This paper presents a novel methodology for safety verification of hybrid systems. For proving that all trajectories of a hybrid system do not enter an unsafe region, the proposed method uses a function of state termed a barrier certificate. The zero level set of a barrier certificate separates the unsafe region from all possible trajectories starting from a given set of initial conditions, hence providing an exact proof of system safety. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes nonlinearity, uncertainty, and constraints can be handled directly within this framework.

Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.

This paper emphasizes the need for methodological frameworks for analysis and design of large scale networks which are independent of specific design innovations and their advocacy, with the aim of making networking a more systematic engineering discipline. Networking problems have largely confounded existing theory, and innovation based on intuition has dominated design. This paper will illustrate potential pitfalls of this practice. The general aim is to illustrate universal aspects of theoretical and methodological research that can be applied to network design and verification. The issues focused on will include the choice of models, including the relationship between flow and packet level descriptions, the need to account for uncertainty generated by modelling abstractions, and the challenges of dealing with network scale. The rigorous comparison of proposed schemes will be illustrated using various abstractions. While standard tools from robust control theory have been applied in this area, we will also illustrate how network-specific challenges can drive the development of new mathematics that expand their range of applicability, and how many enormous challenges remain.

Abstract. We call a hybrid system stable if every trajectory inevitably ends up in a given region. Our notion of stability deviates from classical definitions in control theory. In this paper, we present a model checking algorithm for stability in the new sense. The idea of the algorithm is to reduce the stability proof for the whole system to a set of (smaller) proofs for several one-mode systems. 1

A stability criterion for nonlinear systems, recently derived by the third author, can be viewed as a dual to Lyapunov's second theorem. The criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space and flows along the system trajectories towards the equilibrium. The new

Advances in the direct computation of Lyapunov functions using convex optimization make it possible to efficiently evaluate regions of attraction for smooth nonlinear systems. Here we present a feedback motion planning algorithm which uses rigorously computed stability regions to build a sparse tree of LQR-stabilized trajectories. The region of attraction of this nonlinear feedback policy “probabilistically covers ” the entire controllable subset of the state space, verifying that all initial conditions that are capable of reaching the goal will reach the goal. We numerically investigate the properties of this systematic nonlinear feedback design algorithm on simple nonlinear systems, prove the property of probabilistic coverage, and discuss extensions and implementation details of the basic algorithm. 1

We present an overview of the essential elements of semide nite programming as a computational tool for the analysis of systems and control problems. We make particular emphasis on general duality properties as providing suboptimality or infeasibility certi cates. Our focus is on the exciting developments occurred in the last few years, including robust optimization, combinatorial optimization, and algebraic methods such as sum-of-squares. These developments are illustrated with examples of applications to control systems.

Abstract. The deductive method reduces verification of safety properties of programs to, first, proposing inductive assertions and, second, proving the validity of the resulting set of first-order verification conditions. We discuss the transition from verification conditions to verification constraints that occurs when the deductive method is applied to parameterized assertions instead of fixed expressions (e.g., p0 +p1j +p2k> = 0, for parameters p0, p1, and p2, instead of 3+j-k> = 0) in order to discover inductive assertions. We then introduce two new verification constraint forms that enable the incremental and propertydirected construction of inductive assertions. We describe an iterative method for solving the resulting constraint problems. The main advantage of this approach is that it uses off-the-shelf constraint solvers and thus directly benefits from progress in constraint solving. 1 Introduction The deductive method of program verification reduces the verification of safetyand progress properties to proving the validity of a set of first-order verification conditions [13]. In the safety case, the verification conditions assert thatthe given property is inductive: it holds initially ( initiation), and it is preservedby taking any transition ( consecution). Such an assertion is an invariant of theprogram. In the progress case, the verification conditions assert that a given

The use of the sum of squares decomposition and semidefinite programming have provided an efficient methodology for analysis of nonlinear systems described by ODEs by algorithmically constructing Lyapunov functions. Based on the same methodology we present an algorithmic procedure for constructing Lyapunov-Krasovskii functionals for nonlinear time delay systems described by Functional Differential Equations (FDEs) both for delay-dependent and delay-independent stability analysis. Robust stability analysis of these systems under parametric uncertainty can be treated in a unified way. We illustrate the results with an example from population dynamics.

This paper presents a method for stability analysis of switched and hybrid systems using polynomial and piecewise polynomial Lyapunov functions. Computation of such functions can be performed using convex optimization, based on the sum of squares decomposition of multivariate polynomials. The analysis yields several improvements over previous methods and opens up new possibilities, including the possibility of treating nonlinear vector fields and/or switching surfaces and parametric robustness analysis in a unified way.