Research interests of Rafal Goebel
I am a mathematician. The main areas of my interest are:
- convex analysis, non-smooth analysis, and set-valued analysis;
- generalized dynamical systems, in particular hybrid dynamical systems;
- control theory, including optimal control;
- optimization.
Click here for a brief and informal exposition of
hybrid systems.
Current and recent research topics:
- convex duality ideas in control:
- Hamilton-Jacobi equations of dynamic programming
- value functions for convex problems of optimal control and calculus of variations
- stability and dissipativity properties of linear differential inclusions and their duals
- hybrid dynamical systems / hybrid inclusions:
- generalized time domains, generalized solution concepts
- structural properties of sets of solutions to hybrid inclusions
- stability theory (robustness of stability, converse Lyapunov theorems, invariance principles)
- convex functions and saddle functions:
- dual properties of a convex function and its conjugate
- primal-dual symmetric operations on convex functions and saddle functions
- feedback stabilization:
- optimal feedback for stabilization of constrained linear systems (Linear-Convex Regulator)
- hybrid feedback for robust stabilization of nonlinear control systems
Representative work:
- convex duality concepts in control:
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R. Goebel,
Duality and uniqueness of convex solutions to stationary Hamilton-Jacobi equations,
Transactions of the AMS, Volume 357, 2005, 2165-2186.
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R. Goebel, Convex optimal control problems with smooth Hamiltonians,
SIAM Journal on Control and Optimization, Volume 43, Number 5, 2005, 1787-1811.
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R. Goebel, A. Teel, T. Hu, and Z. Lin,
Conjugate convex Lyapunov functions for dual linear differential inclusions,
IEEE Transactions on Automatic Control,
Volume 51, Issue 4, 2006, 661-666.
- hybrid dynamical systems / hybrid inclusions:
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R. Goebel and A. Teel,
Solutions to hybrid inclusions via set and graphical convergence
with stability theory applications,
Automatica, Volume 42, Issue 4, 2006, 513-696.
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R. Sanfelice, R. Goebel, and A. Teel,
Generalized solutions to hybrid dynamical systems,
ESAIM: Control, Optimisation and Calculus of Variations, to appear.
- convex functions and saddle functions:
-
R. Goebel,
Self-dual smoothing of convex and saddle functions,
Journal of Convex Analysis, Volume 15, Number 1, 2008, to appear.
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R. Goebel and R.T. Rockafellar,
Local strong convexity and local Lipschitz continuity of the gradient of convex functions,
Journal of Convex Analysis, Volume 15, Number 2, 2008, to appear.
- feedback stabilization:
-
R. Goebel,
Stabilizing a linear systems with saturation through optimal control,
IEEE Transactions on Automatic Control,
Volume 50, Issue 5, 2005, 650-655.
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C. Prieur, R. Goebel, and A. Teel,
Hybrid feedback control and robust stabilization of nonlinear systems,
IEEE Transactions on Automatic Control, to appear.