Mı́ry a lineárnı́ maticové nerovnosti v optimálnı́m polynomiálnı́m řı́zenı́ Measures and linear matrix inequalities in polynomial optimal control Summary
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چکیده
Míry a lineární maticové nerovnosti v optimálním polynomiálnímřízení Measures and linear matrix inequalities in polynomial optimal control Summary This lecture describes the application of modern techniques of convex optimization to solve nonconvex nonlinear optimal control problems (OCPs) which may feature oscillation phenomena (chattering control) or concentration phenomena (impulsive control). First, with the help of occupation measures, we reformulate nonconvex non-linear OCPs as convex linear programming (LP) problems on the cone of nonnegative measures. Second, relying on recent results merging techniques of real algebraic geometry , functional analysis (measures, problems of moments) and mathematical programming (semidefinite optimization), we propose a general methodology to solve numerically these infinite-dimensional LP on measures. We describe a hierarchy of convex semidefinite programming (SDP) or linear matrix inequality (LMI) relaxations that can be implemented to solve OCP with asymptotic convergence guarantees. 2 Souhrn Tato přednáška popisuje, jak aplikovat moderní techniky konvexní optimal-izace nařešení problémů optimálníhořízení nelineárních a nekonvexních systémů, které mohou obsahovat jevy oscilace a koncentrace. Nejprve pomocí měr obsazenosti vyjádříme nelineární a nekonvexní problémy optimálníhořízení jako konvexní problémy lineárního programování vkuželu nezáporn´ych měr. Poté na základě nejnovějších v´ysledků, které spojily techniky reálné alge-braické geometrie, funkcionální anal´yzy (míry, problémy momentů) a matem-atického programování (semi-definitní optimalizace), navrhujeme všeobecnou metodologii numerick´ychřešení těchto nekonečněrozměrn´ych problémů lineárního programování měr. Popisujeme hierarchii konvexních semi-definitních problémů neboli lineární maticové nerovnosti, která může b´yt použita křešení problémů optimálníhořízení se zárukou asymptotické konvergence. 3 Klíčová slova: Optimálnířízení, konvexní optimalizace.
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