Vendredi 13 janvier 2017
10H30 Andrei AGRACHEV
Symplectic Geometry of Constrained Optimization and Optimal Control
Abstract: Given a conditional optimization problem, its Lagrange multipliers are points of the cotangent bundles to the space of conditions. Cotangent bundle is endowed with the standard symplectic form, and symplectic geometry of the set of Lagrange multipliers allows to efficiently characterize optimal solutions even for very degenerate problems with complicated constraints. Basic instrument is the Maslov index of a triple of Lagrange subspaces. Being approprially re-arranged, this classical geometric invariant becomes also a powerful analytic tool. As expected, symplectic approach clarifies and unifies many known facts and opens new horizons.
Ref:
Agrachev, A. A.; Gamkrelidze, R. V. Symplectic geometry for optimal control.Nonlinear controllability and optimal control, 263–277, Monogr. Textbooks Pure Appl. Math., 133,Dekker, New York, 1990.
Agrachev, A. A.; Gamkrelidze, R. V. Symplectic methods for optimization and control. Geometry of feedback and optimal control, 19–77, Monogr. Textbooks Pure Appl. Math., 207, Dekker, New York, 1998.
Agrachev, A. A. Constrained optimization and quadratic mappings. SISSA, 2003.
Arnol′d, V. I. Sturm theorems and symplectic geometry. Funktsional. Anal. i Prilozhen. 19 (1985), no. 4, 1–10, 95.
Symplectic Geometry of Constrained Optimization and Optimal Control
Abstract: Given a conditional optimization problem, its Lagrange multipliers are points of the cotangent bundles to the space of conditions. Cotangent bundle is endowed with the standard symplectic form, and symplectic geometry of the set of Lagrange multipliers allows to efficiently characterize optimal solutions even for very degenerate problems with complicated constraints. Basic instrument is the Maslov index of a triple of Lagrange subspaces. Being approprially re-arranged, this classical geometric invariant becomes also a powerful analytic tool. As expected, symplectic approach clarifies and unifies many known facts and opens new horizons.
Ref:
Agrachev, A. A.; Gamkrelidze, R. V. Symplectic geometry for optimal control.Nonlinear controllability and optimal control, 263–277, Monogr. Textbooks Pure Appl. Math., 133,Dekker, New York, 1990.
Agrachev, A. A.; Gamkrelidze, R. V. Symplectic methods for optimization and control. Geometry of feedback and optimal control, 19–77, Monogr. Textbooks Pure Appl. Math., 207, Dekker, New York, 1998.
Agrachev, A. A. Constrained optimization and quadratic mappings. SISSA, 2003.
Arnol′d, V. I. Sturm theorems and symplectic geometry. Funktsional. Anal. i Prilozhen. 19 (1985), no. 4, 1–10, 95.
Vendredi 20 janvier 2017
10H30 Andrei AGRACHEV
Salle 15-16 413
Symplectic Geometry of Constrained Optimization and Optimal Control
Ref: agrachev-2003c
Salle 15-16 413
Symplectic Geometry of Constrained Optimization and Optimal Control
Ref: agrachev-2003c
Vendredi 27 janvier 2017
10h30 Guowey YU
Simple choreographies in the planar Newtonian N-body problem with equal masses
Abstract : In 2000, using variational method, A. Chenciner and R. Montgomery proved the famous "Figure-Eight" solution of the three body problem with equal masses. The remarkable feature of this solution is that all three masses travel on a single loop in the shape of figure 8. For arbitrary N body problem with equal masses, many solutions with the particular feature that all masses travel on a single loop with different shapes were found numerically by C. Simo afterwards and he named them "simple choreographies". Among them, there is a special family called "linear chain", where the single loop looks like a sequence of consecutive bubbles. In this talk, we give a proof of the existence of this family.
Simple choreographies in the planar Newtonian N-body problem with equal masses
Abstract : In 2000, using variational method, A. Chenciner and R. Montgomery proved the famous "Figure-Eight" solution of the three body problem with equal masses. The remarkable feature of this solution is that all three masses travel on a single loop in the shape of figure 8. For arbitrary N body problem with equal masses, many solutions with the particular feature that all masses travel on a single loop with different shapes were found numerically by C. Simo afterwards and he named them "simple choreographies". Among them, there is a special family called "linear chain", where the single loop looks like a sequence of consecutive bubbles. In this talk, we give a proof of the existence of this family.