Speaker:

Prof. A. E. Krener

Abstract:

The Hamilton Jacobi Bellman partial differential equation arises in the solution of optimal control problems. It is a first order, nonlinear, hyperbolic PDE that is very difficult to solve because of the curse of dimensionality. Moreover the solution may not exist in the classical sense, i.e., the solution may not be differentiable everywhere. We describe an approach to approximately solve some of these equations on patches where the solution is smooth.

Speaker:

Prof. M. Arcak

Abstract:

The passivity concept - an abstraction of energy conservation and dissipation in physical systems - has been instrumental in feedback control theory and led to breakthroughs in nonlinear and adaptive control design. In this talk we discuss the use of passivity as a stability test for classes of biochemical reaction networks. The main result determines global asymptotic stability of the network from the diagonal stability of a dissipativity matrix which incorporates information about the passivity properties of the subsystems, the interconnection structure of the network, and the signs of the feedback terms. This stability test encompasses the well-known 'secant criterion' for cyclic networks and extends it to general interconnection structures represented by graphs. An extension to reaction-diffusion PDEs is also discussed. The results are illustrated on MAPK cascade models and on branched interconnection structures motivated by metabolic networks.

Speaker:

Prof. E. P. Ryan

Abstract:

Input-to state stability is a concept that captures "nice" properties of dynamical systems with input (e.g.bounded input implies bounded state, input "eventually small" implies state "eventually small", input convergent to zero implies state convergent to zero). Input-to-state stability (ISS) of a class of differential inclusions is described. Every system in the class is of Lur'e-type: a feedback interconnection of a linear system and a (set-valued) nonlinearity. Applications of the ISS results, in the context of feedback interconnections with a hysteresis operator in the feedback path, are developed.