Miroslav Krstic

Michael Tang, Miroslav Krstic, Jorge Poveda

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.

Iasson Karafyllis, Miroslav Krstic

The paper provides results for the application of boundary feedback control with Zero-Order-Hold (ZOH) to 1-D linear parabolic systems on bounded domains. It is shown that the continuous-time boundary feedback applied in a sample-and-hold fashion guarantees closed-loop exponential stability, provided that the sampling period is sufficiently small. Two different continuous-time feedback designs are considered: the reduced model design and the backstepping design. The obtained results provide stability estimates for weighted 2-norms of the state and robustness with respect to perturbations of the sampling schedule is guaranteed.

Iasson Karafyllis, Miroslav Krstic

For 1-D parabolic PDEs with disturbances at both boundaries and distributed disturbances we provide ISS estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an eigenfunction expansion of the solution, and (ii) a finite-difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation. Finally, the obtained results are applied to quasi-static thermoelasticity models that involve nonlocal boundary conditions. Small-gain conditions that guarantee the global exponential stability of the zero solution for such models are derived.

Rafael Vazquez, Miroslav Krstic

Recently, the problem of boundary stabilization for unstable linear constant-coefficient coupled reaction-diffusion systems was solved by means of the backstepping method. The extension of this result to systems with advection terms and spatially-varying coefficients is challenging due to complex boundary conditions that appear in the equations verified by the control kernels. In this paper we address this issue by showing that these equations are essentially equivalent to those verified by the control kernels for first-order hyperbolic coupled systems, which were recently found to be well-posed. The result therefore applies in this case, allowing us to prove H^1 stability for the closed-loop system. It also shows an interesting connection between backstepping kernels for coupled parabolic and hyperbolic problems.

Iasson Karafyllis, Miroslav Krstic

Robustness is established for the predictor feedback for linear time-invariant systems with respect to possibly time-varying perturbations of the input delay, with a constant nominal delay. Prior results have addressed qualitatively constant delay perturbations (robustness of stability in L2 norm of actuator state) and delay perturbations with restricted rate of change (robustness of stability in H1 norm of actuator state). The present work provides simple formulae that allow direct and accurate computation of the least upper bound of the magnitude of the delay perturbation for which exponential stability in supremum norm on the actuator state is preserved. While prior work has employed Lyapunov-Krasovskii functionals constructed via backstepping, the present work employs a particular form of small-gain analysis. Two cases are considered: the case of measurable (possibly discontinuous) perturbations and the case of constant perturbations.

Iasson Karafyllis, Miroslav Krstic

This paper establishes the equivalence between systems described by a single first-order hyperbolic partial differential equation and systems described by integral delay equations. System-theoretic results are provided for both classes of systems (among them converse Lyapunov results). The proposed framework can allow the study of discontinuous solutions for nonlinear systems described by a single first-order hyperbolic partial differential equation under the effect of measurable inputs acting on the boundary and/or on the differential equation. An illustrative example shows that the conversion of a system described by a single first-order hyperbolic partial differential equation to an integral delay system can simplify considerably the solution of the corresponding robust feedback stabilization problem.

Iasson Karafyllis, Miroslav Krstic, Katerina Chrysafi

For parabolic PDEs, we present a new certainty equivalence-based adaptive boundary control scheme with a least-squares identifier of an event-triggering type, where the triggering is based on the size of the regulation error (as opposed to the identifier updates being triggered by the estimation error, or the control changes being triggered by the regulation error). The scheme guarantees exponential convergence of the state to zero in the L2 norm and a finite-time convergence of the parameter estimates to the true values of the unknown parameters. The scheme is developed for a specific benchmark problem with Dirichlet actuation, where the only unknown parameters are the reaction coefficient and the high-frequency gain. For this specific problem, no existing adaptive scheme can handle the unknown high-frequency gain. An illustrative example allows the comparison with other adaptive control design methodologies.

Iasson Karafyllis, Maria Kontorinaki, Miroslav Krstic

The paper extends a recently proposed indirect, certainty-equivalence, event-triggered adaptive control scheme to the case of non-observable parameters. The extension is achieved by using a novel Batch Least-Squares Identifier (BaLSI), which is activated at the times of the events. The BaLSI guarantees the finite-time asymptotic constancy of the parameter estimates and the fact that the trajectories of the closed-loop system follow the trajectories of the nominal closed-loop system ("nominal" in the sense of the asymptotic parameter estimate, not in the sense of the true unknown parameter). Thus, if the nominal feedback guarantees global asymptotic stability and local exponential stability, then unlike conventional adaptive control, the newly proposed event-triggered adaptive scheme guarantees global asymptotic regulation with a uniform exponential convergence rate. The developed adaptive scheme is tested to a well-known control problem: the state regulation of the wing-rock model. Comparisons with other adaptive schemes are provided for this particular problem.

Iasson Karafyllis, Miroslav Krstic

Implementation is a common problem with feedback laws with distributed delays. This paper focuses on a specific aspect of the implementation problem for predictor-based feedback laws: the problem of the approximation of the predictor mapping. It is shown that the numerical approximation of the predictor mapping by means of a numerical scheme in conjunction with a hybrid feedback law that uses sampled measurements, can be used for the global stabilization of all forward complete nonlinear systems that are globally asymptotically stabilizable and locally exponentially stabilizable in the delay-free case. Special results are provided for the linear time invariant case. Explicit formulae are provided for the estimation of the parameters of the resulting hybrid control scheme.

Miroslav Krstic, Luke Bhan, Yuanyuan Shi

Unlike ODEs, whose models involve system matrices and whose controllers involve vector or matrix gains, PDE models involve functions in those roles functional coefficients, dependent on the spatial variables, and gain functions dependent on space as well. The designs of gains for controllers and observers for PDEs, such as PDE backstepping, are mappings of system model functions into gain functions. These infinite dimensional nonlinear operators are given in an implicit form through PDEs, in spatial variables, which need to be solved to determine the gain function for each new functional coefficient of the PDE. The need for solving such PDEs can be eliminated by learning and approximating the said design mapping in the form of a neural operator. Learning the neural operator requires a sufficient number of prior solutions for the design PDEs, offline, as well as the training of the operator. In recent work, we developed the neural operators for PDE backstepping designs for first order hyperbolic PDEs. Here we extend this framework to the more complex class of parabolic PDEs. The key theoretical question is whether the controllers are still stabilizing, and whether the observers are still convergent, if they employ the approximate functional gains generated by the neural operator. We provide affirmative answers to these questions, namely, we prove stability in closed loop under gains produced by neural operators. We illustrate the theoretical results with numerical tests and publish our code on github. The neural operators are three orders of magnitude faster in generating gain functions than PDE solvers for such gain functions. This opens up the opportunity for the use of this neural operator methodology in adaptive control and in gain scheduling control for nonlinear PDEs.

Iasson Karafyllis, Miroslav Krstic

We present bounded dynamic (but observer-free) output feedback laws that achieve global stabilization of equilibrium profiles of the partial differential equation (PDE) model of a simplified, age-structured chemostat model. The chemostat PDE state is positive-valued, which means that our global stabilization is established in the positive orthant of a particular function space-a rather non-standard situation, for which we develop non-standard tools. Our feedback laws do not employ any of the (distributed) parametric knowledge of the model. Moreover, we provide a family of highly unconventional Control Lyapunov Functionals (CLFs) for the age-structured chemostat PDE model. Two kinds of feedback stabilizers are provided: stabilizers with continuously adjusted input and sampled-data stabilizers. The results are based on the transformation of the first-order hyperbolic partial differential equation to an ordinary differential equation (one-dimensional) and an integral delay equation (infinite-dimensional). Novel stability results for integral delay equations are also provided; the results are of independent interest and allow the explicit construction of the CLF for the age-structured chemostat model.

Iasson Karafyllis, Miroslav Krstic

For nonlinear systems that are known to be globally asymptotically stabilizable, control over networks introduces a major challenge because of the asynchrony in the transmission schedule. Maintaining global asymptotic stabilization in sampled-data implementations with zero-order hold and with perturbations in the sampling schedule is not achievable in general but we show in this paper that it is achievable for the class of feedforward systems. We develop sampled-data feedback stabilizers which are not approximations of continuous-time designs but are discontinuous feedback laws that are specifically developed for maintaining global asymptotic stabilizability under any sequence of sampling periods that is uniformly bounded by a certain "maximum allowable sampling period".

Iasson Karafyllis, Miroslav Krstic

This work studies the design problem of feedback stabilizers for discrete-time systems with input delays. A backstepping procedure is proposed for disturbance-free discrete-time systems. The feedback law designed by using backstepping coincides with the predictor-based feedback law used in continuous-time systems with input delays. However, simple examples demonstrate that the sensitivity of the closed-loop system with respect to modeling errors increases as the value of the delay increases. The paper proposes a Lyapunov redesign procedure which can minimize the effect of the uncertainty. Specific results are provided for linear single-input discrete-time systems with multiplicative uncertainty. The feedback law that guarantees robust global exponential stability is a nonlinear, homogeneous of degree 1 feedback law.

Jie Qi, Jing Zhang, Miroslav Krstic

The recently introduced DeepONet operator-learning framework for PDE control is extended from the results for basic hyperbolic and parabolic PDEs to an advanced hyperbolic class that involves delays on both the state and the system output or input. The PDE backstepping design produces gain functions that are outputs of a nonlinear operator, mapping functions on a spatial domain into functions on a spatial domain, and where this gain-generating operator's inputs are the PDE's coefficients. The operator is approximated with a DeepONet neural network to a degree of accuracy that is provably arbitrarily tight. Once we produce this approximation-theoretic result in infinite dimension, with it we establish stability in closed loop under feedback that employs approximate gains. In addition to supplying such results under full-state feedback, we also develop DeepONet-approximated observers and output-feedback laws and prove their own stabilizing properties under neural operator approximations. With numerical simulations we illustrate the theoretical results and quantify the numerical effort savings, which are of two orders of magnitude, thanks to replacing the numerical PDE solving with the DeepONet.

Rafael Vazquez, Miroslav Krstic

This paper develops an extension of infinite-dimensional backstepping method for parabolic and hyperbolic systems in one spatial dimension with two actuators. Typically, PDE backstepping is applied in 1-D domains with an actuator at one end. Here, we consider the use of two actuators, one at each end of the domain, which we refer to as bilateral control (as opposed to unilateral control). Bilateral control laws are derived for linear reaction-diffusion, wave and 2X2 hyperbolic 1-D systems (with same speed of transport in both directions). The extension is nontrivial but straightforward if the backstepping transformation is adequately posed. The resulting bilateral controllers are compared with their unilateral counterparts in the reaction-diffusion case for constant coefficients, by making use of explicit solutions, showing a reduction in control effort as a tradeoff for the presence of two actuators when the system coefficients are large. These results open the door for more sophisticated designs such as bilateral sensor/actuator output feedback and fault-tolerant designs.

Iasson Karafyllis, Miroslav Krstic

We provide two solutions to the heretofore open problem of stabilization of systems with arbitrarily long delays at the input and output of a nonlinear system using output feedback only. Both of our solutions are global, employ the predictor approach over the period that combines the input and output delays, address nonlinear systems with sampled measurements and with control applied using a zero-order hold, and require that the sampling/holding periods be sufficiently short, though not necessarily constant. Our first approach considers general nonlinear systems for which the solution map is available explicitly and whose one-sample-period predictor-based discrete-time model allows state reconstruction, in a finite number of steps, from the past values of inputs and output measurements. Our second approach considers a class of globally Lipschitz strict-feedback systems with disturbances and employs an appropriately constructed successive approximation of the predictor map, a high-gain sampled-data observer, and a linear stabilizing feedback for the delay-free system. We specialize the second approach to linear systems, where the predictor is available explicitly. We provide two illustrative examples-one analytical for the first approach and one numerical for the second approach.

Iasson Karafyllis, Miroslav Krstic

We present a methodology for the global sampled-data stabilization of systems with a compact absorbing set and input/measurement delays. The methodology is based on the Inter-Sample-Predictor, Observer, Predictor, Delay-Free Controller (ISP-O-P-DFC) scheme and the stabilization is robust to perturbations of the sampling schedule. The obtained results are novel even for the delay-free case.

Kevin Schmidt, Iasson Karafyllis, Miroslav Krstic

For population systems modeled by age-structured hyperbolic partial differential equations (PDEs) that are bilinear in the input and evolve with a positive-valued infinite-dimensional state, global stabilization of constant yield set points was achieved in prior work. Seasonal demands in biotechnological production processes give rise to time-varying yield references. For the proposed control objective aiming at a global attractivity of desired yield trajectories, multiple non-standard features have to be considered: a non-local boundary condition, a PDE state restricted to the positive orthant of the function space and arbitrary restrictive but physically meaningful input constraints. Moreover, we provide Control Lyapunov Functionals ensuring an exponentially fast attraction of adequate reference trajectories. To achieve this goal, we make use of the relation between first-order hyperbolic PDEs and integral delay equations leading to a decoupling of the input-dependent dynamics and the infinite-dimensional internal one. Furthermore, the dynamic control structure does not necessitate exact knowledge of the model parameters or online measurements of the age-profile. With a Galerkin-based numerical simulation scheme using the key ideas of the Karhunen-Loève-decomposition, we demonstrate the controller's performance.

Yuanyuan Shi, Zongyi Li, Huan Yu et al.

State estimation is important for a variety of tasks, from forecasting to substituting for unmeasured states in feedback controllers. Performing real-time state estimation for PDEs using provably and rapidly converging observers, such as those based on PDE backstepping, is computationally expensive and in many cases prohibitive. We propose a framework for accelerating PDE observer computations using learning-based approaches that are much faster while maintaining accuracy. In particular, we employ the recently-developed Fourier Neural Operator (FNO) to learn the functional mapping from the initial observer state and boundary measurements to the state estimate. By employing backstepping observer gains for previously-designed observers with particular convergence rate guarantees, we provide numerical experiments that evaluate the increased computational efficiency gained with FNO. We consider the state estimation for three benchmark PDE examples motivated by applications: first, for a reaction-diffusion (parabolic) PDE whose state is estimated with an exponential rate of convergence; second, for a parabolic PDE with exact prescribed-time estimation; and, third, for a pair of coupled first-order hyperbolic PDEs that modeling traffic flow density and velocity. The ML-accelerated observers trained on simulation data sets for these PDEs achieves up to three orders of magnitude improvement in computational speed compared to classical methods. This demonstrates the attractiveness of the ML-accelerated observers for real-time state estimation and control.

Rafael Vazquez, Miroslav Krstic

An explicit output-feedback boundary feedback law is introduced that stabilizes an unstable linear constant-coefficient reaction-diffusion equation on an $n$-ball (which in 2-D reduces to a disk and in 3-D reduces to a sphere) using only measurements from the boundary. The backstepping method is used to design both the control law and a boundary observer. To apply backstepping the system is reduced to an infinite sequence of 1-D systems using spherical harmonics. Well-posedness and stability are proved in the $H^1$ space. The resulting control and output injection gain kernels are the product of the backstepping kernel used in control of one-dimensional reaction-diffusion equations and a function closely related to the Poisson kernel in the $n$-ball.

Iasson Karafyllis, Maria Kontorinaki, Miroslav Krstic

We provide estimates for the asymptotic gains of the displacement of a vibrating string with endpoint forcing, modeled by the wave equation with Kelvin-Voigt and viscous damping and a boundary disturbance. Two asymptotic gains are studied: the gain in the L2 spatial norm and the gain in the spatial sup norm. It is shown that the asymptotic gain property holds in the L2 norm of the displacement without any assumption for the damping coefficients. The derivation of the upper bounds for the asymptotic gains is performed by either employing an eigenfunction expansion methodology or by means of a small-gain argument, whereas a novel frequency analysis methodology is employed for the derivation of the lower bounds for the asymptotic gains. The graphical illustration of the upper and lower bounds for the gains shows that that the asymptotic gain in the L2 norm is estimated much more accurately than the asymptotic gain in the sup norm.

Iasson Karafyllis, Miroslav Krstic

We construct a family of globally defined dynamical systems for a nonlinear programming problem, such that: (a) the equilibrium points are the unknown (and sought) critical points of the problem, (b) for every initial condition, the solution of the corresponding initial value problem converges to the set of critical points, (c) every strict local minimum is locally asymptotically stable, (d) the feasible set is a positively invariant set, and (e) the dynamical system is given explicitly and does not involve the unknown critical points of the problem. No convexity assumption is employed. The construction of the family of dynamical systems is based on an extension of the Control Lyapunov Function methodology, which employs extensions of LaSalle's theorem and are of independent interest. Examples illustrate the obtained results.

Iasson Karafyllis, Miroslav Krstic

This paper provides global exponential stabilization results by means of boundary feedback control for 1-D nonlinear unstable reaction-diffusion Partial Differential Equations (PDEs) with nonlinearities of superlinear growth. The class of systems studied are parabolic PDEs with nonlinear reaction terms that provide "damping" when the norm of the state is large (the class includes reaction-diffusion PDEs with polynomial nonlinearities). The case of Dirichlet actuation at one end of the domain is considered and a Control-Lypunov Functional construction is applied in conjunction with Stampacchia's truncation method. The paper also provides several important auxiliary results; among which is an extension of Wirtinger's inequality, used here for the construction of the Control Lyapunov functional.

Shumon Koga, Miroslav Krstic

This paper considers a safe trajectory tracking of the Stefan problem with a second-order moving boundary dynamics. The model is given by a parabolic Partial Differential Equation (PDE) defined on a time-varying domain of moving boundary governed by a second-order Ordinary Differential Equation (ODE) associated with the Neumann boundary condition. A feedforward control is designed by a series expansion approach to solve the inverse Stefan problem under given reference trajectory of the moving boundary, and the convergence of infinite series is proven. A trajectory tracking controller is derived based on an energy-shaping, which ensures the safety of the model constraint in the closed-loop system. The closed-loop system is also shown to be globally exponentially stable with respect to the tracking error by performing PDE backstepping transformation and Lyapunov analysis. Numerical simulation illustrates an effective tracking performance of the proposed method under a sinusoidal reference trajectory. Code is released at https://github.com/shumon0423/StefanTracking_ACC2026.git.

Iasson Karafyllis, Michael Malisoff, Miroslav Krstic

We study a feedback stabilization problem for a first-order hyperbolic partial differential equation. The problem is inspired by the stabilization of equilibrium age profiles for an age-structured chemostat, using the dilution rate as the control. Two distinguishing features of the problem are that (a) the PDE has a multiplicative (instead of an additive) input and (b) the state is fed back to the inlet boundary. We provide a sampled-data feedback that ensures stabilization under arbitrarily sparse sampling and that satisfies input constraints. Our chemostat feedback does not require measurement of the age profile, nor does it require exact knowledge of the model.

Iasson Karafyllis, Miroslav Krstic

The paper provides results for the application of boundary feedback control with Zero-Order-Hold (ZOH) to 1-D linear, first-order, hyperbolic systems with non-local terms on bounded domains. It is shown that the emulation design based on the recently proposed continuous-time, boundary feedback, designed by means of backstepping, guarantees closed-loop exponential stability, provided that the sampling period is sufficiently small. It is also shown that, contrary to the parabolic case, a smaller sampling period implies a faster convergence rate with no upper bound for the achieved convergence rate. The obtained results provide stability estimates for the sup-norm of the state and robustness with respect to perturbations of the sampling schedule is guaranteed.

Iasson Karafyllis, Miroslav Krstic

This work provides stability results in the spatial sup norm for hyperbolic-parabolic loops in one spatial dimension. The results are obtained by an application of the small-gain stability analysis. Two particular cases are selected for the study because they contain challenges typical of more general systems (to which the results are easily generalizable but at the expense of less pedagogical clarity and more notational clutter): (i) the feedback interconnection of a parabolic PDE with a first-order zero-speed hyperbolic PDE with boundary disturbances, and (ii) the feedback interconnection, by means of a combination of boundary and in-domain terms, of a parabolic PDE with a first-order hyperbolic PDE. The first case arises in the study of the movement of chemicals underground and includes the wave equation with Kelvin-Voigt damping as a subcase. The second case arises when we apply backstepping to a pair of hyperbolic PDEs that is obtained by ignoring diffusion phenomena. Moreover, the second case arises in the study of parabolic PDEs with distributed delays. In the first case, we provide sufficient conditions for ISS in the spatial sup norm with respect to boundary disturbances. In the second case, we provide (delay-independent) sufficient conditions for exponential stability in the spatial sup norm.

Miroslav Krstic, Iasson Karafyllis, Luke Bhan et al.

Age-structured predator-prey integro-partial differential equations provide models of interacting populations in ecology, epidemiology, and biotechnology. A key challenge in feedback design for these systems is the scalar $ζ$, defined implicitly by the Lotka-Sharpe nonlinear integral condition, as a mapping from fertility and mortality rates to $ζ$. To solve this challenge with operator learning, we first prove that the Lotka-Sharpe operator is Lipschitz continuous, guaranteeing the existence of arbitrarily accurate neural operator approximations over a compact set of fertility and mortality functions. We then show that the resulting approximate feedback law preserves semi-global practical asymptotic stability under propagation of the operator approximation error through various other nonlinear operators, all the way through to the control input. In the numerical results, not only do we learn ``once-and-for-all'' the canonical Lotka-Sharpe (LS) operator, and thus make it available for future uses in control of other age-structured population interconnections, but we demonstrate the online usage of the neural LS operator under estimation of the fertility and mortality functions.

Iasson Karafyllis, Dionysios Theodosis, Miroslav Krstic

In this paper we study the stability properties of the equilibrium point for an age-structured chemostat model with renewal boundary condition and coupled substrate dynamics under constant dilution rate. This is a complex infinite-dimensional feedback system. It has two feedback loops, both nonlinear. A positive static loop due to reproduction at the age-zero boundary of the PDE, counteracted and dominated by a negative dynamic loop with the substrate dynamics. The derivation of explicit sufficient conditions that guarantee global stability estimates is carried out by using an appropriate Lyapunov functional. The constructed Lyapunov functional guarantees global exponential decay estimates and uniform global asymptotic stability with respect to a measure related to the Lyapunov functional. From a biological perspective, stability arises because reproduction is constrained by substrate availability, while dilution, mortality, and substrate depletion suppress transient increases in biomass before age-structure effects can amplify them. The obtained results are applied to a chemostat model from the literature, where the derived stability condition is compared with existing results that are based on (necessarily local) linearization methods.

Luke Bhan, Yuanyuan Shi, Miroslav Krstic

Neural operator approximations of the gain kernels in PDE backstepping has emerged as a viable method for implementing controllers in real time. With such an approach, one approximates the gain kernel, which maps the plant coefficient into the solution of a PDE, with a neural operator. It is in adaptive control that the benefit of the neural operator is realized, as the kernel PDE solution needs to be computed online, for every updated estimate of the plant coefficient. We extend the neural operator methodology from adaptive control of a hyperbolic PDE to adaptive control of a benchmark parabolic PDE (a reaction-diffusion equation with a spatially-varying and unknown reaction coefficient). We prove global stability and asymptotic regulation of the plant state for a Lyapunov design of parameter adaptation. The key technical challenge of the result is handling the 2D nature of the gain kernels and proving that the target system with two distinct sources of perturbation terms, due to the parameter estimation error and due to the neural approximation error, is Lyapunov stable. To verify our theoretical result, we present simulations achieving calculation speedups up to 45x relative to the traditional finite difference solvers for every timestep in the simulation trajectory.

Ji Wang, Miroslav Krstic

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.

Iasson Karafyllis, Miroslav Krstic

Eduardo Sontag and coauthors studied Input-to-Output Stability (IOS) and the output asymptotic gain property. These notions changed control theory and recently had an impact on robust adaptive control through the Deadzone-Adapted Disturbance Suppression (DADS) control scheme. Moreover, recently the notion of IOS was extended to systems described by Partial Differential Equations (PDEs). In this work, we celebrate Eduardo Sontag by combining DADS and IOS for PDEs: we study the partial-state regulation problem for a scalar Ordinary Differential Equation (ODE) which is interconnected with a possibly infinite-dimensional system. In such a case the DADS control scheme can allow an escape from the requirements of the small-gain theorem that is mainly used for partial-state feedback. We show the design procedure of partial-state DADS controllers and we prove robust regulation even in the presence of external inputs (disturbances) without assuming knowledge of any disturbance/parameter bounds. The DADS controller is applied to three different cases of the interconnection of an ODE with an almost completely unknown: (a) heat PDE, (b) transport PDE, and (c) wave PDE with viscous damping. We show that the same DADS controller can achieve robust regulation in all three cases.

Maxence Lamarque, Luke Bhan, Yuanyuan Shi et al.

To stabilize PDEs, feedback controllers require gain kernel functions, which are themselves governed by PDEs. Furthermore, these gain-kernel PDEs depend on the PDE plants' functional coefficients. The functional coefficients in PDE plants are often unknown. This requires an adaptive approach to PDE control, i.e., an estimation of the plant coefficients conducted concurrently with control, where a separate PDE for the gain kernel must be solved at each timestep upon the update in the plant coefficient function estimate. Solving a PDE at each timestep is computationally expensive and a barrier to the implementation of real-time adaptive control of PDEs. Recently, results in neural operator (NO) approximations of functional mappings have been introduced into PDE control, for replacing the computation of the gain kernel with a neural network that is trained, once offline, and reused in real-time for rapid solution of the PDEs. In this paper, we present the first result on applying NOs in adaptive PDE control, presented for a benchmark 1-D hyperbolic PDE with recirculation. We establish global stabilization via Lyapunov analysis, in the plant and parameter error states, and also present an alternative approach, via passive identifiers, which avoids the strong assumptions on kernel differentiability. We then present numerical simulations demonstrating stability and observe speedups up to three orders of magnitude, highlighting the real-time efficacy of neural operators in adaptive control. Our code (Github) is made publicly available for future researchers.

Luke Bhan, Yuexin Bian, Miroslav Krstic et al.

Over the last decade, data-driven methods have surged in popularity, emerging as valuable tools for control theory. As such, neural network approximations of control feedback laws, system dynamics, and even Lyapunov functions have attracted growing attention. With the ascent of learning based control, the need for accurate, fast, and easy-to-use benchmarks has increased. In this work, we present the first learning-based environment for boundary control of PDEs. In our benchmark, we introduce three foundational PDE problems - a 1D transport PDE, a 1D reaction-diffusion PDE, and a 2D Navier-Stokes PDE - whose solvers are bundled in an user-friendly reinforcement learning gym. With this gym, we then present the first set of model-free, reinforcement learning algorithms for solving this series of benchmark problems, achieving stability, although at a higher cost compared to model-based PDE backstepping. With the set of benchmark environments and detailed examples, this work significantly lowers the barrier to entry for learning-based PDE control - a topic largely unexplored by the data-driven control community. The entire benchmark is available on Github along with detailed documentation and the presented reinforcement learning models are open sourced.

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

Due to simplicity and strong stability guarantees, predictor feedback methods have stood as a popular approach for time delay systems since the 1950s. For time-varying delays, however, implementation requires computing a prediction horizon defined by the inverse of the delay function, which is rarely available in closed form and must be approximated. In this work, we formulate the inverse delay mapping as an operator learning problem and study predictor feedback under approximation of the prediction horizon. We propose two approaches: (i) a numerical method based on time integration of an equivalent ODE, and (ii) a data-driven method using neural operators to learn the inverse mapping. We show that both approaches achieve arbitrary approximation accuracy over compact sets, with complementary trade-offs in computational cost and scalability. Building on these approximations, we then develop an output-feedback predictor design for systems with delays in both the input and the measurement. We prove that the resulting closed-loop system is globally exponentially stable when the prediction horizon is approximated with sufficiently small error. Lastly, numerical experiments validate the proposed methods and illustrate their trade-offs between accuracy and computational efficiency.

Luke Bhan, Peter Quawas, Miroslav Krstic et al.

Modern control systems frequently operate under input delays and sampled state measurements. A common delay-compensation strategy is predictor feedback; however, practical implementations require solving an implicit ODE online, resulting in intractable computational cost. Moreover, predictor formulations typically assume continuously available state measurements, whereas in practice measurements may be sampled, irregular, or temporarily missing due to hardware faults. In this work, we develop two neural-operator predictor-feedback designs for nonlinear systems with delayed inputs and sampled measurements. In the first design, we introduce a sampling-horizon prediction operator that maps the current measurement and input history to the predicted state trajectory over the next sampling interval. In the second design, the neural operator approximates only the delay-compensating predictor, which is then composed with the closed-loop flow between measurements. The first approach requires uniform sampling but yields residual bounds that scale directly with the operator approximation error. In contrast, the second accommodates non-uniform, but bounded sampling schedules at the cost of amplified approximation error, revealing a practical tradeoff between sampling flexibility and approximation sensitivity for the control engineer. For both schemes, we establish semi-global practical stability with explicit neural operator error-dependent bounds. Numerical experiments on a 6-link nonlinear robotic manipulator demonstrate accurate tracking and substantial computational speedup of 25$\times$ over a baseline approach.

Bo Wang, Miroslav Krstic

We develop an optimization-free framework for safe stabilization of single-input control-affine nonlinear systems with a given control Lyapunov function (CLF) and a given control barrier function (CBF), where the desired equilibrium lies in the interior of the safe set. An explicit compatibility condition is derived that is necessary and sufficient for the pointwise simultaneous satisfaction of the CLF and CBF inequalities. When this condition holds, two closed-form continuous state-feedback laws are constructed from the Lie-derivative data of the CLF and CBF via standard universal stabilizer formulas, yielding asymptotic stabilization of the origin and forward invariance of the interior of the safe set, without online quadratic programming. The two laws belong to broader families parametrized by a free nondecreasing function, providing additional design flexibility. When the compatibility condition fails, a safety-prioritizing modification preserves forward invariance and drives the state toward the safe-set boundary until a compatible region is reached, whereupon continuity at the origin and asymptotic stabilization are recovered. The framework produces families of explicit constructive alternatives to CLF-CBF quadratic programming for scalar-input nonlinear systems.

Miroslav Krstic

Alberto Isidori's framework of geometric nonlinear control, and particularly of feedback linearization, is the inspiration behind PDE backstepping: apply a transfromation of the state to cast the plant into a canonical form, bring all the non-canonical effects within the "span" of (boundary) control, and close the design with a feedback that makes the closed loop evolve in accordance with well-studied stable dynamics. The specificity of this approach is that, for PDEs, there is not one canonical form (like Brunovsky for ODEs) but the canonical forms are PDE-class-specific. When conducting this process for nonlinear PDEs, where the "transformation of the state" is performed using a nonlinear Volterra series indexed by the spatial variable, enormous technical challenges arise. One has to deal with kernels governed by PDEs on simplex domains growing in dimension to infinity, capture the growth rates of these kernels of the "direct transformation," and conduct the same for the "inverse transformation" without directly studying its Volterra kernels. So far, this agenda has been executed only once, two decades ago: for parabolic PDEs by Vazquez and Krstic [Automatica, 2008]. Generalization attempts have not followed because of the immense complexity involved in feedback-linearizing nonlinear PDEs. In this paper, dedicated to Professor Isidori, we convert the PDE feedback-linearizing methodology of 2008 from the parabolic to a hyperbolic class and, for a transport-adapted subclass of Chen-Fliess series, construct controllers without kernel PDEs.

Maxence Lamarque, Luke Bhan, Rafael Vazquez et al.

To stabilize PDE models, control laws require space-dependent functional gains mapped by nonlinear operators from the PDE functional coefficients. When a PDE is nonlinear and its "pseudo-coefficient" functions are state-dependent, a gain-scheduling (GS) nonlinear design is the simplest approach to the design of nonlinear feedback. The GS version of PDE backstepping employs gains obtained by solving a PDE at each value of the state. Performing such PDE computations in real time may be prohibitive. The recently introduced neural operators (NO) can be trained to produce the gain functions, rapidly in real time, for each state value, without requiring a PDE solution. In this paper we introduce NOs for GS-PDE backstepping. GS controllers act on the premise that the state change is slow and, as a result, guarantee only local stability, even for ODEs. We establish local stabilization of hyperbolic PDEs with nonlinear recirculation using both a "full-kernel" approach and the "gain-only" approach to gain operator approximation. Numerical simulations illustrate stabilization and demonstrate speedup by three orders of magnitude over traditional PDE gain-scheduling. Code (Github) for the numerical implementation is published to enable exploration.

Luke Bhan, Peijia Qin, Miroslav Krstic et al.

Predictor feedback designs are critical for delay-compensating controllers in nonlinear systems. However, these designs are limited in practical applications as predictors cannot be directly implemented, but require numerical approximation schemes, which become computationally prohibitive when system dynamics are expensive to compute. To address this challenge, we recast the predictor design as an operator learning problem, and learn the predictor mapping via a neural operator. We prove the existence of an arbitrarily accurate neural operator approximation of the predictor operator. Under the approximated predictor, we achieve semiglobal practical stability of the closed-loop nonlinear delay system. The estimate is semiglobal in a unique sense - one can enlarge the set of initial states as desired, though this increases the difficulty of training a neural operator, which appears practically in the stability estimate. Furthermore, our analysis holds for any black-box predictor satisfying the universal approximation error bound. We demonstrate the approach by controlling a 5-link robotic manipulator with different neural operator models, achieving significant speedups compared to classic predictor feedback schemes while maintaining closed-loop stability.

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

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.

Luke Bhan, Yuanyuan Shi, Iasson Karafyllis et al.

Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs.

Luke Bhan, Miroslav Krstic, Yuanyuan Shi

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.

Filip Bajraktari, Luke Bhan, Miroslav Krstic et al.

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.

Iasson Karafyllis, Miroslav Krstic

This paper presents a novel methodology for the design of boundary feedback stabilizers for 1-D, semilinear, parabolic PDEs. The methodology is based on the use of small-gain arguments and can be applied to parabolic PDEs with nonlinearities that satisfy a linear growth condition. The nonlinearities may contain nonlocal terms. Two different types of boundary feedback stabilizers are constructed: a linear static boundary feedback and a nonlinear dynamic boundary feedback. It is also shown that there are fundamental limitations for feedback design in the parabolic case: arbitrary gain assignment is not possible by means of boundary feedback. An example with a nonlocal nonlinear term illustrates the applicability of the proposed methodology.

Iasson Karafyllis, Miroslav Krstic

In this work decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the L2 and H1 norms of the solution and discontinuous disturbances are allowed. Although an eigenfunction expansion for the solution is exploited for the proof of the decay estimates, the estimates do not require knowledge of the eigenvalues and the eigenfunctions of the corresponding Sturm-Liouville operator. Examples show that the obtained results can be applied for the stability analysis of parabolic PDEs with nonlocal terms.

Iasson Karafyllis, Miroslav Krstic

For general nonlinear control systems we present a novel approach to adaptive control, which employs a certainty equivalence (indirect) control law and an identifier with event-triggered updates of the plant parameter estimates, where the triggers are based on the size of the plant's state and the updates are conducted using a non-recursive least-squares estimation over certain finite time intervals, with updates employing delayed measurements of the state. With a suitable non-restrictive parameter-observability assumption, our adaptive controller guarantees global stability, regulation of the plant state, and our identifier achieves parameter convergence, in finite time, even in the absence of persistent excitation, for all initial conditions other than those where the initial plant state is zero. The robustness of our event-triggered adaptive control scheme to vanishing and non-vanishing disturbances is verified in simulations with the assistance of a dead zone-like modification of the update law. The major distinctions of our approach from supervisory adaptive schemes is that our approach is indirect and our triggering is related to the control objective (the regulation error). The major distinction from the classical indirect Lyapunov adaptive schemes based on tuning related to the regulation error is that our approach does not involve a complex redesign of the controller to compensate for the detrimental effects of rapid tuning on the transients by incorporating the update law into the control law. Instead, our approach allows for the first time to use a simple certainty equivalence adaptive controller for general nonlinear systems. All proofs are given in a companion paper.

Iasson Karafyllis, Miroslav Krstic

We present the stability analysis for the new regulation-triggered approach to adaptive control introduced in a companion paper. Due to the fact that the closed-loop system is hybrid, our proofs have essential differences from the conventional adaptive control proofs, where the Lyapunov analysis either encompasses the complete closed-loop state or is done in multiple steps through comparison or Gronwall-Bellman lemmas. In addition, we present a convenient algorithm for checking our parameter-observability assumption, which involves repeated Lie derivatives of appropriate vector fields and can be applied to the class of nonlinear control systems for which at most one unknown parameter appears in each differential equation.

Iasson Karafyllis, Miroslav Krstic

Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-to-state stability (ISS) properties and finite gains cannot be established with respect to disturbances at the boundary. This intuition has been reinforced by many unsuccessful attempts, as well as by the success in establishing ISS only with respect to the derivative of the disturbance. Contrary to this intuition, we establish such a result for parabolic PDEs. Our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time-varying subsets of the state space. The obtained results are used for the comparison of the gain coefficients of transport PDEs with respect to inlet disturbances and for the establishment of the ISS property with respect to control actuator errors for parabolic systems under boundary feedback control.