From the very beginning of my university studies, I was exposed to advanced topics in analysis, differential equations, functional analysis, and control theory. I was fortunate to be admitted to a very special programme in applied mathematics at the University of Warsaw, Poland, which combined a rigorous education in mathematics and applications, with a particular emphasis on control theory.

What is control theory of a dynamical system? Traditionally, the classical viewpoint taken in the study of differential equations consisted in the (passive) analysis of the evolution properties displayed by a specific equation, or a class of equations, in response to given data. By contrast, control theory injects an active mode of synthesis in the study of differential equations: it seeks to influence their dynamical evolution by selecting and synthesising suitable data (input functions, or control functions) from within a preassigned class, as to achieve a predetermined desired outcome or performance.

The author studies the properties and mutual relations of various conical approximations of sets in locally convex topological linear spaces. On the basis of their definitions she separates the approximations into two groups - axiomatic and constructive. The former include internal cone, simplicial cone and extremal cone, while the latter include admissible cone, tangent cone, functional decrease cone and radial cones. A separation theorem for internal and extremal cones is presented and employed for deriving necessary conditions of optimality in the spirit of the Dubovitskii-Miljutin formalism.

A finite-difference approximation approach is used to obtain the suboptimal solution of the optimal control problem for systems characterised by nonlinear differential equations with delay. Using the method of Lagrange multipliers, the author discusses the rates of convergence of various suboptimal solutions to the optimal solutions and obtains error estimates for the optimal control, state, and cost functional. The method is constructive and leads, in a natural way, to the development of computational algorithms for the real-time solution of the problem using a computer.

This research was sponsored by the Institute of Automatic Control, Department of Electronics, Technical University, Warsaw, Poland.

This paper deals with regularity properties of optimal solutions of distributed control problems with state constraints. Not only regularity of solutions is an interesting question from the mathematical point of view, but it also provides the mathematical foundations for estimating the error occurring in approximation of control problems. Estimates of such error involve bounds on the derivative of the function being approximated. Therefore, in order to obtain good estimates, certain types of solutions regularity (in the primal and dual control problems) are needed. The required regularity, as a rule, is more restrictive than that imposed by the spaces in which control problems are considered (control functions are usually in $L^{2}$). Also, regularity results give insight into the practicality of approximation schemes. It can indicate which kind of approximation should be applied. Namely, if a function has only the first derivative in $L^{2}$, there is usually no advantage in using piecewise approximating polynomials of degree > 1.

This project is to investigate a coherent set of boundary control problems for systems described by wave-type and plate-type partial differential equations, defined on a bounded domain in higher dimensional space. Both linear and nonlinear dynamics are considered. Problems for investigation include: exact controllability; uniform stabilisation by means either of explicit dissipative feedback operators or else of nondissipative feedback operators based on algebraic Riccati operators; optimal quadratic cost problems and related Riccati equations; asymptotic stability properties for nonlinear models; robustness of asymptotic stability properties under nonlinear structural perturbations; well-posedness of nonlinear wave equations with non-monotone nonlinearities in the Neumann boundary conditions. The emphasis throughout is on the optimal setting where the solutions and their relevant properties are studied in the spaces of optimal regularity, or, alternatively, in the spaces of finite energy. There are many applications for the theoretical developments described above. For instance, the control of robot arms and the reduction of damaging oscillations in large flexible structures on Earth or in orbit, are examples of practical benefits resulting from this work.

This three-year award supports U.S.-France cooperative research in applied mathematics. The investigators are Irena Lasiecka and Roberto Triggiani from the University of Virginia and J P Zolesio and M Souli from the University of Nice. The investigators will utilise partial differential equations, hyperbolic equations and numerical analysis in approaching a number of problems in structural mechanics. The project focuses on elastic structures whose dynamical behaviour is effected by internal damping of vibrations. The objectives are to enhance stability properties by determining the optimal shape or design of the elastic structure so that undesirable vibrations are reduced. The U.S. investigators bring to this collaboration their expertise in mathematical modelling of structural damping, optimum control and related questions and the utilisation of elastic partial differential equations for determining controllability and stabilisation properties. This is complemented by French expertise in shape analysis and optimisation. The research has important applications in the areas of space mechanics and control.

For Contribution to Boundary Control Systems.

For Outstanding Contribution to Nonlinear Mathematical Analysis.

... her fundamental contributions in control and optimisation theory, particularly for dynamical systems governed by partial differential equations and their applications.

My first monograph lecture notes (jointly with R Triggiani) published in 1991 by Springer, provides the first overview of the literature on optimal control theory as applied to classical and physically relevant single partial differential equation models with boundary/point control and/or boundary observation. It axiomatizes the properties and criteria that need to be satisfied for a successful application of optimal-control theory.

After spending 26 years at the University of Virginia (UVA) as a Full Professor and later Commonwealth Professor of Mathematics, an endowed chair, I wanted to expand my academic experience to include different students' environments and academic settings. Compared to UVA, the landscape of education and teaching in Memphis is very different. Many American students at Virginia are groomed in families with educational tradition and ambition. In addition, UVA attracts top International students in mathematics, engineering and sciences. Helping students in Memphis in achieving their academic goals is academically more challenging but also humanly more rewarding. Of course, the fact that Department of Mathematics at the University of Memphis had an excellent reputation in research (with several stellar mathematicians including long time visitor Paul Erdős) and strong tradition of excellence played a dominant role in my decision.

It was a good day for the University of Memphis back in 2013, when Dr Irena Lasiecka decided to leave an endowed professorship at the University of Virginia after 26 years and become a Distinguished University Professor and Chair of Memphis's Department of Mathematical Sciences. According to one of her academic peers, her move "jolted the professional community and instantaneously brought world-wide visibility and reputation to the ... University of Memphis." Lasiecka is a towering figure in her field, combining innovative research, success at recruiting PhD students and post-docs as well as serving the larger mathematics community through a busy schedule of teaching and lecturing all over the world. She is also co-Editor-in-Chief of two academic journals, Applied Mathematics and Optimization, and Evolution Equations and Control Theory.

Lasiecka's area of specialisation is control theory of partial differential equations, an applied mathematical discipline that requires expertise in analysis, partial differential equations, calculus of variations, Riemannian geometry and several applied areas in both engineering and life sciences. "Partial differential equations involve a lot of pure mathematics, and that was what I was trained for very early in my studies," she said. Control theory of partial differential equations remains critical in advancing both aviation and aerospace as well as many other scientific disciplines; Lasiecka's research has been supported for decades by the National Science Foundation, Division of Mathematical Sciences, as well as NASA and the Air Force, among many others. Her research has made her one of the most frequently cited mathematicians in the world.

Explaining control theory, she said, "Your goal is to control something - for example, the temperature in this room or the oscillation of an airplane wing. But the main issue is somehow to understand the phenomena. After you have a good understanding of the model, you try to predict what can happen. Finally, you have to act, and that's exactly when the control comes, okay? You have to do something to achieve the desired target." Or to understand why an aircraft - or spacecraft - might fail in flight. Or a bridge might suddenly fall down. In fact, her presentation slide of the famous 1940 Tacoma Narrows Bridge collapse, caused by wind turbulence, is a favourite. "The study of fluids is related to turbulence ... a similar thing is oscillations, and often you want to reduce the oscillations of a building" or a bridge, as in Tacoma, she said. "There are certain operating structures and physical laws which can be altered or impacted by applications of control. But first you need to understand why it happened. People working in experimental science can provide some prediction. Then we try to confirm or understand it better from the point of view of mathematics. That's exactly what the goal is." In fact, understanding something new and being able to impact a particular problem is the part of Lasiecka's research that she is most proud of. "The example of this bridge, I can tell you that it's a nice part of research that you understand why it collapsed. It was confirmed by physicists and engineers with experimental science. Everything became clear."

Control theory is omnipresent in all aspects of life. Engineering, medicine, life sciences, computer science, and social sciences all benefit from control theory and pose a variety of open problems to be resolved. In the area of big data and artificial intelligence/robotics, one needs to effectively control systems with a reduced number of variables (while retaining relevant information of the system); here, uncertainty is particularly challenging. That refers to modelling, observing, and controlling from remote locations. Closer links with control systems, where big data problems surface to the top of the scientific interests, as well as with the traditional engineering community are expected to continue and grow.