Abstract. 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. The method is also computationally tractable, since barrier certificates can be constructed using the sum of squares decomposition and semidefinite programming. Some examples are provided to illustrate the use of the method.

Abstract-This paper presents a methodology for safety verification of continuous and hybrid systems in the worst-case and stochastic settings. In the worst-case setting, a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region. No explicit computation of reachable sets is required in the construction of barrier certificates, which makes it possible to handle nonlinearity, uncertainty, and constraints directly within this framework. In the stochastic setting, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, barrier certificates can be constructed using convex optimization, and hence the method is computationally tractable. Some examples are provided to illustrate the use of the method.

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SOSTOOLS is a MATLAB toolbox for constructing and solving sum of squares programs. It can be used in combination with semidefinite programming software, such as SeDuMi, to solve many continuous and combinatorial optimization problems, as well as various control-related problems. This paper provides an overview on sum of squares programming, describes the primary features of SOSTOOLS, and shows how SOS-TOOLS is used to solve sum of squares programs. Some applications from different areas are presented to show the wide applicability of sum of squares programming in general and SOSTOOLS in particular.

Abstract-We develop a new method for safety verification of stochastic systems based on functions of states termed barrier certificates. Given a stochastic continuous or hybrid system and sets of initial and unsafe states, our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set, a bound whose validity is proven by the existence of a barrier certificate. For polynomial systems, both the upper bound and its corresponding barrier certificate can be computed using convex optimization, and hence the method is computationally tractable.

Abstract-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 toward the equilibrium. The new criterion has a remarkable convexity property, which in this note is used for controller synthesis via convex optimization. Recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used.

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