# Miroslav Krstić

Miroslav Krstić (Serbian Cyrillic: Мирослав Крстић) is an American control theorist and Distinguished Professor of Mechanical and Aerospace Engineering at the University of California, San Diego (UCSD). Krstić is also the director of the Center for Control Systems and Dynamics at UCSD and a Senior Associate Vice Chancellor for Research. In the list of notable researchers in systems and control, he is the youngest.

## Education

After completing his 5-year BSc degree from University of Belgrade's School of Electrical Engineering in 1989 in the top 1% of his class, and following two years of teaching at University of Belgrade, Krstić moved to the United States for graduate studies in 1991. He wrote his first journal paper a few weeks upon arrival, with a solution that has transformed adaptive control. He received his MSc in electrical engineering in 1992, and his PhD in 1995 (defended in December 1994), from University of California, Santa Barbara with Petar Kokotovic.
Krstić's 455-page PhD dissertation earned the campuswide Lancaster Best Dissertation Award at UC Santa Barbara and was published a few months later, with expansions, by John Wiley and Sons.
Krstić received two Best Student Paper Awards. First, at the 1993 IEEE Conference on Decision and Control, for his paper on the nonlinear swapping approach to adaptive nonlinear control. Second, at the 1996 American Control Conference, for his single-authored paper on invariant manifolds in adaptive control.
Additionally, for single-authored papers written while a PhD student, Krstić earned the O. Hugo Schuck and George S. Axelby outstanding paper awards.

## Faculty career

After receiving his PhD in 1995, and serving as assistant professor at University of Maryland for two years, Krstić was recruited as associate professor by University of California, San Diego (UCSD) in 1997, and promoted to full professor three years later (2000).
Krstić’s grants from those five years include NSF Career, ONR YIP, and PECASE from President Clinton.
Since 2009, Krstić has held the Alspach endowed chair, and in 2015 was promoted to Distinguished Professor.
In 2005, Krstić became the first engineering professor to receive the UC San Diego Chancellor's Award for Research.
In 2008 Krstić founded the Cymer Center for Control Systems and Dynamics and remains its director. Since 2012 he has served as Senior Associate Vice-Chancellor for Research at UCSD.

## Research in Control Theory

Krstić is a co-author of 18 books, about 500 journal papers, and is the highest-published author in both of the flagship control systems journals, Automatica and IEEE Transactions on Automatic Control (according to Scopus), with more than 110 papers in each of the two journals.
* NONLINEAR and ADAPTIVE CONTROL. Krstić’s 1995 book with Kanellakopoulos and Kokotovic, an expanded version of his PhD dissertation, pioneered adaptive stabilization methods for nonlinear systems with unknown parameters and is the second highest-cited control monograph. (The top-cited control monograph is Boyd et al.) Krstić introduced the tuning-function designs, modular designs, nonlinear swapping, passivity-based identifiers, adaptive CLFs and ISS-CLFs, and output-feedback adaptive nonlinear and linear controllers based on backstepping. STOCHASTIC STABILIZATION. Krstić and his student Deng developed stabilizing controllers for stochastic nonlinear systems, introduced ISS-CLFs to stochastic systems, and designed differential game controllers that achieve, with probability one, peak-to-peak gain function assignment with respect to unknown noise covariance.
* EXTREMUM SEEKING. Krstić pioneered, for general nonlinear dynamical systems, extremum seeking (ES) as an approach for real-time model-free optimization. To establish stability and performance guarantees, he introduced a combination of averaging and singular perturbation techniques to establish exponential stability. Among Krstić’s advances of ES is source seeking for autonomous vehicles, model-free Nash equilibrium seeking for non-cooperative games, Newton-based ES for model-free assignment of convergence rate, model-free stabilization by minimum-seeking of CLFs, and ES for maps with large delays and PDEs with Oliveira. STOCHASTIC AVERAGING AND STOCHASTIC EXTREMUM SEEKING. In introducing stochastic ES, Krstić and his postdoc Liu generalized stochastic averaging theorems by removing the restrictions of global Lipschitzness, global exponential stability of the average system, vanishing noise, and finiteness of time horizon.
* PDE BACKSTEPPING. Krstić generalized backstepping control from ODEs to PDEs. His IFAC Chestnut prize-winning book with his student Smyshlyaev provides an accessible introduction to PDE backstepping. PDE backstepping uses explicit and invertible Volterra-type integral transformations, with spatial integration kernels governed by linear PDEs of Goursat type on triangular domains. His general methodology stabilizes PDEs of parabolic and hyperbolic types, as well as of higher orders in space (Korteweg-De Vries, Schroedinger, Kuramoto-Sivashinsky, beams, etc.). For PDEs with unknown parameters, Krstić developed adaptive controllers. Krstic applied PDE backstepping to traffic flows with his student Yu and to additive manufacturing with his student Koga. CONTROL OF NAVIER-STOKES SYSTEMS. For turbulent fluids, including electrically conducting flows with Maxwell’s PDEs (magnetohydrodynamic/MHD flows), Krstić and his student Vazquez developed flow control designs. ISS FOR PDEs. Despite the unboundedness of input operators in PDEs with boundary inputs, Krstić and Karafyllis established ISS of PDEs, developed small-gain theorems for PDEs, and enabled analysis of interconnected PDEs from different classes.
* PREDICTORS FOR NONLINEAR DELAY SYSTEMS. In his single-authored 2009 Birkhäuser book Krstić extended his hyperbolic PDE results to nonlinear ODEs with delays. He introduced nonlinear predictor operators of the infinite-dimensional delay state and launched control of interconnected PDE-ODE and PDE-PDE systems. In three subsequent books, Krstić and collaborators generalized the predictors to time- and state-dependent delays, to delay-adaptive control for unknown delays, and to sampled-data implementation.
* PRESCRIBED-TIME CONTROL. Krstić introduced time-varying techniques for the design and analysis of controllers and observers that achieve stabilization in user-prescribed time, independent of initial conditions, and even in the presence of deterministic disturbances and stochastic disturbances.
* SAFE, NON-OVERSHOOTING NONLINEAR CONTROL. In his 2006 paper, Krstić pioneered a backstepping procedure for guaranteeing what is now referred to as "safety" and back then as "non-overshooting" control. He provided designs for CBFs of high relative degree and for achieving, for systems with unmatched disturbances, what is now referred to as input-to-state safety (ISSf). He extended deterministic non-overshooting control to nonlinear systems with stochastic disturbances. He extended his prescribed-time (PT) idea from stabilization to safety, introducing PT safety filters, which reduce the restrictiveness of conventional exponential safety filters. He converted his results on inverse optimal stabilization to inverse optimal safe control, where a safety filter simultaneously maximizes safety and liveness, over the entire infinite time horizon. He generalized safety control from ODEs to PDEs.
* MACHINE LEARNING FOR PDE CONTROL. In his 2023 IEEE Bode Lecture, Krstić introduced deep neural operators for off-line learning of PDE backstepping designs for hyperbolic and parabolic PDEs, to enable online use of gain-scheduling backstepping for nonlinear PDEs and adaptive backstepping for PDEs with unknown parameters. He pioneered a roadmap for both the approximation theory for gain kernel PDEs and for stability guarantees under neural network approximations of the PDE backstepping controllers.
An enumerated and explained list of the most well known topics that Krstić has pioneered is in the section "Concepts and methods introduced by Krstić" below.
In control systems, Krstić is among the highest-cited researchers, with a Google Scholar h-index over 135. Among the living mechanical and aerospace engineers in the U.S., Krstić is among the ten highest cited according to Research.com and Google Scholar.
Krstić is Editor-in-Chief of Systems & Control Letters and has been senior editor in Automatica and IEEE Transactions on Automatic Control.

## Industry Work

Krstić has impacted technology development in extreme ultraviolet lithography in semiconductor manufacturing, advanced arresting gear on the newest aircraft carrier class Gerald Ford, the ChemCam laser-based spectroscopy on NASA Mars rover Curiosity, charged particle accelerators, oil drilling, nuclear fusion, and in Lithium-ion battery management systems.
CHIP PHOTOLITHOGRAPHY: Cymer Inc., a San Diego company with which Krstić co-founded the Cymer Center for Control Systems and Dynamics in 2008, employed Krstić’s extremum seeking (ES) technology in 2012 to stabilize extreme ultraviolet (EUV) light sources. This boosted chip density 220-fold: from 193 nm resolution to 13 nm. Krstić’s 2002 discrete ES algorithm is the basis of the seminal 2013 US Patent 8598552B1, by his 4 ES trainees (Drs. Frihauf, Riggs, Graham, Dunstan), who transitioned Krstić's ES technology as employees at Cymer. Shortly upon stabilizing EUV with extremum seeking, Cymer was acquired by ASML, for $3.7B. EUV is a $10B/yr industry in 2024. EUV is used by Intel, IBM, Samsung, and TSMC.
AIRCRAFT CARRIERS: In 2014-2019, employed by General Atomics Electromagnetic Systems (San Diego) as a consultant, Krstić led his 4 former PhD trainees, hired by GA (Drs. G. Prior, N. Ghods, P. Frihauf, C. Kinney), in the control design and performance analysis for electromagnetic advanced arresting gear (AAG). Their controllers now manage all arrestments on the aircraft “supercarrier” USS Gerald R. Ford (CVN-78). In the US Navy 2020 video the controllers by Krstić's team manage the landings of F/A-18 Super Hornet, C-2A Greyhound, and F-14 Tomcat aircraft. Since 2020, these controllers have enabled tens of thousands of aircraft landings on USS Ford, including over 10,000 landings in 2024 alone. GA’s AAG is due to be also installed on USS Kennedy (CVN-79) and Enterprise (CVN-80).
ACCELERATORS: Krstić and his students introduced the ES methodology to the field of charged particle accelerators. His PhD graduate Dr. A. Scheinker implemented ES on the Los Alamos Lab's 1-km LANSCE accelerator, at several other U.S. Department of Energy labs (Lawrence Berkeley, Stanford Linear Accelerator, Argonne), and at other world-leading accelerators (CERN in Switzerland and Germany's Elektronen-Synchrotron DESY). ES cuts accelerator re-tuning after upgrade from weeks to minutes.
MARS ROVER: The MS thesis supervised by Krstić supplied the auto-focus algorithm for Mars Rover Curiosity’s ChemCam System, which performs chemical testing of Martian rocks.
OTHER DEVELOPMENTS FOR INDUSTRY AND GOVERNMENT: Statoil (Norway) and Krstić, with collaborators, implemented his adaptive PDE backstepping observers for downhole pressure on a 700-meter underbalanced drilling oil rig. Krstić, Bosch, and his students, under ARPA-E contract, developed Li-ion battery estimators to reduce charging time to 15 min. With General Atomics, Krstić designed controllers for D-IIID tokamak fusion reactor. With Livermore Lab, Krstić implemented his ES to optimize HCCI engines. With Ford, Krstić and his student developed a PDE controller for automotive catalytic converters. Krstić and his student Krieger employed at Northrop-Grumman developed ES for endurance maximization for UAVs (such as, e.g., the Global Hawk). With United Technologies, Krstić implemented ES to stabilize combustion and compressor instabilities in Pratt & Whitney's jet engines. For US Navy, Krstić developed controls for air-cushioned ship called T-craft.

## Awards

In the list of eminent researchers in systems and control, Krstić is one of the recipients of the highest number of lifetime achievement awards. His awards include
* IEEE Hendrik W. Bode Lecture Prize
* AACC Richard E. Bellman Control Heritage Award
* ASME Rufus Oldenburger Medal
* SIAM W. T. and Idalia Reid Prize
* A. V. "Bal" Balakrishnan Award for Research in Mathematics of Systems (inaugural recipient)
* AACC John R. Ragazzini Award
* IFAC Harold Chestnut Control Engineering Textbook Prize
* IFAC TC Award on Non-Linear Control Systems
* IFAC TC Distributed Parameter Systems Ruth F. Curtain Award (inaugural recipient)
* IFAC TC Award on Adaptive and Learning Systems
* ASME Henry Paynter Outstanding Investigator Award
* ASME Nyquist Lecture
* IEEE Control Systems Society Distinguished Member Award
* First engineering recipient of the UC San Diego Chancellor's Associates Award for Excellence in Science & Engineering Research (immediately following the Nobel laureate Roger Tsien)
* Presidential Early Career Award for Scientists and Engineers
* Office of Naval Research Young Investigator Award
* National Science Foundation Career Award
* George Axelby Outstanding Paper Award (IEEE Transactions on Automatic Control)
* O. Hugo Schuck Best Paper Award (1996 and 2019)
* Foreign Member, Serbian Academy of Sciences and Arts
Krstić has contributed to control systems in electrical, mechanical, and aerospace engineering, as well as in mathematics and physics. As a result, he is Fellow of seven scholarly societies in those disciplines: Institute of Electrical and Electronics Engineers, International Federation of Automatic Control, Society for Industrial and Applied Mathematics, American Society of Mechanical Engineers, Institution of Engineering and Technology, American Association for the Advancement of Science as well as an associate fellow of the American Institute of Aeronautics and Astronautics.
For launching several new control system directions, Krstić has been recognized by International Federation of Automatic Control (IFAC) with a trifecta of triennial technical awards: IFAC TC Award on Non-Linear Control Systems, IFAC TC Distributed Parameter Systems Ruth F. Curtain Award, and IFAC TC Award on Adaptive and Learning Systems. Each of the three areas is large, with a decades-long IFAC symposium series. Krstić is the only researcher to receive such triennial awards for lifetime achievement in more than one controls area.
For his 50th birthday, as a tribute to his legacy, Krstić's colleagues published a monograph on nonlinear delay systems.

## Books

* Nonlinear and Adaptive Control Design (1995), co-authored with Ioannis Kanellakopoulos and Petar Kokotovic; John Wiley and Sons. ISBN 0-471-12732-9
* Stabilization of Nonlinear Uncertain Systems (1998), co-authored with Hua Deng; Springer. ISBN 1-85233-020-1
* Flow Control by Feedback (2002), co-authored with Ole Morten Aamo; Springer. ISBN 1-85233-669-2
* Real-Time Optimization by Extremum Seeking Feedback (2003), co-authored with Kartik B. Ariyur; John Wiley and Sons. ISBN 0-471-46859-2
* Control of Turbulent and Magnetohydrodynamic Channel Flows (2007), co-authored with Rafael Vazquez; Birkhauser. ISBN 978-0-8176-4698-1
* Boundary Control of PDEs: A Course on Backstepping Designs (2008), co-authored with Andrey Smyshlyaev; SIAM. ISBN 978-0-89871-650-4
* Delay Compensation for Nonlinear, Adaptive, and PDE Systems (2009); Birkhauser. ISBN 978-0-8176-4698-1
* Adaptive Control of Parabolic PDEs (2010), co-authored with Andrey Smyshlyaev; Princeton University Press. ISBN 978-0691142869
* Stochastic Averaging and Stochastic Extremum Seeking (2012), co-authored with Shu-Jun Liu; Springer. ISBN 978-1-4471-4086-3
* Nonlinear Control Under Nonconstant Delays (2013), co-authored with Nikolaos Bekiaris-Liberis; SIAM. ISBN 978-1-61197-284-9
* Predictor Feedback for Delay Systems: Implementations and Approximations (2017), coauthored with Iasson Karafyllis; Birkhauser, ISBN 978-3-319-42377-7
* Model-Free Stabilization by Extremum Seeking (2017), co-authored with Alexander Scheinker; Springer. ISBN 978-3-319-50790-3
* Input-to-State Stability for PDEs (2018), co-authored with Iasson Karafyllis; Springer. ISBN 978-3-319-91011-6
* Delay-Adaptive Linear Control (2019), co-authored with Yang Zhu; Princeton University Press. ISBN 9780691202549
* Materials Phase Change PDE Control & Estimation: From Additive Manufacturing to Polar Ice (2020), co-authored with Shumon Koga; Springer. ISBN 978-3-030-58490-0
* PDE Control of String-Actuated Motion (2022); co-authored with Ji Wang, Princeton University Press. ISBN 9780691233499
* Extremum Seeking through Delays and PDEs (2022), co-authored with Tiago Roux Oliveira, SIAM. ISBN 978-1-61197-734-9
* Traffic Congestion Control by PDE Backstepping (2023), co-authored with Huan Yu, Birkhäuser. ISBN 978-3-031-19345-3

## Concepts and methods introduced by Krstić
### Adaptive Nonlinear Control

Source:
* tuning-function design
** adaptive backstepping with a single parameter estimator, for unmatched parametric uncertainties
* modular designs
** combine any parameter estimator with any ISS controller
* nonlinear swapping
** stability analysis with nonlinear filter-based gradient and least-squares parameter estimators for nonlinear systems
* passivity-based identifiers
** identifiers with observers and nonlinear damping
* adaptive CLFs and ISS-CLFs
** general frameworks for Lyapunov and ISS-based adaptive nonlinear control
* output-feedback nonlinear and linear adaptive backstepping
** adaptive observer-based controllers with K-filters

### Stochastic Nonlinear Stabilization

Source:
* stochastic backstepping
** backstepping employing Itô calculus for continuous-time stochastic systems; ensures stability in probability
* noise-to-state stability (NSS)
** ISS in probability with respect to unknown covariance of noise
* inverse optimal differential games w.r.t. noise covariance
** optimal assignment of integral gains from covariance to state

### Extremum Seeking

* stability of extremum seeking for general nonlinear dynamical systems
** analysis via singular perturbations and averaging of reduced model
* source seeking
** search/navigation in space for GPS-denied autonomous vehicles and robots
* Nash equilibrium seeking
** ES in non-cooperative game multi-agent setting
* Newton-based ES
** for model-free assignment of convergence rate and for equalizing convergence across input channels of multivariable maps; with inversion of Hessian estimate using Riccati ODE
* model-free stabilization with ES
** by minimum-seeking of CLFs (with Scheinker)
* ES for maps with large delays and PDEs
** employing PDE backstepping (with Oliveira)
* stochastic extremum seeking
** for ES with random walk perturbations, like employed by E.Coli bacteria (with Liu)
* generalized stochastic averaging
** without restrictions (of global Lipschitzness, global exp. stability of average system, vanishing noise)

### PDE Backstepping

Source:
* backstepping transformations, kernel PDEs
** transformations into desirable target PDEs
** analysis of Goursat-form PDEs for gain kernels (with Smyshlyaev)
* backstepping for parabolic and hyperbolic PDEs
** designs of full-state stabilizing feedback law and convergent observers
* backstepping for PDE-ODE and PDE-PDE cascades
** for cascades like ODEs with parabolic PDE input dynamic and reaction-diffusion PDEs with input delays
* adaptive PDE backstepping
** for PDEs (parabolic and hyperbolic) with unknown functional parameters, using Lyapunov, swapping, and passive estimators
* traffic flow stabilization
** control of ARZ PDEs (with Yu)
* additive manufacturing
** control of Stefan PDEs (with Koga)

### Navier-Stokes Stabilization and Mixing

* Backstepping for turbulent flows and MHD systems (with Vazquez)
** Stabilizing controllers and observers at high Reynolds and Hartmann numbers
* Fluid mixing by optimal de-stabilization

### ISS for PDEs

Source:
* ISS to boundary inputs (with Karafyllis)
* small-gain theorems for PDEs
** for parabolic and hyperbolic PDEs

### Predictors for Nonlinear Delay Systems

* nonlinear predictors
** including predictors for state-dependent delays (with Bekiaris-Liberis)
* delay-adaptive control
** for linear systems with unknown delays and other plant parameters (with Bresch-Pietri and Zhu)
* approximate nonlinear predictors
** for real-time and sampled-data implementation (with Karafyllis)

### Prescribed-Time Control

* PT stabilization
** backstepping for driving state to setpoint by user-prescribed time regardless of initial condition (with Song)
* PT observers
** for state estimate convergence in arbitrary finite time (with Holloway)
* stochastic PT control
** for PT stabilization in probability (with W. Li)

### Non-overshooting and Safe Nonlinear Control

* high relative degree CBFs
** recursive backstepping design of CBFs without shrinking the safe set
* ISSf gain assignment
** backstepping design of non-overshooting controllers that assign a desired input-to-state safety gain function
* stochastic non-overshooting control
** to guarantee safety at least in the mean
* prescribed-time safety filters
** to let the state reach the safety boundary by the time the prohibition on the unsafe set is lifted (as in robot-human handover)
* inverse optimal safety filters
** safety filters with safety-maximization and liveness-maximization along entire time horizon, under deterministic and stochastic disturbances
* safe control for PDEs
** for Stefan, liquid-tank, gas-piston, and chemostat (population dynamics) PDEs

### Deep Neural Operators for PDE Control

* universal approximability theory for backstepping kernel PDEs
* stability guarantees under ML approximation of PDE backstepping
** for hyperbolic and parabolic PDEs