# Probabilistic Reachability for Stochastic Hybrid Systems: Theory, Computations, and Applications

### Alessandro Abate

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2007-132

November 7, 2007

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-132.pdf

Stochastic Hybrid Systems are probabilistic models suitable at describing the dynamics of variables presenting interleaved and interacting continuous and discrete components.

Engineering systems like communication networks or automotive and air traffic control systems, financial and industrial processes like market and manufacturing models, and natural systems like biological and ecological environments exhibit compound behaviors arising from the compositions and interactions between their heterogeneous components. Hybrid Systems are mathematical models that are by definition suitable to describe such complex systems.

The effect of the uncertainty upon the involved discrete and continuous dynamics---both endogenously and exogenously to the system---is virtually unquestionable for biological systems and often inevitable for engineering systems, and naturally leads to the employment of stochastic hybrid models.

The first part of this dissertation introduces gradually the modeling framework and focuses on some of its features. In particular, two sequential approximation procedures are introduced, which translate a general stochastic hybrid framework into a new probabilistic model. Their convergence properties are sketched. It is argued that the obtained model is more predisposed to analysis and computations.

The kernel of the thesis concentrates on understanding the theoretical and computational issues associated with an original notion of probabilistic reachability for controlled stochastic hybrid systems. The formal approach is based on formulating reachability analysis as a stochastic optimal control problem, which is solved via dynamic programming. A number of related and significant control problems, such as that of probabilistic safety, are reinterpreted with this approach. The technique is also computationally tested on a benchmark case study throughout the whole work. Moreover, a methodological application of the concept in the area of Systems Biology is presented: a model for the production of antibiotic as a component of the stress response network for the bacterium Bacillus subtilis is described. The model allows one to reinterpret the survival analysis for the single bacterial cell as a probabilistic safety specification problem, which is then studied by the aforementioned technique.

In conclusion, this dissertation aims at introducing a novel concept of probabilistic reachability that is both formally rigorous, computationally analyzable and of applicative interest. Furthermore, by the introduction of convergent approximation procedures, the thesis relates and positively compares the presented approach with other techniques in the literature.

**Advisor:** S. Shankar Sastry

BibTeX citation:

@phdthesis{Abate:EECS-2007-132, Author = {Abate, Alessandro}, Title = {Probabilistic Reachability for Stochastic Hybrid Systems: Theory, Computations, and Applications}, School = {EECS Department, University of California, Berkeley}, Year = {2007}, Month = {Nov}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-132.html}, Number = {UCB/EECS-2007-132}, Abstract = {Stochastic Hybrid Systems are probabilistic models suitable at describing the dynamics of variables presenting interleaved and interacting continuous and discrete components. Engineering systems like communication networks or automotive and air traffic control systems, financial and industrial processes like market and manufacturing models, and natural systems like biological and ecological environments exhibit compound behaviors arising from the compositions and interactions between their heterogeneous components. Hybrid Systems are mathematical models that are by definition suitable to describe such complex systems. The effect of the uncertainty upon the involved discrete and continuous dynamics---both endogenously and exogenously to the system---is virtually unquestionable for biological systems and often inevitable for engineering systems, and naturally leads to the employment of stochastic hybrid models. The first part of this dissertation introduces gradually the modeling framework and focuses on some of its features. In particular, two sequential approximation procedures are introduced, which translate a general stochastic hybrid framework into a new probabilistic model. Their convergence properties are sketched. It is argued that the obtained model is more predisposed to analysis and computations. The kernel of the thesis concentrates on understanding the theoretical and computational issues associated with an original notion of probabilistic reachability for controlled stochastic hybrid systems. The formal approach is based on formulating reachability analysis as a stochastic optimal control problem, which is solved via dynamic programming. A number of related and significant control problems, such as that of probabilistic safety, are reinterpreted with this approach. The technique is also computationally tested on a benchmark case study throughout the whole work. Moreover, a methodological application of the concept in the area of Systems Biology is presented: a model for the production of antibiotic as a component of the stress response network for the bacterium Bacillus subtilis is described. The model allows one to reinterpret the survival analysis for the single bacterial cell as a probabilistic safety specification problem, which is then studied by the aforementioned technique. In conclusion, this dissertation aims at introducing a novel concept of probabilistic reachability that is both formally rigorous, computationally analyzable and of applicative interest. Furthermore, by the introduction of convergent approximation procedures, the thesis relates and positively compares the presented approach with other techniques in the literature.} }

EndNote citation:

%0 Thesis %A Abate, Alessandro %T Probabilistic Reachability for Stochastic Hybrid Systems: Theory, Computations, and Applications %I EECS Department, University of California, Berkeley %D 2007 %8 November 7 %@ UCB/EECS-2007-132 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-132.html %F Abate:EECS-2007-132