1 Obituary to A. J. Pritchard1

On the 12th of August 2007 Tony Pritchard, professor of mathematics at the University of Warwick, died suddenly at the age of 70 from a heart attack while on holiday with his family on the island of Lanzarote. Both of the authors have had the privilege of working with him for many years. We have lost both a colleague and a friend and the following notes are dedicated to his memory.

1.1 A. J. Pritchard: biographical notes2 Anthony J. Pritchard was born on 5th March 1937 in Pon- typridd in South Wales. Although most of his upbringing and all his school education was in Bristol, England, he always regarded himself as Welsh. This feeling was reinforced when, during his university days in London, he paired up for life with Buddug, who was born in the same year in Den- bighshire, North Wales, and spoke Welsh ﬂuently. He took a B.Sc. degree in mathematics at Kings College, London in 1958. After only a year as a Ph.D. student at Imperial Col- lege, he became the proud father of a baby daughter Sˆıan. His new ﬁnancial responsibilities dictated that he leave in 1959 with a Diploma of Imperial College to take up a teaching post at Filton high school in Bristol. By 1964 three more daughters Catrin, Rhian and Ceri, had been born in quick succession, resulting in a bevy of 4 girls under the age of 5.

1R. F. Curtain, Department of Mathematics, University of Gronin- gen, P.O. Box 800, 9700 AV Groningen, The Netherlands, and D. Hinrichsen, Institut f¨urDynamische Systeme, Universit¨atBremen, Postfach 330 440, D 28334 Bremen, Germany 2Acknowledgements: We are grateful to Buddug Pritchard for helpful comments on an earlier draft of these biographical notes. We also thank Abdelhaq El Ja¨ıfor providing us some details about his collaboration with Tony.

1 In September 1966 he was appointed as a lecturer at the Department of Mathematics, Rugby College of Engineering Technology, which gave him the opportunity to take up mathematics once again. Under the supervision of Patrick Parks, he started on a Ph.D. in applied mathematics at the new university of Warwick and soon after, in September 1968, recognizing his scientific strength the founding professor of engineering, Arthur Shercliff, appointed him to the post of lecturer in the engineering department. The founding professor of mathematics, Christopher Zee- man, decreed that applied mathematicians did not belong in the mathematics department, but close to the applications of mathematics. So the applied mathematicians were distributed between the engineering and physics de- partments. Following an enquiry into the state of applied mathematics in the UK by the Royal Society, Larry Markus and Patrick Parks submitted a proposal to the SERC to set up a Control Theory Centre at the University of Warwick within the Schools of Mathematics and Engineering Sci- ence. The aim was to stimulate research and study among British scientists of the mathematical and theoretical aspects of control science. The proposal was successful and in 1970 the Control Theory Centre was born. The three founding members were the director, Larry Markus (professor of mathematics at the university of Minnesota and the university of Warwick), the assistant director, Patrick Parks, and Tony. In 1971 the first research fellows, David Wishart and Ruth Curtain, were appointed and in addition, there were funds for visitors and for an annual conference. The first in 1971 was the ”IMA Conference on Control of Systems described by Partial Differential Equations”. This relatively new research area was to be the central theme of the Control Theory Centre with Tony as the main catalyst. Since Larry Markus only spent the summer at the Univer- sity of Warwick, much of the administration was carried out by Patrick and Tony. In 1974 Larry passed the directorship on to Patrick Parks and in 1976 Tony took over the directorship. The centre developed rapidly into one of the main hubs of research in control theory in the UK and attracted international experts to the annual conferences and as summer visitors. However, its international recog- nition stemmed from the ”Infinite Dimensional Linear Sys- tems Theory” approach to analyzing the control of partial and delay differential equations. In particular, the monograph bearing that title [B4] (jointly authored by Tony and Ruth Curtain) became the calling card of the centre. The location of the Control Theory Centre in a specially built penthouse on top of the engineering building helped to create a close-knit group. The atmosphere was stimulating

2 intellectually and at the same time supportive and friendly with regular social evenings at the homes of the members. Tony was actively involved in the research and social ac- tivities of the research fellows and he supervised several Ph.D. students in the area of control of partial differential equations. Under his leadership research collaboration with Rome and Toulouse were initiated in 1977. As a result of these collaborations international visitors were regularly on the scene in the Control Theory Centre, so creating an ideal intellectual climate for the Ph.D. students and research fellows. Many of the early research fellows David Wishart, Ruth Curtain, Roberto Triggiani, Andrew Plant, Andrew Wirth, Akira Ichikawa, Eugene Ryan, Jerzy Zabczyk, Steve Banks went on to be professors in the UK and overseas. On the national level the Control Theory Centre promoted the use of functional analysis in control engineering by giving a number of short courses on the sub- ject (see also the book [B5]). During a stay at the LAAS in Toulouse in 1979 Tony met Abdelhaq El Ja¨ıwho later returned to the university of Ra- bat in Morocco. Through Abelhhaq Tony became involved in developing the research programme in control there. He was very influential in the control group through his series of lecture courses on control to advanced students, his supervision of doctoral students and his participation in the jury in doctoral examinations. Many of these students are now professors at universities in Rabat, Casablanca, Tang- iers, Meknes and Marrakech. In 1985 Addelhaq became a professor at the university of Perpignan, France, and the collaboration with Tony continued and culminated in the monograph [B2]. In 1976 Tony and Ruth Curtain had been invited by Diederich Hinrichsen to present their approach to infinite dimensional systems theory in a lecture series at the Uni- versity of Bremen. This was the beginning of a collaboration that lasted more than thirty years. A first joint project financed by the EEC was a teaching project and resulted in Lecture Notes for a joint introductory course on Mathe- matical Systems Theory at the Universities of Warwick and Bremen, 1977-79. After Ruth Curtain moved to the Univer- sity of Groningen in 1977 Peter Crouch joined the Control Theory Centre. In 1980 the collaboration was extended to the University of Groningen with the aim of setting up joint advanced lecture courses on control. The financial support of the EEC facilitated several exchange visits of staff and doctoral students between the universities and the organiza- tion of joint workshops in Edzell, Scotland, 1980 and 1981, which were attended by leading systems theorists from Eu- rope and Israel.

3 After the end of the EEC funded teaching projects, Tony and Diederich Hinrichsen continued their collaboration and began to pursue the project of writing a comprehensive textbook on mathematical systems theory based on the lecture notes developed earlier. Long term exchange visits during the years 1983-85 were made possible by the financial support of the universities of Bremen and Warwick, and by funds of the SERC. Out of this collaboration the second main research theme in Tony’s career emerged, ”Ro- bust stability and robust control”. The research was supported by two grants of the European Community, the first one (1986-90) obtained jointly with Ruth Curtain (Gronin- gen) and the second (1991-94) jointly with Ruth Curtain and Jan Willems (Groningen). Apart from the direct ben- efits for the research of the three groups these European projects had two important side-effects for each of them. Firstly, it enabled them to purchase a workstation which, for the first time, made experimental work possible and also greatly enhanced the communication between them via email. Secondly, it facilitated an exchange of ideas and personal contacts among the doctoral students and postdocs during the frequent short term exchange visits between the three groups, the annual workshops and additional conferences. Usually the workshops took place in quiet hotels in the Dutch countryside, following the model of the workshops in Edzell, Scotland, and this helped to build personal relationships which are so important for successful scientific collaboration. Tony liked to reminisce on these days observing how the cooperation between the young scientists from Germany, the Netherlands and the UK continued long after the EEC-projects had ended and many of them had become professors. In his view the “twinning proposals” of the eighties and nineties were a very efficient way of promot- ing scientific cooperation in Europe and he often expressed his regret that the EU no longer supported such medium size European research projects. During all this time Tony and Diederich Hinrichsen continued to work on the book, although writing was some- times interrupted for whole years by research questions aris- ing from the book and other research projects. The chapters grew and grew and in the late nineties it became clear that the material had to be divided into two volumes. The first volume dedicated to systems analysis was finally published 2005 in the Springer “Texts in Applied Mathematics” Se- ries. It contained most of the robustness results obtained previously. The second volume has not been completed and about 500 pages dealing with control aspects of systems theory (controllability, observability, model reduction, Riccati equations and H∞ optimal control theory) remain to be

4 published. During all these years, Tony fulfilled his duties as a lecturer and since 1983 as professor of applied mathematics. Over the years the philosophy towards applied mathematics at Warwick had changed and the control theory centre moved to the mathematics department in 1986 with Diet- mar Salamon taking up the lectureship vacated by Peter Crouch. Tony was a born lecturer. He enjoyed performing with an almost messianic zeal to rows and rows of students and continued to give a course on control theory even after retirement. The three main loves and passions in his life were family, mathematics and sport. Over the years Tony participated in many different sports, but he excelled most in football and running. After retiring from playing football in his for- ties, he took up running seriously and competed in several marathons. At the age of 45 he ran the London marathon in just over 3 hours and at the age of 60 he could still run 10 km in 36 minutes. Some of his friends would say he was addicted to running. What is less well-known about Tony is that he was a bon vivant, who enjoyed cooking and serving the evening meal with a good wine to family and friends. Tony and Buddug were very hospitable and many colleagues will remember with pleasure being invited to their home during a visit to the university or after a conference or workshop. The delicious food, French wine was typically accompanied by lively discussions about politics, social questions, as well as the arts and of course, sports, especially rugby. With the arrival of their first grandchildren Buddug en Tony spent most weekends with their daughters and their families, enjoying their role of grandparents. Tony kept up with the friends of his youth, and he loved travelling with his family and friends. It was usually he who took the ini- tiative and organized, in the most efficient way, trips to France, Portugal, Spain, the Lake District and Wales. In 2000, to celebrate the new millenium, he set out for a 7 days walk along Offa’s dyke, mostly solo, but on sections of the route he was accompanied by various friends and family members. Tony had a real zest for life and was always op- timistic. He will be dearly missed by his wife, 4 daughters, nine grandchildren and many friends and colleagues.

1.2 A. J. Pritchard: a survey of his research Tony started his scientiﬁc career with studies in stability analysis [1]-[6], [62]-[64]. His Ph.D. was on the stabil-

5 ity of p.d.e.’s (partial differential equations) using a Lia- punov approach [B1]. The emphasis was on the construction of Liapunov functionals for distributed parameter systems (mainly from fluid dynamics), but the method developed for these systems was also applied to obtain new results on stability boundaries and domains of attraction for classical nonlinear second order systems [4]-[6]. The work on the stability of p.d.e.’s led naturally to considering stabilization by feedback control ([7], [9]) and, in particular, to the study of the classic linear quadratic control problem. In 1966 Lions had solved this problem for parabolic p.d.e.’s using a variational approach. Tony chose to use the semigroup approach which had been adopted in the mid-sixties by Fattorini for studying the controllability of p.d.e.’s. The 1969 paper [3] was one of the first to solve the classic linear quadratic control problem for abstract systems described by strongly continuous semigroups on a Hilbert space. The solution used a dynamic programming approach and resulted in a differential Riccati equation analogous to the one for ordinary differential equations. In [11] the time- varying integral and differential Riccati equations were derived. Probably the most important was the work on the time-varying linear quadratic optimal control problem described by evolution operators. This led to the introduction of the new concepts of almost-strong evolution operators and quasi-evolution operators in [12]. The abstract semigroup (evolution operator) approach allows for a natural generalization of the finite dimensional concepts of stability, controllability, observability, stabilizability, detectability ([14], [19]) and the corresponding methods of control ([3], [11]–[13], [15], [17], [18]). The advantage of the semigroup approach is that it applies in a unified way to linear systems described by ordinary differential equations, partial differential equations, stochastic systems and delay equations. These early results on infinite-dimensional systems were included in the monograph [B4], which aimed at summarizing the main developments in infinite-dimensional linear systems theory, both deterministic and stochastic and in illustrating the theory with several examples of p.d.e.’s, stochastic systems and delay equations. It was regularly cited in journal articles for many years. One disadvantage of this abstract approach compared with the variational approach is that the formulation of control on the boundary leads to the introduction of unbounded operators that are mathematically difficult to handle. This arises, for example, if one applies a force or a torque at the end of a beam instead of in its interior or if the control action is delayed in time. Some results on unbounded control action for partial differential equations were included in [B4], but it was a theme

6 that Tony pursued for some time in [13], [14], [17], [16] and [18]. These results motivated Dietmar Salamon, a former Ph.D. student of D. Hinrichsen, to refine the theory of unbounded control action in the context of delay equations. In the joint paper [27] these ideas came together, providing a complete theory for linear quadratic control problems for the class that became known as “Pritchard-Salamon systems”. These systems include systems described by delay equations and many systems described by p.d.e.’s. Other themes Tony pursued were the application of the semigroup approach to nonlinear systems (see [16], [20], [21], [47]) and realization theory (see [22]). With Abdelhaq El Ja¨ı(Perpignan) he studied problems of sensing and control for distributed parameter systems, the location of sensors and actuators and introduced new concepts such as regional controllability, spreadability, remediability and vulnerability (see [B5],[28], [48]). At the end of the seventies the control techniques of linear state space theory (”geometric control theory” and linear quadratic control) met with increasing criticism. The main point was that – in contrast to traditional frequency based methods – state space methods largely failed to deal with the problems of modeling errors and unknown dis- turbances. In the early eighties Tony became interested in these problems and looked for methods of dealing with parameter uncertainty within a state space framework. To- gether with Diederich Hinrichsen around 1984/85 he developed the concept of stability radius [25], which measures the distance of a stable system with respect to some given norm from the set of unstable systems. They were primarily interested in real parameter perturbations of linear state space systems, but to characterize the corresponding stability radius proved to be a difficult problem. However, allowing complex perturbations greatly simplified the problem and in [26] Tony characterized the complex stability radius via a parametrized algebraic Riccati equation which led to efficient and reliable algorithms. In the real case computable formulae were only obtained for rank 1 perturbations. Nonetheless, this was sufficient to derive explicit formulae for the real stability radius of arbitrary uncertain Hurwitz polynomials with affinely perturbed coefficient vec- tors [89],[94],[41]. The concept of stability radius led naturally to a robust control problem, namely that of maximising the stability radius for a given perturbation class by means of static linear state or dynamic output feedback. As early as 1985 Tony was experimenting with Riccati equations in order to solve this problem and he came close to a complete solution using Tichonov regularization. Later it became clear that the

7 problem that Tony investigated in 1985 was equivalent to a singular version of the H∞-optimal control problem. This is remarkable since the first treatment of the regular H∞- problem by means of Riccati equations appeared in the liter- ature only at the end of the eighties. At that time Tony was primarily interested in robustness analysis and concentrated his efforts on the extension of the concept of stability radii to new classes of systems and perturbations. Together with Diederich Hinrichsen he analyzed the effect of extended perturbation classes on stability radii [26],[97],[40] and showed that the stability radii of time-invariant linear systems with respect to complex time-invariant, time- varying and nonlinear time-varying perturbations coincide, whereas analogous results for real perturbations do not hold. Only at the end of the eighties, Tony returned to the control problem of maximizing the complex stability radius by output feedback [92],[36],[38]. To extend the system class for which stability radii could be determined proved to be a much harder problem than to extend the perturbation class. Still today the problem of de- termining the stability radius of time-varying linear systems is unsolved, even in the complex case. In [33],[45],[46],[47] Tony and his co-authors derived lower bounds for the stability radii of finite and infinite-dimensional time-varying linear systems. Surprisingly, for stochastic parameter perturbations, both of deterministic and of stochastic linear systems, the stability radius problem admitted a complete solution. In a series of papers Tony and his coauthors derived computable formulae for the corresponding stability radii using a Ric- cati type rational matrix equation as the principal tool and showed that in this case the real and complex stability radii coincide [42],[52],[57]. Moreover, for deterministic and stochastic linear systems, they were able to characterize the stability radii which can be achieved by static linear state or dynamic output feedback [43],[52]-[55]. While analyzing the real stability radius of non-normal systems, Tony and Diederich Hinrichsen introduced at the beginning of the nineties the concept of spectral value sets, see [99],[39]. These sets are by definition the union of the spectra of all perturbed system matrices where the norm of the perturbations is less than a given uncertainty bound. Spectral value sets have a close relationship with pseu- dospectra which were studied at about the same time in numerical analysis and fluid mechanics. However, the concept of pseudospectrum was based on complex perturbations whereas spectral value sets were initially introduced for real perturbations and this greatly complicated the analysis. For complex perturbations it was easy to obtain com-

8 putable formulae which permitted the visualization of spectral value sets on the computer screen. In [56] Tony and his co-authors developed a general framework for computing spectral value sets of a large class of infinite dimensional linear systems with respect to unbounded complex operator perturbations. Most of the above papers dealt with uncertain systems with “full block uncertainty”, i.e. the perturbation matrices vary in a full (real or complex) matrix space. Highly structured perturbations cannot be represented in this way and require, in general, the application of µ-analysis. How- ever, µ-values are usually not computable and this reduces the usefulness of formulae which are based on µ-values. In recent years, motivated by Gershgorin’s inclusion theorem, Tony became interested in perturbations of diagonal matrices whose entries are perturbed at an arbitrary set of diagonal and/or off-diagonal positions. For these highly structured perturbations of diagonal matrices he and his coauthors obtained computable formulae which showed that the classical inclusion regions of Brauer and of Brualdi are tight. The results can be extended to block-diagonal matrices [59] and provide exact computable formulae for spectral value sets and stability radii of interconnected systems with uncertain couplings. In [59] only time-invariant parameter perturbations were considered. The question whether a diagonal system can be destabilized by smaller, but time-varying perturbations was left as an open problem. Dealing with this problem Tony returned to one of his favorite research themes from where his scientific career had started, the construction of Lia- punov functions. A complete solution was obtained in his last two papers. In [60] he and Diederich Hinrichsen derived computable formulae for the stability radius of diagonal matrices with respect to time-varying parameter perturbations of arbitrarily prescribed structure. Surprisingly, if the diagonal matrix is real, the stability radii with respect to time-invariant and time-varying parameter perturbations are equal, whereas this does not hold for systems with complex diagonal entries. For block-diagonal systems the sit- uation is more complicated. In this case the two stability radii are generically dinstinct even if the underlying data are all real. Nevertheless, computable formulae have been obtained for both stability radii, and necessary and sufficient conditions for their equality have been found. Tony was working on the final revision of the paper containing these results [61] during his last days in Lanzarote.

1.3 Ph.D. students • K. Parker 1975

9 • W. Blakely 1977 • Jenny Pollock 1980 • Michael Chapman 1981 • K.Magnussen 1982 • Neil Carmichael 1982 • Stefan Szumko 1987 • Stuart Townley 1987 • Marcos Botelho 1994 • Neil Evans 1999 • Noel Mc William 2003

2 Publications of A. J. Pritchard

2.1 Books B1 A. J. Pritchard, Stability and control of distributed parameter systems, Ph.D. thesis, University of Warwick, June, 1970. B2 R. F. Curtain and A. J. Pritchard, Functional analysis in modern applied mathematics. Mathematics in Science and Engi- neering, Vol. 132. Acad. Press, London, 1977. B3 A. J. Pritchard and S. P. Banks (eds.), Proc. Second IFAC Symp. on Control of distributed parameter systems (Coventry 1977), Pergamon Press, Oxford, 1978. B4 R. F. Curtain and A. J. Pritchard, Infinite dimensional linear systems theory. Lecture Notes in Control and Information Sciences 8. Springer-Verlag, Berlin, 1978. B5 A. El Ja¨ıand A. J. Pritchard, Capteurs et actionneurs dans l’analyse des systèmesdistribués. RMA3-Masson, 1986. Also published in English: Sensors and controls in the analysis of distributed systems. Translated from the French by Catrin Pritchard and Rhian Pritchard. Ellis Horwood Ltd. – John Wiley & Sons, New York, 1988. B6 D. Hinrichsen and A. J. Pritchard, Mathematical systems theory. I. Modelling, state space analysis, stability and robustness. Texts in Applied Mathematics 48, Springer-Verlag, Berlin, 2005.

2.2 Papers Published in International Jour- nals 1. A. J. Pritchard, A study of two of the classical problems of hy- drodynamic stability by the Liapunov method. J. Inst. Math. Appl. 4: 78–93, 1968. 2. A. J. Pritchard, Stability boundaries for a two dimensional panel. AIAAJ 6: 1590–92, 1969. 3. A. J. Pritchard, Stability and control of distributed parameter systems. Proc. Inst. Elec. Engrs. 8, 1433–38, 1969. 4. A. J. Pritchard, Stability of the damped Mathieu equation. Electron. Lett. 5: 700–701, 1969.

10 5. A. J. Pritchard, On nonlinear stability theory. Quart. Appl. Math. 27: 531–536, 1969/70. 6. A. J. Pritchard, Stability and stabilization of second-order systems. J. Inst. Math. Appl. 7: 348–360, 1971. 7. M. J. E. Mayhew and A. J. Pritchard, Feedback from discrete points for distributed parameter systems. Int. J. Control 14: 619–630, 1971. 8. M. J. E. Mayhew and A. J. Pritchard, A reduction in the Riccati equation for distributed-parameter systems. Electron. Lett. 7, no. 19: 628–629, 1971. 9. A. J. Pritchard and K. T. Parker, Simplified Liapunov matrix equation with application to the control of distributed param- eters systems. Electron. Lett. 7, no. 194, 1971. 10. A. J. Pritchard and K. T. Parker, A lower bound for the cost functional for control problems in Hilbert space. J. Inst. Math. Appl. 13: 97–106, 1974. 11. R. F. Curtain and A. J. Pritchard, The infinite-dimensional Riccati equation. J. Math. Anal. and Appl. 46: 43-57, 1974. 12. R. F. Curtain and A. J. Pritchard, The infinite-dimensional Riccati equation for systems defined by evolution operators. SIAM J. Control Optim. 14, no. 5: 951–983, 1976. 13. R. F. Curtain and A. J. Pritchard, An abstract theory for unbounded control action for distributed parameter systems. SIAM J. Control Optim. 15, no. 4: 566–611, 1977. 14. A. J. Pritchard and A. Wirth, Unbounded control and obser- vation systems and their duality. SIAM J. Control Optim. 16, no. 4: 535–545, 1978. 15. A. J. Pritchard, Infinite-dimensional linear systems theory. Proc. Inst. Electr. Engrs. 126, no. 6: 641–648, 1979. 16. A. Ichikawa and A. J. Pritchard, Existence, uniqueness, and stability of nonlinear evolution equations. J. Math. Anal. Appl. 68, no. 2: 454–476, 1979. 17. J. Pollock and A. J. Pritchard, The infinite time quadratic cost control problem for distributed systems with unbounded control action. J. Inst. Math. Appl. 25, no. 3: 287–309, 1980. 18. J. Pollock and A. J. Pritchard, The tracking problem for distributed systems with unbounded control action. J. Inst. Math. Appl. 25, no. 4: 387–396, 1980. 19. A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinite-dimensional systems. SIAM Rev. 23, no. 1: 25–52, 1981. 20. A. J. Pritchard, Observers and minimum-energy controls for nonlinear systems. Proc. IEE-D 128, no. 5: 185–187, 1981. 21. N. Carmichael, A. J. Pritchard and M.D. Quinn, State and parameter estimation for nonlinear systems. Appl. Math. Op- tim. 9, no. 2: 133–161, 1982/3. 22. M. J. Chapman and A. J. Pritchard, Infinite-dimensional linear realization theory. IMA J. Appl. Math. 29, no. 3: 287–308, 1982. 23. A. J. Pritchard, The application of pseudo-inverses and fixed point theorems in systems theory. Bull. Inst. Math. Appl. 19, no. 5-6: 110–115, 1983.

11 24. R. C. Grimmer and A. J. Pritchard, Analytic resolvent operators for integral equations in Banach space. J. Differential Equations 50, no. 2: 234–259, 1983. 25. D. Hinrichsen and A. J. Pritchard, Stability radii of linear systems. Systems Control Lett. 7, no. 1: 1–10, 1986. 26. D. Hinrichsen and A. J. Pritchard, Stability radius for structured perturbations and the algebraic Riccati equation. Sys- tems Control Lett. 8, no. 2: 105–113, 1986. 27. A. J. Pritchard and D. Salamon, The linear quadratic control problem for infinite-dimensional systems with unbounded input and output operators. SIAM J. Control Optim. 25, no. 1: 121– 144, 1987. 28. A. El Ja¨ıand A. J. Pritchard, Sensors and actuators in distributed systems. Internat. J. Control 46, no. 4: 1139–1153, 1987. 29. M. J. Chapman, A. J. Pritchard and S. Townley, Modelling linear distributed parameter systems. Systems Sci. 13, no. 3- 4: 37–50, 1987. 30. A. J. Pritchard and S. Townley, Robust compensator design via structured stability radii. Systems Control Lett. 11, no. 1: 33– 37, 1988. 31. G. A. Medrano-Cerda and A. J. Pritchard, Parameter estimation algorithms and robust identifiability. Internat. J. Control 48, no. 5: 1915–1936, 1988. 32. A. J. Pritchard and S. Townley, Robustness of linear systems. J. Differential Equations 77, no. 2: 254–286989. 33. D. Hinrichsen, A. Ilchmann and A. J. Pritchard, Robustness of stability of time-varying linear systems. J. Differential Equa- tions 82, no. 2: 219–250, 1989. 34. D. Hinrichsen and A. J. Pritchard, An improved error estimate for reduced-order models of discrete-time systems. IEEE Trans. Automat. Control 35, no. 3: 317–320, 1990. 35. D. Hinrichsen and A. J. Pritchard, A note on some differ- ences between real and complex stability radii. Systems Control Lett. 14, no. 5: 401–408, 1990. 36. D. Hinrichsen, A. J. Pritchard and S. Townley, A Riccati equation approach to maximizing the complex stability radius by state feedback. Internat. J. Control 52, no. 4: 769–794, 1990. 37. G. Da Prato, A. J. Pritchard and J. Zabczyk, On minimum energy problems. SIAM J. Control Optim. 29, no. 1: 209–221, 1991. 38. A. J. Pritchard and S. Townley, Robustness optimization for uncertain infinite-dimensional systems with unbounded inputs. IMA J. Math. Control Inform. 8, no. 2: 121–133, 1991. 39. D. Hinrichsen and A. J. Pritchard, On spectral variations under bounded real matrix perturbations. Numer. Math. 60, no. 4: 509–524, 1992. 40. D. Hinrichsen and A. J. Pritchard, Destabilization by output feedback. Diff. Integral Equations 5, no. 2: 357–386, 1992. 41. D. Hinrichsen and A. J. Pritchard, Robustness measures for linear systems with application to stability radii of Hurwitz and Schur polynomials. Internat. J. Control 55, no. 4: 809– 844, 1992.

12 42. A. El Bouhtouri and A. J. Pritchard, Stability radii of linear systems with respect to stochastic perturbations. Systems Control Lett. 19, no. 1: 29–33, 1992. 43. A. El Bouhtouri and A. J. Pritchard, A Riccati equation approach to maximizing the stability radius of a linear system by state feedback under structured stochastic Lipschitzian perturbations. Systems Control Lett. 21, no. 6: 475–484, 1993. 44. J.A.M. Felipe de Souza and A. J. Pritchard, A note on overall observability: the joint problem of state estimation and parameter identification. Mat. Apl. Comput. 12, no. 1: 19–31, 1993. 45. D. Hinrichsen and A. J. Pritchard, Robust exponential stability of time–varying linear systems under time–varying parameter perturbations. Internat. J. Robust and Nonlinear Control 3: 63-83, 1993. 46. D. Hinrichsen and A. J. Pritchard, Robust stability of linear evolution operators on Banach spaces. SIAM J. Control Op- tim. 32, no. 6: 1503–1541, 1994. 47. B. Jacob, V. Dr˘aganand A. J. Pritchard, Infinite-dimensional time-varying systems with nonlinear output feedback. Integral Equations Operator Theory 22, no. 4: 440–462, 1995. 48. A. El Ja¨ı, M. C. Simon, E. Zerrik and A. J. Pritchard, Re- gional controllability of distributed parameter systems. Inter- nat. J. Control 62, no. 6: 1351–1365, 1995. 49. Z. Emirsajlow, A. J. Pritchard and S. Townley, On structured perturbations for two classes of linear infinite-dimensional systems. Dynam. Control 6, no. 3: 227–261, 1996. 50. A. J. Pritchard, The Aizerman conjecture. Math. Today 32, no. 7-8: 115–117, Southend-on-Sea, 1996. 51. A. J. Pritchard and Yuncheng You, Causal feedback optimal control for Volterra integral equations. SIAM J. Control Op- tim. 34, no. 6: 1874–1890, 1996. 52. D. Hinrichsen and A. J. Pritchard, Stability radii of systems with stochastic uncertainty and their optimization by output feedback. SIAM J. Control Optim. 34, no. 6: 1972–1998, 1996. 53. D. Hinrichsen and A. J. Pritchard, Stochastic H∞. SIAM J. Control Optim. 36, no. 5: 1504–1538, 1998. 54. A. El Bouhtouri, D. Hinrichsen and A. J. Pritchard, On the disturbance attenuation problem for a wide class of time-invariant linear stochastic systems. Stochastics and Stochastics Reports 65, no. 3-4: 255–297, 1999. 55. A. El Bouhtouri, D. Hinrichsen and A. J. Pritchard, H∞-type control for discrete-time stochastic systems. Internat. J. Ro- bust and Nonlinear Control 9, no. 13: 923–948, 1999. 56. E. Gallestey, D. Hinrichsen and A. J. Pritchard, Spectral value sets of closed linear operators. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456, no. 1998: 1397–1418, 2000. 57. A. El Bouhtouri, D. Hinrichsen and A. J. Pritchard, Stability radii of discrete-time stochastic systems with respect to block diagonal perturbations. Automatica 36, no. 7: 1033–1040, 2000. 58. A. Bernoussi, A. El Ja¨ıand A. J. Pritchard, Spreadability and evolving interfaces. Internat. J. Systems Sci. 32, no. 10: 1217– 1232, 2001.

13 59. M. Karow, D. Hinrichsen and A. J. Pritchard, Interconnected systems with uncertain couplings: explicit formulae for µ-values, spectral value sets, and stability radii. SIAM J. Control Op- tim. 45, no. 3: 856–884, 2006. 60. D. Hinrichsen and A. J. Pritchard, Gershgorin-Brualdi perturbations and Riccati equations. Lin. Algebra and its Appl. (Spe- cial Issue in honor of Paul Fuhrmann) 425: 374–403, 2007. 61. D. Hinrichsen and A. J. Pritchard, Composite systems with uncertain couplings of ﬁxed structure: Scaled Riccati equations and the problem of quadratic stability. To be submitted.

2.3 Papers Published in Books and Con- ference Proceedings 62. A. J. Pritchard and P. C. Parks, On the construction and use of Liapunov functionals, Proc. IVth IFAC Congress (Warsaw 1969). 63. A. J. Pritchard, Stability and control of distributed parameter systems governed by wave equations. Proc. IFAC Symp. on Distributed parameter systems (Banff 1971). 64. A. J. Pritchard and P. C. Parks, Stability analysis in structural dynamics using Liapunov functions, Proc. Symp. on Nonlinear dynamics (Loughborough 1972), Paper 4. Also published in J. of Sound and Vibration, 1972. 65. R. F. Curtain and A. J. Pritchard, The infinite-dimensional Riccati equation. Proc. IMA Conf. Recent developments in modern control theory (Bath 1972). 66. A. J. Pritchard, Sensitivity analysis for evolution equations in Hilbert space. Proc. Third IFAC Symp. on Sensitivity, adap- tivity and optimality (Ischia 1973), pp. 107–113, Instr. Soc. Amer., Pittsburg, Pa., 1973. 67. A. J. Pritchard, Control theory and applications. Control theory and topics in functional analysis (Internat. Sem. Inter- nat. Centre Theoret. Phys., Trieste 1974), Vol. I, pp. 179–222, Internat. Atomic Energy Agency, Vienna, 1976. 68. A. J. Pritchard and P. E. Crouch, Modelling and control of distributed parameter systems. Proc. IFIP Conf. on Modelling and Optimisation (Nice 1975). 69. A. J. Pritchard and W. T. F. Blakeley, Perturbation results and their applications to problems in structural dynamics. Lect. Notes in Math. 503, pp. 438–449, Springer, Berlin, 1976. 70. R. F. Curtain and A. J. Pritchard, A semigroup approach to infinite-dimensional systems theory. Proc. Conf. Recent theoretical developments in control (Leicester 1976), Acad. Press, London, pp. 19–48, 1980. 71. A. J. Pritchard, Control theory and applications. Control The- ory and Topics in Functional Analysis. Internat. Atomic En- ergy Commission. Vienna 1976. 72. A. J. Pritchard and E. P. Ryan, Control and identification of distributed parameter systems. Proc. IFIP Conf. Distr. parameter systems: modelling and identification (Rome 1976), pp. 348– 369, Lect. Notes in Control Information Sci. 1, Springer, Berlin, 1978.

14 73. R. F. Curtain and A. J. Pritchard, A semigroup approach to infinite-dimensional system theory. Proc. Conf. Recent theoretical developments in control (Leicester 1976), pp. 19–47, Acad. Press, London, 1978. 74. W. T. F. Blakeley and A. J. Pritchard, Stability of abstract evolution equations. Proc. Second IFAC Sympos. on Control of distributed parameter systems (Coventry 1977), pp. 121–132, Pergamon Press, Oxford, 1978. 75. N. Carmichael and A. J. Pritchard, Optimal scheduling for cancer radiotherapy. Proc. First World Conf. on Mathematics at the Service of Man (Barcelona 1977). 76. N. Carmichael and A. J. Pritchard, Optimal scheduling for the treatment of cancer. Proc. IMA Conf. on Analysis and optimisation of stochastic systems (Oxford 1978), Acad. Press, London, pp. 511–521, 1980. 77. A. J. Pritchard, Control of abstract evolution equations. Conf. on Recent Advances in Diff. Equations (Trieste 1978). 78. L. Bogataj and A. J. Pritchard, Sensitivity analysis for control problems described by delay equations. Proc. MECO Conf. (Athens 1978). 79. K. Magnusson and A. J. Pritchard, Local controllability with applications to delay equations. Proc. Conf. Nonlinear Semi- groups and Functional Diff. equations (Graz 1979). 80. M. J. Chapman and A. J. Pritchard, Realisation theory for a class of distributed parameter systems. Proc. MTNS (Delft 1979). 81. K. Magnusson and A. J. Pritchard, Local exact controllability of nonlinear evolution equations. Recent advances in diff. equations (Trieste 1978), pp. 271–280, Acad. Press, London, 1981. 82. A. J. Pritchard, The application of a fixed point theorem to nonlinear systems theory. Third IMA Conf. on Control Theory (Sheffield 1980), pp. 775–792, Acad. Press, London, 1981. 83. N. Carmichael, A. E. Fincham and A. J. Pritchard, Nonlinear observers with application to gas distribution networks. IEE Conf. on Control and Applications (Univ. Warwick 1981). 84. N. Carmichael, A. J. Pritchard and M. D. Quinn, Control and state-estimate of nonlinear systems. Evolution equations and their applications (Schloss Retzhof 1981), pp. 30–51, Research Notes in Math. 68, Pitman, Boston–London, 1982. 85. M. J. Chapman and A. J. Pritchard, Finite-dimensional com- pensators for nonlinear infinite-dimensional systems. Proc. Con- trol theory for distributed parameter systems and applications (Vorau 1982), pp.60–76, Lect. Notes in Control and Inform. Sci. 54, Springer, Berlin, 1983. 86. A. J. Pritchard, Nonlinear infinite-dimensional systems theory. Proc. Workshop Control of distributed parameter systems (Toulouse 1982), pp. 7–12, IFAC, Laxenburg, 1983. 87. A. J. Pritchard and D. Salamon, The linear quadratic optimal control problem for infinite-dimensional systems with unbounded input and output operators. Proc. Workshop Infinite- dimensional systems (Schloss Retzhof 1983), pp. 187–202, Lect. Notes in Math. 1076, Springer, Berlin, 1984.

15 88. K. Magnusson, A. J. Pritchard and M. D. Quinn, The application of fixed point theorems to global nonlinear controllability problems. Mathematical control theory, Banach Center Publ. 14, PWN, Warsaw, pp.319–344, 1985. 89. D. Hinrichsen and A. J. Pritchard, On the robustness of root locations of polynomials under complex and real perturbations. Proc. 27th IEEE Conf. Decision and Control (Austin 1988), pp.1410–1414, 1988. 90. D. Hinrichsen and A. J. Pritchard, New robustness results for linear systems under real perturbations. Proc. 27th IEEE Conf. Decision and Control (Austin 1988), pp.1375–1379, 1988. 91. D. Hinrichsen and A. J. Pritchard, The complex stability radius of time-varying linear systems. Proc. 5th IMA Conf. Mathe- matics of Control Theory (Glasgow 1988), pp.281–290, Claren- don Press, Oxford, 1992. 92. D. Hinrichsen and A. J. Pritchard, Parametrized Riccati equations and the problem of maximizing the complex stability radius. Proc. Workshop The Riccati equation in control, systems, and signals, Pitagora Editrice Bologna, pp.136–142, 1989. 93. D. Hinrichsen and A. J. Pritchard, Robustness of stability of linear state space systems with respect to time-varying, nonlinear and dynamic perturbations. Proc. 28th Conf. Decision and Control (Tampa 1989), pp.52–53, 1989. 94. D. Hinrichsen and A. J. Pritchard, An application of state space methods to obtain explicit formulae for robustness measures of polynomials. Proc. Workshop Robustness in identification and control (Torino 1988), pp.183–206, Plenum, New York, 1989. 95. D. Hinrichsen and A. J. Pritchard, Robustness measures for linear state space systems under complex and real parameter perturbations. Proc. Workshop Perspectives in control theory (Sielpia 1988), pp.56–74, Birkhäuser, Boston, 1990. 96. A. J. Pritchard and S. Townley, Real stability radii for infinite- dimensional systems. Robust control of linear systems and nonlinear control (Amsterdam 1989), 635–646, Birkhäuser, Boston, 1990. 97. D. Hinrichsen and A. J. Pritchard, Real and complex stability radii: a survey. Control of uncertain systems (Bremen 1989), pp.119–162, Birkhäuser, Boston, 1990. 98. D. Hinrichsen and A. J. Pritchard, Robust stability of linear time–varying systems with respect to multi-perturbations. Proc. European Control Conf. (Grenoble 1991), pp.1366-1371, 1991. 99. D. Hinrichsen and A. J. Pritchard, On the robustness of stable discrete-time linear systems. Proc. New trends in systems theory (Genoa 1990), pp. 393–400, Birkhäuser,Boston, 1991. 100. A. J. Pritchard, Introduction to semigroup theory. Analy- sis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems (Sophia-Antipolis 1992), pp.1–22, Lect. Notes in Control and Inform. Sci. 185, Springer, Berlin, 1993. 101. A. El Ja¨ıand A. J. Pritchard, Regional controllability of distributed systems. Analysis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems (Sophia-Antipolis 1992), pp.326–335, Lect. Notes in Con- trol and Inform. Sci. 185, Springer, Berlin, 1993.

16 102. D. Hinrichsen and A. J. Pritchard, Robust stability of time– invariant infinite dimensional systems with respect to time– varying perturbations. Proc. 2nd European Control Conf. (Gro- ningen 1993), pp.1089-1093, 1993. 103. R.F. Curtain and A. J. Pritchard, Robust stabilization of infinite- dimensional systems with respect to coprime factor perturbations. Differential equations, dynamical systems, and control science, pp.437–456, Lect. Notes in Pure and Appl. Math. 152, Dekker, New York, 1994. 104. D. Hinrichsen, and Pritchard, A. J. Stability of uncertain systems. Proc. MTNS (Regensburg 1993), Vol. I Key Invited Lec- tures, pp.159–182, Akademie-Verlag, Berlin, 1994. 105. D. Hinrichsen and A. J. Pritchard, Stability margins for systems with deterministic and stochastic uncertainty. Proc. 33rd Conf. Decision and Control. Florida, pp.3825-3826, 1994. 106. D. Hinrichsen and A. J. Pritchard, Stability radii for infinite dimensional systems with stochastic uncertainty. Proc. 3rd. European Control Conf. (Rome 1995), pp.3291-3293, 1995. 107. D. Hinrichsen, A. J. Pritchard, Maximization of stability radii via dynamic output feedback for systems with stochastic uncertainty. Proc. 3rd. European Control Conf. (Rome 1995), pp.984-988, 1995. 108. A. El Bouhtouri, D. Hinrichsen and A. J. Pritchard, Attenua- tion of multiperturbed stochastic systems by output feedback. Proc. 3rd Internat. Symp. Methods and Models in Automation and Robotics (Miedzyzdroje 1996), vol. 1, pp.367–372, 1996. 109. D. Hinrichsen and A. J. Pritchard, A Riccati equation for stochastic H∞. Proc. 4th. European Control Conf. (Brussels 1997), FR-A-L-2, 1997. 110. E. Gallestey, D. Hinrichsen and A. J. Pritchard. On the spectral value set problem in infinite dimensions. Proc. 4rd Inter- nat. Symp. Methods and Models in Automation and Robotics (Miedzyzdroje 1997), pp.109–114, 1997. 111. D. Hinrichsen, A. J. Pritchard and A. El Bouhtouri, Riccati approach to disturbance attenuation of discrete time stochastic systems. Proc. MTNS (Padua 1998), pp.641 – 644, Il Poligrafo, Padua, 1998. 112. E. Gallestey, D. Hinrichsen and A. J. Pritchard, Approximation of spectral value sets. Proc. MTNS (Padua 1998), pp.463–466, Il Poligrafo, Padua, 1998. 113. E. Gallestey, D. Hinrichsen and A. J. Pritchard, Spectral value sets of infinite-dimensional systems. Open problems in mathematical systems and control theory, pp.109–113, Springer, Lon- don, 1999. 114. D. Hinrichsen and A. J. Pritchard, Robust stability of linear stochastic systems. Open problems in mathematical systems and control theory, pp.125–129, Springer, London, 1999. 115. D. Hinrichsen and A. J. Pritchard, On the transient behaviour of stable linear systems. Proc. MTNS (Perpignan 2000), CD- ROM, 2000. 116. D. Hinrichsen, E. Plischke and A. J. Pritchard, Liapunov and Riccati equations for practical stability. Proc. 6th European Control Conf. (Porto 2001), CD-ROM, 2001.

17 117. A. J. Pritchard, Transitory behavior of uncertain systems. Proc. Workshop Advances in mathematical systems theory (Borkum 1999), pp.1–17, Birkh¨auser, Boston, 2001.