrimental to them or advantageous to the deceiver. We consider model-free Nash-equilibrium-seeking for non-cooperative games with asymmetric information and introduce model-free deceptive algorithms with stability guarantees. In the simplest algorithm, the deceiver includes in his action policy the victim's exploration signal, with an amplitude tuned by an integrator of the regulation error between the deceiver's actual and desired payoff. The integral feedback drives the deceiver's payoff to the payoff's reference value, while the victim is led to adopt a suboptimal action, at which the pseudogradient of the deceiver's payoff is zero. The deceiver's and victim's actions turn out to constitute a “deceptive” Nash equilibrium of a different game, whose structure is managed — in real time — by the deceiver. We examine quadratic, aggregative, and more general games and provide conditions for a successful deception, mutual and benevolent deception, and immunity to deception (for a “non-generic” set of payoff functions). Stability results are established using techniques based on averaging and singular perturbations. Among the examples in the paper is a microeconomic duopoly in which the deceiver induces in the victim a belief that the buyers disfavor the deceiver more than they actually do, leading the victim to increase the price above the Nash price, and resulting in an increased profit for the deceiver and a decreased profit for the victim. A study of the deceiver's integral feedback for the desired profit reveals that, in duopolies with equal marginal costs, a deceiver that is greedy for very high profit can attain any such profit, and pursue this with arbitrarily high integral gain (impatiently), irrespective of the market preference for the victim.

For quasilinear parabolic partial differential equations (PDEs) that exhibit finite-time blow up in open loop, i.e., under null boundary conditions, we provide an estimate of the region of attraction under cubic feedback laws applied at the boundary, using boundary measurements. We guarantee: 1-L 2 and H 1 exponential stability of the origin with an estimate of the region of attraction. 2-Convergence of the H 2 and the C 1 norms of the solutions to zero. 3-Existence and uniqueness of complete classical solutions. 4-Positivity of the solutions starting from positive initial conditions. Unlike existing approaches, our framework handles nonlinear state-dependent diffusion, convection, and (destabilizing) reaction. The cubic terms are used to enlarge our estimate of the region of attraction. The size of the region of attraction is shown, in many cases, to grow unboundedly as diffusion increases. Finally, our controllers can be implemented as Neumann, Dirichlet, or mixed-type boundary conditions.

This note studies the distributed feedback optimization for linear multi-agent systems without precise knowledge of cost functions and agent dynamics. The goal is to regulate the outputs of the agents toward an unknown minimizer of a sum of local costs. To achieve this, distributed reference signals are combined with an extremum seeking mechanism to search for the minimizer. Meanwhile, each agent steers its output toward the designed reference signal using a learning-based adaptive optimal tracker. The entire process relies only on measurements of local costs and input-state data along the agents' trajectories. Moreover, the overall feedback loop has three time scales: tracking and consensus of the reference signals are the fastest, periodic sinusoidal perturbation is the medium, and optimization of the global cost is the slowest. Through this time-scale separation, the closed-loop system is guaranteed to be practically exponentially stable at an equilibrium of interest, along with the convergence of the output of each agent to a small neighborhood of the desired minimizer. A numerical example of robotic networks demonstrates the efficacy of the proposed method.

In the theory of multi-agent systems, deception refers to the strategic manipulation of information to influence the behavior of other agents, ultimately altering the long-term dynamics of the entire system. Recently, this concept has been examined in the context of model-free Nash equilibrium seeking (NES) algorithms for noncooperative games. Specifically, it was demonstrated that players can exploit knowledge of other players' exploration signals to drive the system toward a ``deceptive" Nash equilibrium, while maintaining the stability of the closed-loop system. To extend this insight beyond the duopoly case, in this paper we conduct a comprehensive study of deception mechanisms in N-player oligopoly markets. By leveraging the structure of these games and employing stability techniques for nonlinear dynamical systems, we provide game-theoretic insights into deception and derive specialized results, including stability conditions. These results allow players to systematically adjust their NES dynamics by tuning gains and signal amplitudes, all while ensuring closed-loop stability. Additionally, we introduce novel sufficient conditions to demonstrate that the (practically) stable equilibrium point of the deceptive dynamics corresponds to a true Nash equilibrium of a different game, which we term the ``deceptive game." Our results show that, under the proposed adaptive dynamics with deception, a victim firm may develop a distorted perception of its competitors' product appeal, which could lead to setting suboptimal prices.