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The Discovery of a solution to a problem is often more difficult than verifying the correctness of a given solution. For example, solving a given Sudoku puzzle is at times very difficult, whereas verifying the correctness of any given solution of a Sudoku puzzle very easy. This difference in discovery and verification is formally captured by the classes P and NP: P contains all problems whose solutions can be efficiently discovered, and NP contains all problems whose solutions can be efficiently verified.

It is widely believed that P does not equal NP, but there is no proof known despite intensive efforts over the last 40 years. The reason for this belief is that discovery often requires an intuitive leap that cannot be captured as a mechanical process, while verification is a mechanical process. There will be very interesting consequences of P being equal to NP. For example, diseases can be eradicated (for any disease, its cure can be found efficiently), and e-commerce will cease to exist (codes used for hiding sensitive information can be efficiently decoded).

During the attempts to prove the separation of P and NP, two major barriers have been discovered. These barriers disallow a number of proof strategies for separating the two classes. In this talk, I will introduce the problem and discuss attempts to solve it.

The key components of feedback control systems - sensors, actuators, computation, power, and communication - are continually becoming smaller, lighter, more robust, higher performance, and less expensive. By using appropriate algorithms and system architectures, it is thus becoming possible to “close the loop” on almost any machine and to create new capabilities that fully exploit their dynamic potential. Flying platforms are particularly well-positioned to take advantage of these technological advancements. This talk will feature our research in the area of acrobatic flight.

We discuss nonlinear control problems in which the flow of information between the plant and the controller is limited. Specific focus is on such effects as quantization, time delays, and external disturbances. Our main technical tools are input-to-state stability. Lyapunov functions, and hybrid systems. The overall goal is to build a general theoretical framework for capturing the interplay between control and communication in the context of nonlinear control systems.

Biological systems make use of feedback in an extraordinary number of ways, on scales ranging from molecules to cells to organisms to ecosystems. In this talk, I will discuss the use of concepts from control and dynamical systems in the analysis and design of biological feedback circuits at the molecular level. After a brief survey of relevant concepts from synthetic biology, I will present some recent results that combine modeling, identification, design, and experimental implementation of biological feedback circuits.

These results include the use of intrinsic noise for system identification in transcriptional regulatory networks, analysis of the role of multiple feedback loops in providing robust behavior (ultrasensitivity and bimodality), development of feedback circuits for rate regulation and event detection, and the use of time delay as a means of designing biomolecular feedback dynamics. Using these results as examples, I will discuss some of the open problems and research challenges in the area of feedback control using biological circuits.